Macroscopic quantum resonators (MAQRO): 2015 Update
Rainer Kaltenbaek,
1,
∗
Markus Arndt,
1
Markus Aspelmeyer,
1
Peter F. Barker,
2
Angelo Bassi,
3
James Bateman,
4
Kai Bongs,
5
Sougato Bose,
2
Claus Braxmaier,
6, 7
ˇ
Caslav Brukner,
1, 8
Bruno Christophe,
9
Michael Chwalla,
10
Pierre-Fran ̧cois Cohadon,
11
Adrian M. Cruise,
5
Catalina Curceanu,
12
Kishan Dholakia,
13
Klaus D ̈oringshoff,
14
Wolfgang Ertmer,
15
Jan Gieseler,
16
Norman G ̈urlebeck,
6
Gerald Hechenblaikner,
17
Antoine Heidmann,
11
Sven
Herrmann,
6
Sabine Hossenfelder,
18
Ulrich Johann,
19
Nikolai Kiesel,
1
Myungshik Kim,
20
Claus L ̈ammerzahl,
6
Astrid
Lambrecht,
11
Michael Mazilu,
13
Gerard J. Milburn,
21
Holger M ̈uller,
22
Lukas Novotny,
16
Mauro Paternostro,
23
Achim Peters,
14
Igor Pikovski,
24
Andr ́e Pilan-Zanoni,
10
Ernst M. Rasel,
15
Serge Reynaud,
11
C. Jess Riedel,
25
Manuel
Rodrigues,
9
Lo ̈ıc Rondin,
16
Albert Roura,
26
Wolfgang P. Schleich,
26, 27
J ̈org Schmiedmayer,
28
Thilo Schuldt,
7
Keith
C. Schwab,
29
Martin Tajmar,
30
Guglielmo M. Tino,
31
Hendrik Ulbricht,
4
Rupert Ursin,
8
and Vlatko Vedral
32, 33
(MAQRO Consortium, names after first author sorted alphabetically)
1
Vienna Center for Quantum Science and Technology, University of Vienna, Austria
2
Department of Physics and Astronomy, University College London, United Kingdom
3
Department of Physics, University of Trieste, and INFN, Italy
4
Physics and Astronomy, University of Southampton, United Kingdom
5
School of Physics and Astronomy, University of Birmingham, United Kingdom
6
Center of Applied Space Technology and Micro Gravity (ZARM), University of Bremen, Germany
7
German Aerospace Center (DLR), Institute for Space Systems, Bremen, Germany
8
Institute of Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Austria
9
ONERA, The French Aerospace Lab, Chˆatillon, France
10
Airbus Defence & Space Friedrichshafen, Germany
11
Laboratoire Kastler Brossel, UPMC-Sorbonne Universit ́es, CNRS,
ENS-PSL Research University, Coll`ege de France, Paris, France
12
Laboratori Nazionali di Frascati dell’INFN, Italy
13
School of Physics and Astronomy, University of St. Andrews, United Kingdom
14
Institut f ̈ur Physik, Humboldt-Universit ̈at zu Berlin, Germany
15
Institut f ̈ur Quantenoptik, Leibniz Universit ̈at Hannover, Germany
16
Photonics Laboratory, ETH Zurich, Zurich, Switzerland
17
European Southern Observatory, Munich, Germany
18
Nordita, KTH Royal Institute of Technology and Stockholm University, Sweden
19
Airbus Defence & Space Friedrichshafen, Immenstaad, Germany
20
QOLS, Blackett Laboratory, Imperial College London, United Kingdom
21
ARC Centre for Engineered Quantum Systems, University of Queensland, Australia
22
Department of Physics, University of California, Berkeley, USA
23
Centre for Theoretical Atomic, Molecular and Optical Physics,
School of Mathematics and Physics, Queen’s University, Belfast, United Kingdom
24
ITAMP, Harvard-Smithsonian Center for Astrophysics, USA
25
Perimeter Institute for Theoretical Physics, Canada
26
Institute f ̈ur Quantenphysik, Universit ̈at Ulm, Germany
27
Texas A & M University Institute for Advanced Study,
Institute for Quantum Science and Engineering and Department of Physics and Astronomy, Texas A & M University, USA
28
Vienna Center for Quantum Science and Technology, Technical University of Vienna, Austria
29
Applied Physics, California Institute of Technology, USA
30
Institute of Aerospace Engineering, Technische Universit ̈at Dresden, Germany
31
Dipartimento di Fisica e Astronomia and LENS, Universit ́a di Firenze, INFN, Italy
32
Atomic and Laser Physics, Clarendon Laboratory, University of Oxford, United Kingdom
33
Center for Quantum Technologies, National University of Singapore, Republic of Singapore
Do the laws of quantum physics still hold for macroscopic objects – this is at the heart of
Schr ̈odinger’s cat paradox – or do gravitation or yet unknown effects set a limit for massive particles?
