††thanks: Corresponding author<EMAIL_ADDRESS>Current address: Research
Laboratory of Electronics, MIT-Harvard Center for Ultracold Atoms, Department
of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA.

# Realization of a complete Stern-Gerlach interferometer:
Towards a test of quantum gravity

Yair Margalit  Or Dobkowski  Zhifan Zhou  Omer Amit  Yonathan Japha  Samuel
Moukouri  Daniel Rohrlich  Department of Physics, Ben-Gurion University of the
Negev, Be’er Sheva 84105, Israel  Anupam Mazumdar  Van Swinderen Institute,
University of Groningen, 9747 AG Groningen, The Netherlands  Sougato Bose
Department of Physics and Astronomy, University College London, Gower Street,
WC1E 6BT London, United Kingdom  Carsten Henkel  Institute of Physics and
Astronomy, University of Potsdam, Germany  Ron Folman  Department of Physics,
Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel

###### Abstract

The Stern-Gerlach effect, discovered a century ago, has become a paradigm of
quantum mechanics. Surprisingly there has been little evidence that the
original scheme with freely propagating atoms exposed to gradients from
macroscopic magnets is a fully coherent quantum process. Specifically, no
full-loop Stern-Gerlach interferometer has been realized with the scheme as
envisioned decades ago. Furthermore, several theoretical studies have
explained why such an interferometer is a formidable challenge. Here we
provide a detailed account of the first full-loop Stern-Gerlach interferometer
realization, based on highly accurate magnetic fields, originating from an
atom chip, that ensure coherent operation within strict constraints described
by previous theoretical analyses. Achieving this high level of control over
magnetic gradients is expected to facilitate technological as well as
fundamental applications, such as probing the interface of quantum mechanics
and gravity. While the experimental realization described here is for a single
atom, future challenges would benefit from utilizing macroscopic objects doped
with a single spin. Specifically, we show that such an experiment is in
principle feasible, opening the door to a new era of fundamental probes.

## Introduction

The discovery of the Stern-Gerlach (SG) effect [1, 2] was followed by ideas
concerning a full-loop SG interferometer (SGI) consisting of a beam of atoms
exposed to field gradients from macroscopic magnets [3]. However, starting
with Heisenberg, Bohm and Wigner [4] a coherent SGI was considered impractical
because it was thought that the macroscopic device could not be accurate
enough to ensure a reversible splitting process, namely, a complete overlap in
position and momentum of the two interferometric paths. Bohm, for example,
noted that the magnet would need to have “fantastic” accuracy [5]. Englert,
Schwinger and Scully analyzed the problem in more detail and coined it the
Humpty-Dumpty (HD) effect [6, 7, 8, 9]. They too concluded that for
significant coherence to be observed, exceptional accuracy in controlling
magnetic fields would be required. Indeed, while atom interferometers based on
light beam-splitters enjoy the quantum accuracy of the photon momentum
transfer [10], the SGI magnets not only have no such quantum discreteness,
they also suffer from inherent lack of flatness due to Maxwell’s equations.
Later work added the effect of dissipation and suggested that only low-
temperature magnetic field sources would enable an operational SGI [11].
Claims have even been made that no coherent splitting is possible at all [12].

Figure 1: The longitudinal full-loop SGI ($z$ position vs. time). The figure
is plotted in the center-of-mass frame. The interferometer operates for a
duration of 2T (on the order of a few hundred $\mu s$), and consists of four
magnetic gradient pulses (purple columns). The states
$|1\rangle\equiv|F=2,m_{F}=1\rangle$ and $|2\rangle\equiv|2,2\rangle$ are
defined along the $\tilde{z}$ axis in the Bloch sphere (different from real-
space coordinates). The signal is made of spin population fringes. The
experiment starts with a spin in the $\tilde{x}$ direction, and the final
measurement is again of the spin in the $\tilde{x}$ direction. The latter is
performed by mapping the spins from $\tilde{x}$ to $\tilde{z}$ with a $\pi/2$
rotation and applying a SG gradient to separate the populations before taking
an image. The same configuration may be used for a macroscopic-object
interferometer, see Section: Testing Different Aspects of Gravity.

Here we detail the realization of a full-loop SGI, in which magnetic field
gradients act on the atom during its flight through the interferometer, as
originally envisioned. We obtain a high full-loop SGI contrast of 95% with a
spin interference signal. We achieve this utilizing the highly accurate
magnetic fields of an atom chip [13]. Following the footsteps of impressive
endeavors [14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 22], and additional recent
scientific advancements with spatial SGIs [25, 26], this is, to the best of
our knowledge, the first realization of a complete SGI interferometer
analogous to that originally envisioned.

Achieving this high level of control over magnetic gradients may facilitate
fundamental research as well as technological applications. SG interferometry
with mesoscopic objects has been suggested as a compact detector for space-
time metric and curvature [27], possibly enabling detection of gravitational
waves. It has also been suggested as a probe for the quantum nature of gravity
[28]. Such SG capabilities may also enable searches for exotic effects like
the fifth force, or the hypothesized self-gravitation interaction [29, 30,
31]. In the following we show that such an experiment is in principle
feasible.

We note that the realization presented here has already enabled the
construction of a unique matter-wave interferometer whose phase scales with
the cube of the time the atom spends in the interferometer [32]. This
realization has been suggested as an experimental test for the Einstein’s
equivalence principle when extended to the quantum domain [36].

High magnetic stability and accuracy may also make possible technological and
metrological applications such as large-area atom interferometry [33],
sensitive probing of electron transport, e.g., squeezed currents [34], as well
as nuclear magnetic resonance and compact accelerators [35]. We note that as
the SGI makes no use of light, it may serve as a high-precision surface probe
at short distances for which administering light is hard. In addition, our
atom chip setup is compatible with cryogenic environments, and may hence probe
cold surfaces, while laser light may cause unwanted heating.

Finally, let us emphasize that a full-loop scheme, with its spin-population
observable, as described here, has several advantages for macroscopic-object
interferometry, over other suggestions which utilize a spatial interference
pattern as their signal. First, a spin-population observable requires no high-
resolution imaging. For massive objects this may be crucial. Second, as shown
in [32], a SGI can achieve a $T^{3}$ scaling of the phase accumulation,
enabling high sensitivity. Furthermore, the SGI allows us to apply magnetic
forces throughout the interferometer, and a significant splitting may be
achieved in a few ms or so. Last, and perhaps most importantly, the spin-
population observable does not require free propagation to develop, in
contrast to a spatial interference signal. These reasons provide a crucial
advantage when taking into account the high decoherence rate expected for
macroscopic objects.

## The full-loop SG interferometer

A schematic of the full-loop interferometer is presented in Fig. 1. The full-
loop SGI utilizes four magnetic gradient regions or pulses: splitting,
stopping, accelerating back and stopping, the latter two closing a loop in the
space-time diagram. To the best of our knowledge, all previous SG realizations
used only the first one or two operations, thereby realizing at most a ‘half
loop’. The entanglement of spin with the spatial degrees of freedom persists
throughout the SGI, and the magnetic moment associated with the spin is used
to control the external degrees of freedom, using magnetic gradients and the
SG effect. The SGI actively recombines the wavepackets in both position and
momentum, and uses the spin state of the recombined wavepacket as an
interference signal. This is in contrast to the spatial fringe SG
interferometer realized in our previous work [25, 26] which consists only of
splitting and stopping the wavepackets (thus corresponding to a half loop, to
which we added spin mixing to allow for spatial fringes to form, in analogy to
a double-slit experiment).

The recombination in the full-loop SGI, is in fact required to be a time-
reversal operation of the splitting process, such that the final two magnetic
gradients exactly undo the first two. In order to obtain high coherence (or
contrast) in the output of a spatial interferometer, one must apply stable and
accurate operations on the atom, such that the final relative distance between
the wavepackets’ centers, $\Delta z(2T)$, and the final relative momentum
between the centers, $\Delta p(2T)$, are minimized, where $2T$ is the
interferometer duration. Inaccuracy of the magnetic field gradient throughout
the particle’s trajectory gives rise to imperfect overlap, either in position
or momentum, and will cause a decay in the resulting interferometric contrast.

The difficulty in maintaining spin coherence due to inaccuracy is illustrated
by the following simple argument relating to the momentum splitting (as argued
by Heisenberg and others [6, 4, 37]). In order to achieve macroscopic
splitting of the wavepackets using a differential force $F$ acting for a
duration $T_{1}$, the relative momentum between the split wavepackets $FT_{1}$
must be much bigger than the initial wavepacket width in momentum
$\sigma_{p}$, i.e. $FT_{1}\gg\sigma_{p}$. As each part of the wavepacket
samples a different part of the potential, it acquires a different phase. The
linear phase spread over the wavepacket is thus given by
$|\delta\phi|=|\delta(ET_{1}/\hbar)|=|(\partial E/\partial
z)\sigma_{z}T_{1}/\hbar|=FT_{1}\sigma_{z}/\hbar\gg\sigma_{p}\sigma_{z}/\hbar$,
where $\sigma_{z}$ is the initial wavepacket width in position ($\sigma_{p}$
and $\sigma_{z}$ are both defined in the $z$ direction), and $E$ is the atom’s
energy due to the magnetic field. By invoking the uncertainty principle, one
finds that $\delta\phi$ may not be made small, further complicating the
recombination, where a successful recombination requires maximizing the
overlap integral of the two wavepackets. In other words, large splitting
requires large relative momentum in units of the internal momentum width,
which corresponds to a large phase spread over the wavepacket (due to the
relation between momentum and phase - $e^{ipz/\hbar}$). To achieve high
coherence, this phase spread originating from momentum splitting has to be
undone. If the size of the two wavepackets is the same, minimizing $\Delta p$
ensures to some degree that the phase profile of both wavepackets is the same,
which is sufficient. Note that in a practical experiment, the phase pattern is
actually more complex and harder to match due to the curvature of the magnetic
potential.