What is the fundamental relation between quantum physics and gravity? Ground-based experiments
addressing these questions may soon face limitations due to limited free-fall times and the quality
of vacuum and microgravity. The proposed mission MAQRO may overcome these limitations and
allow addressing those fundamental questions. MAQRO harnesses recent developments in quantum
optomechanics, high-mass matter-wave interferometry as well as state-of-the-art space technology to
push macroscopic quantum experiments towards their ultimate performance limits and to open new
horizons for applying quantum technology in space. The main scientific goal of MAQRO is to probe
the vastly unexplored “quantum-classical” transition for increasingly massive objects, testing the
predictions of quantum theory for truly macroscopic objects in a size and mass regime unachievable
in ground-based experiments. The hardware for the mission will largely be based on available space
arXiv:1503.02640v1 [quant-ph] 9 Mar 2015
2
technology. Here, we present the MAQRO proposal submitted in response to the (M4) Cosmic
Vision call of the European Space Agency for a medium-size mission opportunity with a possible
launch in 2025.
I. EXECUTIVE SUMMARY
A. What are the fundamental physical laws of the universe?
The laws of quantum physics challenge our understanding of the nature of physical reality and of space-time,
suggesting the necessity of radical revisions of their underlying concepts. Experimental tests of quantum phenomena,
such as quantum superpositions involving massive macroscopic objects, provide novel insights into those fundamental
questions. MAQRO allows entering a new parameter regime of macroscopic quantum physics addressing some of
the most important questions in our current understanding of the basic laws of gravity and of quantum physics of
macroscopic bodies.
B. Fundamental science and technology pathfinder
The main scientific objective of MAQRO is to test the predictions of quantum theory in a hitherto inaccessible
regime of quantum superpositions of macroscopic objects that contain up to 10
10
atoms. This is achieved by com-
bining techniques from quantum optomechanics, matter-wave interferometry and from optical trapping of dielectric
particles. MAQRO will test quantum physics in a parameter regime orders of magnitude beyond existing ground-
based experimental tests – a realm where alternative theoretical models predict noticeable deviations from the laws
of quantum physics[1–3]. These models have been suggested to harmonize the paradoxical quantum phenomena both
with the classical macroscopic world[4–7] and with notions of Minkowski space-time[8–10]. MAQRO will, therefore,
enable a direct investigation of the underlying nature of quantum reality and space-time, and it may pave the way
towards testing the ultimate limit of matter-wave interference posed by space-time fluctuations[11, 12]. Recent works
showed that MAQRO might even allow testing certain models of dark matter[13, 14]. In contrast to collapse models,
even standard quantum theory, in the presence of gravitation, predicts decoherence for spatially extended, massive
superpositions[15, 16]. While this is not applicable in a microgravity setting, ground-based tests in this direction may
benefit from the technology development necessary for MAQRO.
By pushing the limits of state-of-the-art experiments and by harnessing the space environment for achieving the re-
quirements of high-precision quantum experiments, MAQRO may prove a pathfinder for quantum technology in space.