The precision with which one has to recombine the wavepackets is set by the
spatial coherence length $l_{z}$ and momentum coherence width $l_{p}$, which
we use as a phenomenological model to describe the loss of contrast. These
measures of coherence are inversely proportional to the momentum and position
uncertainties of the atom, $\sigma_{p}$ and $\sigma_{z}$, and may be defined
as [38]

$l_{z}=\frac{\hbar}{\sigma_{p}},\quad l_{p}=\frac{\hbar}{\sigma_{z}},$ (1)

where the momentum and position uncertainties satisfy the uncertainty relation
$\sigma_{p}\sigma_{z}\geq\hbar/2$. If the two paths at the output port of the
interferometer (time $t=2T$) are displaced by a distance $\Delta z(2T)$ the
contrast is expected to drop as $C\propto\exp[-\frac{1}{2}\left(\Delta
z(2T)/l_{z}\right)^{2}]$. Equivalently, if the two paths are displaced in
momentum by $\Delta p(2T)$ the contrast reduces as
$C\propto\exp[-\frac{1}{2}\left(\Delta p(2T)/l_{p}\right)^{2}]$.

In the case of a minimal-uncertainty Gaussian wavepacket with a negligible
expansion rate, the loss of contrast (or coherence) $C$ due to inaccuracy of
the recombination process is quantified by the HD equation, which is given by
[7]

$C=\exp\left[-\frac{1}{8}\left(\frac{\Delta
z(2T)}{\sigma_{z}}\right)^{2}-\frac{1}{8}\left(\frac{\Delta
p(2T)}{\sigma_{p}}\right)^{2}\right].$ (2)

(Note a factor $1/2$ in our definition of $\Delta z$ and $\Delta p$ relative
to the original definition). This equation is the result of calculating the
overlap integral between the Gaussian wavefunctions at the end of the
interferometer. In order to keep the contrast close to unity, a ‘microscopic’
level of accuracy is required at the end of the interferometer, described by
the relations $|\Delta z(2T)|\ll\sigma_{z}$ and $|\Delta p(2T)|\ll\sigma_{p}$.
Quantitatively, to maintain a contrast of $\simeq 0.99$ using
$FT_{1}/\sigma_{p}=10^{3}$, it turns out that one must control the fields with
an accuracy of $\delta F/F=10^{-5}$ [6, 7], a formidable technical challenge.
Addressing this challenge is expected to open the door to a wide variety of
new experiments for technology and fundamental studies.

## Experiment

Our experiment begins by releasing a BEC of about $10^{4}$ 87Rb atoms from a
magnetic trap below an atom chip [13]. We initially prepare the BEC in the
state $|F,m_{F}\rangle=|2,2\rangle$, and then create a superposition of the
two spin states $|F,m_{F}\rangle=|2,2\rangle\equiv|2\rangle$ and
$|2,1\rangle\equiv|1\rangle$ by applying a $\pi/2$ RF pulse. These two states
constitute an effective two-level system, as all other states in the $F=2$
manifold are pushed out of resonance by the nonlinear Zeeman shift generated
using an external bias field (see Methods for more details). To avoid
dephasing of the spin superposition due to noise in the bias fields, we add
$\pi$ pulses giving rise to an echo sequence (see Methods). The full-loop SGI
is then applied by using a series of four magnetic gradient pulses (gradients
along the axis of gravity, $z$, see Fig. 1), which are generated by running
currents on the atom chip (more details on the setup can be found in Ref.
[26]). The first pulse, of duration $T_{1}$, splits the superposition into two
momentum components, which then freely propagate during a delay time $T_{d1}$.
The wavepackets are then stopped relative to one another (pulse duration
$T_{2}$), accelerated back (pulse duration $T_{3}$), and after a second delay
time $T_{d2}$ are stopped again (pulse duration $T_{4}$), ideally when overlap
in space and momentum is maximal. As the direction of acceleration in the
second and third pulses is opposite to that of the first and fourth pulses, we
name the $T_{2}$ and $T_{3}$ pulses, reverse pulses. As the $T_{3}$ and
$T_{4}$ pulses are required to undo the splitting in position and momentum
created by the $T_{1}$ and $T_{2}$ pulses, $T_{3}$ and $T_{4}$ are named the
recombination pulses. We obtain the population signal with the help of a
second $\pi/2$ pulse followed by a spin population measurement. We measure the
visibility by scanning the phase $\phi$ of the second $\pi/2$ pulse (Fig. 1),
and observe the contrast of the resulting population fringes.

Figure 2: Full-loop optimization procedure: population output as a function of
the reverse pulse duration $T_{2}+T_{3}$, for $T_{1}=T_{4}=6\,\mu$s,
$T_{d1}=300\,\mu$s, and a relative acceleration $a=635$ m/s2. The population
oscillates around the optimal point as expected by a simplified model of a
Gaussian times a sine. The peak of the Gaussian envelope corresponds to the
time at which the wavepackets’ overlap integral at the end of the
interferometer is maximized for the given parameters. The sine function
corresponds to the added phase between the two interferometer arms, per unit
time of reversing pulse.