For example, quantum optomechanics is already proving a useful tool in high-precision experiments on Earth[17].
MAQRO may open the door for using such technology in future space missions.
C. A unique environment for macroscopic quantum experiments
In ground-based experiments, the ultimate limitations for observing macroscopic quantum superpositions are vi-
brations, gravitational field-gradients, and decoherence through interaction with the environment. Such interactions
comprise, e.g., collisions with background gas as well as scattering, emission and absorption of blackbody radiation.
The spacecraft design of MAQRO allows operating the experimental platform in an environment offering a unique
combination of microgravity (
<
∼
10
−
9
g), low pressure (
<
∼
10
−
13
Pa) and low temperature (
<
∼
20 K). This allows suf-
ficiently suppressing quantum decoherence for the effects of alternative theoretical models to become experimentally
accessible, and to observe the evolution of macroscopic superpositions over free-fall times of about 100 s.
D. The case for space
The main reasons for performing MAQRO in space are the required quality of the microgravity environment
(
<
∼
10
−
9
g), the long free-fall times (100 s), the high number of data points required (up to
∼
10
4
per measurement
∗
Corresponding author: rainer.kaltenbaek@univie.ac.at
3
run), and the combination of low pressure (
<
∼
10
−
13
Pa) and low temperature (
<
∼
20 K) while having full optical access.
These conditions cannot be fulfilled with ground-based experiments.
E. Technological heritage for MAQRO
MAQRO benefits from recent developments in space technology. In particular, MAQRO relies on technologi-
cal heritage from LISA Pathfinder (LPF)[18], the LISA technology package (LTP)[19], GAIA[20], GOCE[21, 22],
Microscope[23, 24] and the James Webb Space Telescope (JWST)[25]. The spacecraft, launcher, ground segment and
orbit (L1/L2) are identical to LPF.
The most apparent modifications to the LPF design are an external, passively cooled optical instrument thermally
shielded from the spacecraft, and the use of two capacitive inertial sensors from ONERA technology. In addition, the
propulsion system will be mounted differently to achieve the required low vacuum level at the external subsystem, and
to achieve low thruster noise in one spatial direction. The additional optical instruments and the external platform
will reach TRL 5 at the start of the BCD phases. For all other elements, the TRL is 6-9 because of the technological
heritage from LPF and other missions.
F. Alternative mission scenarios
Implicit strengths of MAQRO are its relatively low weight and power consumption such that MAQRO’s scientific
instrument can, in principle, be combined on the same spacecraft with other missions that have similar requirements in
precision and orbit. An example could be sun-observation instruments benefiting from an L1 orbit. Another example
could be a combination with the ASTROD I mission or similar mission concepts fulfilling the orbit requirements of
MAQRO.
G. Technological Readiness & the MAQRO consortium
Since its original proposal as an M3 mission in 2010[26], MAQRO has made significant progress in technology
development[27] and in its support within the scientific community. In 2013, we formed the MAQRO consortium,
now consisting of more than 30 groups from the UK, Germany, Italy, France, Austria, Switzerland, the US, Australia
and Sweden.
MAQRO benefits from significant technological progress made since 2010. The TRLs of several core technologies
increased from initial concepts to TRL 3-5. In particular, research groups within the MAQRO consortium have
successfully demonstrated cavity cooling of trapped nanospheres[28, 29], feedback cooling [30, 31], optical trapping of
nanospheres in high vacuum[32], and hybrid optical & Paul trapping of nanospheres[29, 33]. Moreover, optomechanical
cooling close to the quantum ground state was successfully demonstrated[34–36]. Detailed thermal studies of the
MAQRO shield design showed the feasibility of achieving the temperature and vacuum technical requirements of
MAQRO[37]. A more detailed thermal study showed even better results[38]. A collaboration of the University of
Vienna, the University of Bremen and Airbus Defence &Space, successfully implemented a high-finesse, adhesively
bonded optical cavity using space-proof glue and ultra-low-expansion (ULE) material[39]. The same technology is
currently in use to implement a high-finesse test cavity with the same specifications as needed for MAQRO. Based
on recent theoretical studies[40], the design of MAQRO was adapted for preparing macroscopic superpositions with
state-of-the-art non-linear-optics and laser technology[41] also benefiting from recent advances in the single-mode
transmission of deep-UV light[42]. In this way, a central drawback of the initial MAQRO proposal (the need for low
power, extremely short-wavelength light) could be resolved.
II. SCIENCE CASE
Do the laws of quantum physics remain applicable without modification even up to the macroscopic level? This
question lies at the heart of Schr ̈odinger’s famous gedankenexperiment (thought experiment) of a dead-and-alive
cat[43]. Matter-wave experiments have confirmed the predictions of quantum physics from the microscopic level of
electrons[44, 45], atoms and small molecules[46] up to massive molecules with up to 10
4
atomic mass units (amu)[47].
Still, experiments are orders of magnitude from where alternative theories predict deviations from quantum physics[3,
48].
4
Using ever more massive test particles on Earth may soon face principal limitations because of the limited free-
fall times as well as the limited quality of microgravity environments achievable on Earth. Currently, it is assumed
that this limit will be reached for interferometric experiments with particles in the mass range between 10
6
amu and
10
8
amu[40]. These limitations may be overcome by harnessing space as an experimental environment for high-mass
matter-wave interferometry[26]. At the same time, quantum optomechanics provides novel tools for quantum-state
preparation and high-sensitivity measurements[49]. The mission proposal MAQRO combines these aspects in order
to test the foundations of quantum physics in a parameter regime many orders of magnitude beyond current ground-
based experiments, in particular, for particle masses in the range between 10
8
amu and 10
11
amu. This way, MAQRO
will not only significantly extend the parameter range over which quantum physics can be tested. It will also allow
for decisive tests of a number of alternative theories, denoted as “collapse models” predicting notable deviations from
the predictions of quantum theory within the parameter regime tested.
An important feature of MAQRO is that the parameter range covered has some overlap with experiments that
should be achievable on ground even before a possible launch of MAQRO. This allows cross-checking the performance
of MAQRO and to provide a fail-safe in case the predictions of quantum physics should fail already for masses
between 10
6
amu and 10
8
amu. In this case, MAQRO would not allow for observing matter-wave interference due to
the presence of strong, non-quantum decoherence. For this reason, the MAQRO instrument is designed for allowing
three modes of operation for testing quantum physics over a wide parameter range even in the presence of strong
decoherence:
•
Non-interferometric tests of collapse models
The stochastic momentum transfer in collapse models can lead to heating of the centre-of-mass motion of
trapped nanospheres[50, 51]. This can, in principle, be observed by comparing the measured noise spectra with
theoretical predictions[52].
•
Deviations from quantum physics in wave-packet expansion
As in the frequency-based non-interferometric approach above, this method is based on the stochastic momentum
transfer due to collapse mechanisms. In particular, the momentum transfer leads to a random walk resulting in
an increased rate for the expansion of wavepackets[50, 52, 53].
•
High-mass matter-wave interferometry
This central experiment of MAQRO is based on the original M3 proposal[26]. It has been adapted for harnessing
the successful technique of Talbot-Lau interferometry, which currently holds the mass record for matter-wave
interferometry[47]. The goal is to observe matter-wave interferometry with particles of varying size and mass,
comparing the interference visibility the predictions of quantum theory and the predictions of alternative theo-
retical models.
In particular, the non-interferometric tests and observing wave-packet expansion will allow for performing tests in the
presence of comparatively strong decoherence mechanisms. If these two tests show agreement with the predictions
of quantum physics, MAQRO’s scientific instrument can then be used for performing matter-wave interferometry to
test for smaller deviations from quantum physics.