In order to minimize $\Delta z(2T)$ and $\Delta p(2T)$ (inaccuracies in the
final recombination), and thus maximize the visibility of the interference
signal, we must optimize the experimental parameters. In Fig. 2 we present an
example optimization run, in which we fix the durations of the first and last
gradient pulses $T_{1}$ and $T_{4}$ and also the durations of the delay times
$T_{d1}$ and $T_{d2}$ (usually $T_{1}=T_{4}$ and $T_{d1}=T_{d2}=T_{d}$ to
begin with). We measure the output population as a function of the duration
$T_{2}+T_{3}$ of the second and third gradient (reverse) pulses, while keeping
the total duration $T_{2}+T_{3}+T_{d2}$ constant. The data nicely fit a
Gaussian envelope times a sine function. The peak of the envelope corresponds
to the point in which the overlap integral is maximal. Ideally for linear
magnetic gradients, one would expect the peak overlap to occur when the
sequence is symmetric i.e. $T_{1}+T_{4}=T_{2}+T_{3}$ (assuming
$T_{d1}=T_{d2}$). However the non-linearity of the magnetic potential created
by the chip wires in the $z$ direction [26], together with the wavepackets’
movement with respect to the chip, breaks the symmetric timing diagram. The
optimal duration of the gradient pulses depends on the delay times, the
initial distance from the chip, and the specific scheme used (see Methods
section for more details about the optimization procedure). Generally, one
needs to optimize two independent parameters - one to minimize the final
relative position $\Delta z(2T)$, and one to minimize the final relative
momentum $\Delta p(2T)$.

While, as noted in the introduction, accuracy is the main challenge, we also
need to address the issue of stability, whereby temporal fluctuations may give
rise to dephasing, decoherence and drifts. Even in the absence of decoherence,
noise may cause the interference phase to jitter from one experimental shot to
the next (e.g. due to a fluctuating bias field), thus dephasing the averaged
phase, or preventing recombination altogether in the case of a different noise
(e.g. fluctuating gradients). In this work we achieve high stability by
utilizing an atom chip [13] with several advantages, including strong magnetic
gradients so that the experimental duration is very short and consequently the
interaction with external noise is brief, and very low inductance so that the
gradients can be switched in micro-seconds. In addition, the structure and
position of the magnet are very precise as it is made of a near-perfect wire.
Also, care was taken to reduce a wide range of hindering effects. For example,
a novel method is used to reduce the effect of current fluctuations on the
chip by utilizing a 3-wire configuration which produces a quadrupole and
exposes the wavepackets to a weaker magnetic field from the chip, while
maintaining strong gradients. Using these advantages, we have been able to
show low phase noise [26] (standard deviation as low as 0.1 rad of the spatial
fringe SGI), demonstrating the stability of the apparatus.

## Results

We now validate that we are able to successfully recombine in position and
momentum, as both can cause loss of visibility. First, in Fig. 3, we present
the loss of coherence due to the first magnetic pulse alone (i.e. by setting
$T_{2},\,T_{3},\,T_{4}=0$), giving rise to orthogonality in momentum. The
contrast drops to $1/\sqrt{e}$ at $l_{p}/m=0.12$ mm/s momentum splitting.
Next, we apply a much stronger momentum splitting of $\Delta p/m=2.6$ mm/s,
and show in Fig. 4 (blue data) that we are able to undo this orthogonality by
a second gradient pulse, which stops the relative motion of the wavepackets.
However, as we extend the delay time $T_{d1}$ the wavepackets start to split
in space and we observe a decay in visibility which cannot be restored using
$T_{2}$ alone, as the distance between the wavepackets becomes larger than
their coherence length $l_{z}$. We validate that this loss of visibility is
mainly due to spatial splitting, by optimizing $T_{2}$ for each value of
$T_{d1}$, such that maximal visibility is achieved (see Methods for details).

Figure 3:  Single kick effect on the contrast: population as a function of the
applied velocity difference $\Delta v$ between the wavepakcets for a single
pulse of duration $T_{1}$ (where we set $T_{2}=T_{3}=T_{4}=0$). As the
splitting is increased, the population decays to 50% - corresponding to zero
contrast. In this measurement we take care that the spatial separation is as
small as possible ($\Delta z\leq 50$ nm) such that the decay due to spatial
splitting is negligible. The velocity difference is calculated according to
$\Delta v=aT_{1}$, where $a=59.5$ m/s2 is the applied relative acceleration.
The data are fitted to a Gaussian times a sine function, and the fit returns a
momentum coherence width of $l_{p}/m=0.12\pm 0.03$ mm/s, where $m$ is the
atom’s mass. To the best of our knowledge, this is the first direct
measurement of the momentum coherence width of a BEC (see previous results
using neutrons [39] and atomic beams [40]).