A. Non-interferometric tests of quantum physics
The vast majority of the proposals for the test of collapse models put forward so far is based on interferometric
approaches in which massive systems are prepared in large spatial quantum superposition states. In order for such
tests to be effective, the superposition has to be sufficiently stable in time to allow for the performance of the necessary
measurements. Needless to say, these are extremely demanding requirements from a practical viewpoint. Matter-
wave interferometry and cavity quantum optomechanics are generally considered as potentially winning technological
platforms in this context, and considerable efforts have been made towards the development of suited experimental
configurations using levitated spheres or gas-phase molecular or metallic-cluster beams. Alternatively, one might
adopt a radically different approach and think of non-interferometric strategies to achieve the goal of a successful test.
MAQRO offers the opportunity for exploring one such possibility by addressing the influences that collapse models
(or in general, any non-linear effect on quantum systems) have on the spectrum of light interacting with a radiation-
pressure-driven mechanical oscillator in a cavity-optomechanics setting. The overarching goal of this part of MAQRO
is to affirm and consolidate novel approaches to the revelation of deviations from standard quantum mechanics in
ways that are experimentally viable and open up unforeseen perspectives in the quest at the center of the MAQRO
endeavours.
A benchmark in this sense will be provided by the assessment of the CSL model through a non-interferometric
approach. In particular, we will take advantage of the fact that the inclusion of the CSL mechanism in the dynamics
of a harmonic oscillator results in an extra line-broadening effect that can be made visible from its density noise
5
spectrum. By bypassing the necessity of preparing, manipulating, and sustaining the quantum superposition state
of a massive object, the proposed scheme would be helpful in bringing the goal of observing collapse-induced effects
closer to the current experimental capabilities.
The equation of motion of the optomechanical system (regardless of its embodiment) in the presence of the CSL
mechanism can be cast in the form given in equation (1)
∂
∂t
ˆ
O
=
i
̄
h
[
ˆ
H,
ˆ
O
]
+
i
̄
h
[
ˆ
V
t
,
ˆ
O
]
+
ˆ
N
,
(1)
where
ˆ
O
is an operator of the system,
ˆ
H
is the Hamiltonian of the mechanical oscillator coupled to the cavity
light field,
ˆ
N
embodies all the relevant sources of quantum noise affecting the system, and
ˆ
V
t
is a stochastic linear
potential (linked directly to the position of the harmonic oscillator) that accounts for the effective action of the
CSL mechanism[51]. It can be shown that such a potential is zero-mean and delta-correlated, and thus embodies a
source of white noise that adds up to the relevant noise mechanisms affecting the optomechanical system, namely the
damping of the optical cavity and the Brownian motion (occurring at temperature
T
) of the mechanical oscillator.
A lengthy calculation based on the study, in the frequency domain, of the fluctuation operators of both the optical
and mechanical system, leads to the following expression for the density noise spectrum of the mechanical system’s
position fluctuation:
S
(
ω
) =
2
α
2
s
̄
h
2
κχ
2
(
∆
2
+
κ
2
+
ω
2
)
+ ̄
hmω
[
(
∆
2
+
κ
2
−
ω
2
)
2
+ 4
κ
2
ω
2
]
[
γ
m
coth(
βω
) +
Y
]
∣
∣
∣
2
α
2
s
∆ ̄
hχ
2
+
m
(
ω
2
−
ω
2
m
−
i
γ
m
ω
)
[
∆
2
+ (
κ
+ i
ω
)
2
]
∣
∣
∣
2
,
(2)
where
α
s
is the steady-state amplitude of the cavity field,
κ
is the cavity damping rate,
χ
is the optomechanical
coupling rate. ∆ is the detuning between the cavity field and an external pump,
m
is the mass of the mechanical
oscillator,
γ
m
is the mechanical damping rate,
ω
m
is the mechanical frequency, and
β
is the inverse temperature of
the system. Finally, we have introduced:
Y
=
λ
√
̄
h
mω
m
,
(3)
where
λ
is the CSL coefficient. In our numerical simulations of the observability of the effects, we have used the
value of such parameter achieved by assuming Adler’s estimate of the CSL mechanism’s strength. Quite evidently,
the CSL mechanism enters into the expression of the density noise spectrum as an extra thermal-like line broadening
contribution. While being formally rather appealing, this elegant result also suggests the strategy to implement in
order to observe the collapse model itself, and identifies the challenges that have to be faced, namely a cold enough
mechanical system that lets the
Y
-dependent term dominate over the temperature-determined one. Our numerical
estimate shows that, indeed, it is possible to pinpoint the effects of the CSL contribution in a parameter regime
currently available in optomechanical labs. Figure 1 shows a typical result achieved by using the parameters stated
in Ref. [51].