Using this method we are able to also accurately determine $l_{z}$. To
describe the expected loss of contrast with increasing spatial splitting, we
fit the blue data with a Gaussian times a sine function of the form
$P_{1}=A\exp(-\frac{1}{2}T_{d}^{2}/\tau^{2})\sin[\delta\phi(T_{d})+\phi_{0}]+c$,
where $\delta\phi(T_{d})$ is the accumulated phase difference, which contains
terms proportional to $T_{d}$ and $T_{d}^{2}$ [32], $\phi_{0}$ is a constant
phase shift, $A$ is the amplitude, $c$ is a constant, and $\tau$, the decay
constant, is essentially the coherence time (as $T_{d}$ is much larger than
the pulse time). Using the value of $\tau$ we calculate the spatial coherence
length $l_{z}$ and obtain $l_{z}=0.5\pm 0.07\,\mu$m (see Methods). Note that
the blue data decay to less than the expected 50% value, probably due to
imperfect RF $\pi/2$ pulses, affecting the state preparation and population
measurement.

We are now ready to recombine the two wave packets. We add the recombining
pulses $T_{3}$ and $T_{4}$ to the previous sequence, and generate the red data
in Fig. 4, where the value of $T_{4}$ is also optimized in a similar manner
for maximal visibility. One can clearly see the Gaussian decay of the
visibility due to spatial splitting in the first (blue) data, and the revival
of the visibility due to the successful recombination of the spatially-split
wavepackets (red data). At $T_{d1}=T_{d2}=350\,\mu$s, for example, the blue
data decay to 16% of their original amplitude, while the red data show no
decay at this time scale. The red data are fitted with a similar function as
that used for the blue data, but without the decaying Gaussian term (as the
decay is not visible in this range), namely
$P_{1}=A\sin[\delta\phi(T_{d})+\phi_{0}]+c$.

In summary, we have clearly shown the successful recombination in momentum and
position, thus realizing a complete SGI. We have also measured $l_{p}$ and
$l_{z}$, and the observed visibility is proof that the accuracy of our
recombination in momentum and position is better than these coherence scales
(see Methods for more discussion on the results).

Figure 4:  Full-loop recombination: population in the $m_{F}=1$ state as a
function of delay times $T_{d}$ for two different sequences, using
$T_{1}=5.4\,\mu$s and a relative acceleration $a=481$ m/s2. Blue data is an
optimized splitting ($T_{1}$) and stopping ($T_{2}$) sequence without
recombination (i.e. $T_{3},\,T_{4}=0$). The optimization procedure ensures
that the visibility loss due to momentum splitting is minimized for every
value of $T_{d1}$, meaning we see visibility loss mainly due to spatial
splitting (as it becomes large relative to the coherence length). The red data
use the same parameters of $T_{1},\,T_{d1}$, and $T_{2}$, but add the
recombining pulses. Here we use $T_{d1}=T_{d2}=T_{d}$, and they are scanned
together. Solid lines are fits to the data (see text for details); dashed
lines represent the envelope of the fit, demonstrating the visibility as a
function of the delay time. Note that the red data seems to start with a lower
contrast. This effect of flatness and “increasing contrast” is expected by the
theory [32], and originates from a non-linear phase term. One can clearly see
that for long delay times, the spatial splitting reduces the the visibility of
the interference. The recombination then revives the visibility, demonstrating
successful recombination in both position and momentum.

Figure 5: Analysis of accuracy: (a) Spin population visibility vs. maximal
separation $\Delta z(T)$. Experimental parameters are $T_{1}=2-30\,\mu$s,
$T_{d1}\simeq T_{d2}=50-160\,\mu$s. (b) Visibility vs. maximal momentum
splitting $\Delta p(T_{1})$. Experimental parameters are $T_{1}=2-6\,\mu$s,
$T_{d1}\simeq T_{d2}=164-367\,\mu$s. In both (a) and (b), the relative
acceleration is varied in the range $a=635-2641\,$m/s2. As expected, coherence
decays as we increase spatial and momentum splitting. A detailed quantitative
analysis could not explain the observed slope (see discussion in text). In
both panels, the visibility is normalized to the Ramsey visibility without
splitting (i.e., no magnetic gradients), typically $\sim 90\%$. Insets show
the raw data for two of the data points, namely the population fringe
generated using a phase scan of the second $\pi/2$ pulse, and a fit to a sine
wave used to extract the contrast (red line, see Methods). Data points in (a)
and (b) are of different data sets.

## Limits on Accuracy

To test the limits of our accuracy, in Fig. 5 we plot the visibility as a
function of the maximum splitting in momentum $\Delta p(T_{1})$ and in
position $\Delta z(T)$. In order to try and maximize the splitting, we utilize
several different configurations: we invert the sign of the relative
acceleration by reversing the sign of the currents in the chip wires, while in
other sequences we keep the same currents while inverting the spins with the
help of $\pi$ pulses (see Methods). We also utilize a variety of magnetic
gradient magnitudes by varying the current on the chip, and scan both the
splitting gradient pulse duration $T_{1}$, and the delay time between the
pulses $T_{d}$. Each point in Fig. 5 was taken using different parameters, and
was optimized independently. For weak splitting we observe high visibility
($\sim 95\%$), while for a momentum splitting equivalent to 1 $\hbar k$
(optical photon recoil on the rubidium D2 line, $\hbar k/m=5.8\,$mm/s, where
$m$ is the atom’s mass) the visibility is still high ($\sim 75\%$) indicating
that the magnetic field accuracy allowed reversing the splitting to a high
degree. The quoted values are normalized to that of a pure Ramsey sequence,
i.e. a sequence without any magnetic gradients, in order to cancel technical
effects (see Methods). The visibility as a function of maximum splitting is
qualitatively the same for space and momentum splitting, and shows a decrease
in visibility as splitting is increased.