At the present state, this non-interferometric approach has not been investigated in sufficient detail in the context
of MAQRO. While this does not impede the main science goals of MAQRO, we plan nevertheless to investigate this
non-interferometric method more closely during the study phase of MAQRO. It may offer the attractive possibility
to supplement the results of the other two experiments (subsections II B and II C).
B. Deviations from quantum physics in wave-packet expansion
Most forms of decoherence can be described as resulting from the interaction of a quantum system with its
environment[54]. Examples are elastic and inelastic scattering as well as emission of massive particles or radiation[55].
All of these interactions result in a change of momentum, eventually leading to dephasing and decoherence of quantum
states. In a paper by Collett and Pearle[50], it was shown that decoherence mechanisms assumed in collapse models
also lead to momentum transfer. That means, even in the absence of standard decoherence mechanisms, collapse
models may result in a random walk due to stochastic momentum transfer. This random walk can, in principle, be
observed when comparing the expansion rate of a quantum wave packet with the predictions of quantum theory as
well as with the predictions of alternative models. Apart from the original suggestion for such an experiment[50],
there have also been more recent suggestions to observe this effect using free-falling or optically trapped, dielectric
particles[52, 53].
6
FIG. 1.
Broadening of noise power spectra.
Comparison between the density noise spectrum of the mechanical position
fluctuation operators with (solid red line) and without (dashed black line) the influence of the CSL mechanism obtained using
Adler’s estimate of the CSL coupling strength and a mechanical oscillator of 15 ng. The inset shows an analogous study for
m
= 150 ng (figure from Ref. [51]).
Even if there is no decoherence, the width of a quantum wave packet will expand over time according to the
Schr ̈odinger equation. The square of the width of the wave packet
w
s
(
t
)
2
evolves according to the following relation:
w
s
(
t
)
2
=
〈
ˆ
x
2
(
t
)
〉
s
=
〈
ˆ
x
2
(0)
〉
+
t
2
m
2
〈
ˆ
p
2
(0)
〉
.
(4)
Here, the subscript “s” denotes evolution according to Schr ̈odinger’s equation,
m
is the mass of the particle, the
angular brackets denote the expectation value for a given quantum state, ˆ
x
denotes the position operator, and ˆ
p
denotes the momentum operator. Equation (4) relates the width of the wave packet at time
t
with the initial width
of the wave packet and the initial width of the momentum distribution.
In the presence of decoherence, the width of the wave packet increases more quickly:
w
(
t
)
2
=
〈
ˆ
x
2
(
t
)
〉
=
w
s
(
t
)
2
+
2Λ ̄
h
2
3
m
2
t
3
.
(5)
Here, Λ is a parameter governing the strength of decoherence mechanisms. The width of the wave packet is not an
observable – it has to be inferred from the statistical distribution of many measurements[56]. If we assume that we
perform
N
measurements of the particle position and if the result of the j-th measurement is
x
j
, for large
N
, the
width of the wave packet can be approximated as:
w
=
1
√
N
−
1
√
√
√
√
N
∑
j
=1
x
2
j
.
(6)
Given that the error of each position measurement is ∆
x
j
=
σ
, the error of our estimate of the width of the wave
packet will be:
∆
w
=
σ
√
N
−
1
≈
σ
√
N
,
(7)
where the approximation holds for large
N
.