As we know the potential created by the chip, the wavepackets’ positions and
widths can be calculated in every step of the experiment. The HD equation (Eq.
2) should thus, in principle, allow for a quantitative calculation of the
contrast using the experimental parameters. However, the equation makes two
simplifying assumptions: first, that the wavepackets do not expand during the
propagation through the interferometer, which does not hold in our case.
Second, it assumes that there is no non-linear phase imprinted on the
wavepackets (i.e. magnetic gradients are linear). In our case, phase non-
linearity originates from both the curvature of the magnetic gradient
generated by the chip, and from conversion of the mean-field potential energy
of the BEC released from trap to kinetic energy (causing a quadratically-
varying phase to evolve [41]). In order to account for these effects and try
to quantitatively explain the loss of contrast in the full-loop SGI, we have
developed a generalized wavepacket model for studying coherence of matter-wave
interferometers [42]. While we have simulated our experimental conditions with
care, and while such simulations accurately describe our previous
interferometry results (e.g. [25, 43, 32, 44, 26]), the coherence drop
observed in the full-loop experiment is not well described by the generalized
wavepacket model [42], and neither by the HD model. Using the experimental
parameters to calculate the visibility, both models predict values which are
significantly higher than those observed in the experiment. We therefore leave
the quantitative comparison to future work.

In order to give limits on our accuracy in recombining the wavepackets, we use
the experimentally measured coherence scales as a figure of merit, and examine
the ratio between the achieved maximum splittings and the coherence scales. We
are able to recombine wavepackets with a maximum spatial splitting of ${\Delta
z(T)}/l_{z}=3.9$ before visibility goes to zero, and a maximum momentum
splitting of ${\Delta p(T_{1})}/l_{p}=158$. These results indicate that we are
more successful in recombining in momentum than in space. The reason for this
difference is yet unclear.

## Testing different aspects of gravity

The idea of using the SG interferometer with a macroscopic object as a probe
for gravity has been described in detail in several theoretical works [27, 28,
45, 46, 47, 36]. These works detail a wide range of experiments from detection
of gravitational waves to the testing of the quantum nature of gravity. Here,
we discuss the feasibility of such an experiment using the full-loop SGI, as
described above for an atom, and show that such an experiment using a
macroscopic body is also feasible. As before, our interference observable is
oscillations of spin population, rather than spatial fringes (density
modulations). This observable, as demonstrated in the atomic SGI described
above, has the advantages that there is no requirement for long evolution
times in order to allow the spatial fringes to develop, there is no need for
high resolution imaging to resolve the spatial fringes, and the phase
accumulation may be fast (e.g. $T^{3}$ [32]). Let us note that there are other
proposals to realize a spatial superposition of macroscopic objects [48, 49],
but they do not rely on the SG effect, and they utilize spatial rather than
spin population fringes.

As a specific example, in the following we consider an object comprising
$10^{6}$ atoms. Let us first emphasize that even prior to any probing of
gravity, a successful SGI will already achieve 3 orders of magnitude more
atoms than the state of the art in macroscopic-object interferometry [50],
thus contributing novel insight to the foundations of quantum mechanics.
Another contribution to the latter would be the ability to test continuous
spontaneous localization (CSL) models (e.g. [51] and references therein).

When considering gravity, the first contribution of such a massive-object SGI
would simply be to measure small $g$ (the local gravitational acceleration).
Such an experiment would first of all enable to verify the created
superposition of the macroscopic object [52, 45, 53]. A second contribution of
such a SGI in the field of gravity would be in testing modifications to
gravity at short ranges (also known as the fifth force), as one of the SGI
paths may be brought close to a massive object, thus allowing to probe gravity
at short distances [54]. This is a regime in which light-based interferometers
would be difficult to use due to diffraction and light scattering near the
surface. Once SGI technology allows for large masses, a third contribution
will be the testing of hypotheses concerning gravity self-interaction [29, 30,
31], and when large-area SGI with large masses is available, a fourth
contribution would be to detect gravitational waves [27]. Finally, it is
claimed that placing two such SGIs in parallel next to each other will allow
probing the quantum nature of gravity [28, 47]. Let us emphasize, that
although high accelerations may be obtained with multiple spins, in the
following we discuss only the case of a macroscopic object with a single spin,
as the observable of such a quantum-gravity experiment is entanglement between
two spins, and averaging over many spins may wash out the signal. Furthermore,
a multi-spin SGI would give rise to multiple trajectories.