The mode of operation of this experiment is to determine the wave-packet size as a function of time
t
, and to compare
these measurements with the predictions of quantum physics using equation (5). In this way, we can experimentally
determine the decoherence parameter Λ and compare it with the predictions of quantum physics. The more Λ deviates
from the value predicted by quantum physics, the easier it will be to discern by measuring the wave-packet expansion.
For simplicity, let us assume that we have a well isolated quantum system, i.e., quantum physics predicts Λ = 0
or at least much smaller than the deviation we want to measure. The minimum Λ we can distinguish experimentally
7
from the case of no decoherence is:
Λ
>
Λ
min
= 3
m
2
σw
s
(
t
)
√
N
−
1 ̄
h
2
t
3
.
(8)
We can relate this minimum decoherence parameter to a decoherence rate Γ =
r
2
c
Λ by introducing a representative
length scale
r
c
= 100 nm. This is a typical length scale for the experiments in MAQRO and also the same as the
length scale chosen in the collapse model of Ghirardi, Rimini and Weber[4]:
Γ
min
= Λ
min
r
2
c
= 3
m
2
σw
s
(
t
)
r
2
c
√
N
−
1 ̄
h
2
t
3
.
(9)
FIG. 2.
Comparison of
Λ
min
(solid, black) with the decoherence rates predicted for the CSL model with
λ
= 2
.
2
×
10
−
17
Hz
(black, dashed), the quantum-gravity model of Ellis et al (blue, long dashed), the model of Di ́osi & Penrose (red, dot-dashed),
and the model of K ́arolyh ́azy (green, dotted). Where models predict a higher decoherence rate than Γ
min
, one can, in principle,
distinguish them from the predictions of quantum physics..
In Figure 2, we compare the predictions of several collapse models with that minimum, discernible decoherence rate
Γ
min
. The figure shows that, by investigating wave-packet expansion, MAQRO can, in principle, perform decisive
tests of the CSL model even with the originally suggested parameters[7, 50], and MAQRO could test the quantum
gravity model of Ellis and others[57, 58]. However, the plot also illustrates that wave-packet expansion will neither
allow testing the model of K ́arolyh ́azy nor that of Di ́osi-Penrose.
In order to estimate the values plotted in Figure 2, we assumed that we let the wave-packet expand for a maximum
of 100 s, and that we collect at most
N
= 24
×
10
3
data points to experimentally estimate the decoherence parameter.
The number of data points was chosen in order to limit the integration time to at most four weeks. Moreover, we
assumed our test particle to initially be in a thermal state of a harmonic oscillator – with a mechanical frequency
ω
= 10
5
rad
/
s, an average occupation number of 0
.
3, and that we can determine the particle position with an accuracy
of 100 nm. Because the mechanical frequency for an optically trapped particle only depends on the mass density and
the material’s dielectric constant, the mechanical frequency is roughly constant for the particles chosen for MAQRO.
The occupation number, however, is assumed to be inversely proportional to the mass of the test particle because it
depends on the optomechanical coupling achievable.
Because testing quantum physics using wave-function expansion was first introduced for the CSL model[50], and
because the CSL model represents a rather general, heuristic approach to collapse models, we will now discuss the
prerequisites for testing the CSL model in the context of MAQRO. The CSL model depends on two parameters,
a
and
λ
, where
a
= 100 nm defines the typical length scale at which the CSL model predicts a transition from quantum
to classical behaviour. For
λ
, which predicts the rate of decohering events on the microscopic level, a wide variety of
values have been suggested, ranging from 2
.
2
×
10
−
17
Hz[7, 50] to 10
−
8
Hz[59]. The smaller one assumes the value of
λ
, the smaller the deviation from quantum physics. Using equation (9), we can now estimate the smallest value of
λ
that MAQRO would allow detecting. In particular, we get:
λ
min
= 4
a
2
(
m
p
m
)
2
f
(
r
a
)
−
1
Λ
min
> m
2
p
f
(
r
a
)
−
1
12
a
2
σw
s
(
t
)
√
N
−
1 ̄
h
2
t
3
,
(10)