We focus solely on the required SG parameters, which include the necessary
accelerations and accuracy in relation to the required splitting and coherence
length, respectively. Other issues such as material science (e.g. clean
surfaces against patch potentials, spin contaminations, or spin coherence time
near the surface), or the very good vacuum which will be required, will not be
dealt with here. We will also not deal with the issue of the Casimir-Polder or
diamagnetic interactions that may arise. These issues have been dealt with
extensively in previous works [55, 28]. While we believe all discussed
applications noted here, including the two parallel interferometers for the
quantum-gravity experiment, may be done in the longitudinal configuration
presented in the experimental part of this paper, transverse (i.e. 2D)
interferometers may also be realized with the same techniques discussed here.
Both the 1D and 2D interferometers involve similar operations and present the
same challenges, and in the following we will not discuss the differences
between them.

We consider, for example, a nano-diamond composed of $10^{6}$ carbon atoms
(corresponding to a sphere radius of 11 nm) with a single NV spin embedded in
it. Coherence time of 1 ms has been demonstrated at room temperature [56], and
0.6 s while cooling to 77 K [57]. The total interferometer time of the
proposed experiment will have to be shorter than these times. Utilizing well-
known NV techniques (e.g. see for example our own work [58, 59]), we do not
see any fundamental spin-related obstacles.

The experimental procedure is the same as for the atomic full-loop SGI
demonstrated above, with the required adjustments for manipulating the nano-
diamond. The experiment begins by trapping and cooling the center-of-mass
motion of the nano-diamond [60, 61]. We note that to the best of our
knowledge, ground-state cooling of nano-diamonds is yet to be achieved [62].
We then release the nano-diamond from the trap, and prepare it in a spin
superposition of the $\pm 1$ spin projections using a $\pi/2$ microwave pulse
(a two-photon transition). Placing the nano-diamond 1 $\mu$m away from a
$1\times 1\,\mu$m wire on an atom chip, carrying 1 A of current (with a
current density of $10^{8}$ A/cm2, achievable with CNT embedded Cu wires
[63]), we get a magnetic gradient of $8.7\times 10^{4}$ T/m. For $10^{6}$
carbon atoms, the acceleration for a single spin (1 Bohr magneton) is
$a=\mu_{B}\partial_{z}B/m=81\,$m/s2. We then apply a microwave $\pi$ pulse to
inverse the relative acceleration, and apply the stopping and reversing pulses
$T_{2}$ and $T_{3}$. After another $\pi$ pulse, we apply $T_{4}$, and stop the
relative motion (see Fig. 1 for description of pulses). For total
interferometer times of $T=(0.1,\,1,\,10,\,100)$ ms, we get maximum splittings
of $\Delta z=a(T/4)^{2}=5\times(10^{-8},\,10^{-6},\,10^{-4},\,10^{-2})$ m.
Maintaining a constant acceleration for the long durations requires placing
current-carrying wires along the wavepackets’ trajectories such that they are
always in a region of strong gradient (e.g. as previously suggested in Ref.
[27]), or realizing the experiment in a very strong quadrupole field created
by coils (e.g. superconducting). We note that if the nano-diamond is very
close to the wire, one of the spin states will have to be $0$ so that it does
not crash into the wire. The spin state is then read out by standard NV
optical techniques. Let us briefly recall that the phase is predicted to be
independent of the initial motional condition [45].

The crucial parts of the experiment are the ground-state cooling and the
recombination. The required recombination accuracy is on the order of the
coherence length, given by the harmonic oscillator length
$\sqrt{\hbar/m\omega}$, which for the considered object is assumed to be about
0.1 nm (assuming ground-state cooling with $\omega/2\pi=80\,$kHz, as in Ref.
[60]). This value can be increased by adiabatically lowering the trap
frequencies after cooling. We do not take into account some works which claim
that techniques exist with which the coherence length may be increased even
further [64, 49], as these are done for spatial interference fringes, where
the “local” coherence length is what matters [64]. In contrast, what matters
for the full-loop SGI is the overlap integral, for which the coherence length
only depends on the initial momentum width, namely on the ground state energy
when cooling to the ground state. This recombination accuracy represents the
main technical challenge, as it must be better than the achieved coherence
length. In the experimental results presented above for an atomic SGI, high
visibility is achieved for a 700 nm coherence length, which allows us to
assume we have obtained a recombination accuracy on the order of 100 nm (when
the maximal splitting was an order of magnitude larger). Improving this
recombination accuracy by 3 orders of magnitude (while maintaining the same
ratio to the maximal splitting) is well within reach by utilizing better
current sources with less current and time jitters. (We note that at present
we use current sources with instabilities far above shot-noise.)

Let us briefly also touch upon the topic of decoherence due to external and
internal degrees of freedom, namely, which-path information due to some
scattering event between the environment and the nano-diamond. The information
may be encoded in the environment (i.e. the scattered particle, equivalent to
the interaction with some noise with a correlation length smaller than the
spatial splitting [66]), or in the internal degrees of freedom. For example,
as explained in Ref. [67], any object with excited internal degrees of freedom
may emit radiation which would localize it. Other sources of decoherence may
originate from any differential interaction the nano-diamond suffers between
the two paths.

As a simple example of lack of decoherence due to the internal degrees of
freedom, we can consider the atom interferometer demonstrated in this paper.
The atom has many internal degrees of freedom such as those of the electrons
or the nucleus. Decoherence can occur due to the emergence of which-path
information, or in other words orthogonality. However, if the inner degrees of
freedom do not develop some orthogonality along the two paths, then they are
irrelevant for the interferometer, as no which-path information is encoded in
them. What can be relevant is some differential interaction between those
internal degrees of freedom and the environment which can create such
orthogonality, where by differential we mean that the interaction with the
environment affecting the internal degrees of freedom, in a coordinate system
relative to the c.o.m, is different for the two paths of the interferometer.
In the case of atom interferometry, it seems that external perturbations are
negligible with respect to the atom’s bare Hamiltonian.

In contrast, with the nano-diamond the differential interaction between the
internal degrees of freedom and the environment may not be negligible. For
example, a background gas collision can introduce rotation or phonons in the
nano-diamond, in only one of the paths. The collision can even be with a
cosmic muon. Furthermore, an external magnetic/electric/EM field that creates
rotation or phonons, or some spin flip of a contamination spin, just in one of
the paths (due to a small correlation length), would give rise to
orthogonality. A more subtle process of decoherence would be the excitation of
phonons, solely due to the magnetic force acting on the NV centers. Here the
sudden (non-adiabatic) force acting on the spin would create a phonon in the
lattice (this may be estimated through a Debye-Waller factor). As the
acceleration in the two paths is different, the excited phonons would be
different, giving rise to orthogonality. Even if the acceleration is symmetric
but opposite in sign, the opposite $k$-vector of the identical phonons would
create orthogonality. To maintain adiabaticity, one may have to resort to
engineering a gradual increase of the magnetic gradient within the pulse
duration.

Calculating the cross-section/probability for such events and the amount of
orthogonality they create is a daunting task. Some of these calculations have
already been done in Refs. [28, 49] and references therein. In any case, we do
not see a clear correlation between these cross sections and the internal
temperatures, and so a priori there is no clear need for the cooling of the
internal degrees of freedom or the environment, as long as the wavelength of
the blackbody radiation is larger than the splitting induced in the SGI. If,
however, for some reason, cooling of the internal degrees of freedom is
required, the experiment may be done in a dilution fridge cryostat, with which
the atom chip technology is compatible. Additional methods to avoid
decoherence, such as dynamic decoupling, have been suggested as well [68]. In
any case, the fact that the SGI can achieve large spatial splitting in a very
short time, as presented above, is a crucial advantage when addressing the
issue of decoherence.

Finally, let us emphasize that the SGI also has several clear advantages over
laser-pulse interferometers in the context of such an experiment: (a) Laser-
pulse matter-wave beam splitters require an appropriate internal transition to
coherently scatter photons; this demand severely restricts the applicability
of these splitters to macroscopic objects. The SGI merely requires a magnetic
moment, which enables to readily achieve the magneto-mechanical coupling. (b)
No heating is generated on the macroscopic object, since no light is
scattered. The scattering of light may also severely reduce the amount of
light interacting with the two-level system embedded in the object. Lack of
light scattering also suppresses the decoherence rate. (c) In all experiments
not dealing with the entanglement, as the number of spins can be increased
with the macroscopic object’s mass (typical concentrations are in the parts
per million), the acceleration is independent of the mass, enabling to split
large masses, or to achieve large angle interferometry for small masses.

## Conclusion

To conclude, we have demonstrated for the first time a full-loop SGI,
consisting of freely propagating atoms exposed to magnetic gradients, as
originally envisioned decades ago. We have unambiguously shown recombination
in both momentum and position. We have shown that SG splitting may be realized
in a highly coherent manner with macroscopic magnets without requiring
cryogenic temperatures or magnetic shielding. Furthermore, we have analyzed
the limits of our system’s accuracy.

We briefly compare our experiments to state-of-the-art SG interferometry [14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. While these longitudinal beam
experiments did show spin-population interference fringes, the experiments
presented here are very different. Most importantly, as explained in [19] and
[22], the full-loop configuration was never realized, as only splitting and
stopping operations were applied (i.e., no active recombination); namely,
wavepackets exit the interferometer with the same separation as the maximal
separation achieved within.

Finally, achieving this high level of control over magnetic gradients may
facilitate fundamental research as well as technological applications.
Specifically, we show that in principle, full-loop SG interferometry with a
macroscopic body is feasible. It may be used to test the foundations of
quantum theory, as well as to probe exotic forces such as the fifth force. It
may be developed as a gravitational sensor, serve to test exotic gravitational
models such as self-interaction, and finally, may probe the quantum nature of
gravity.

## Methods

###  .1 Detailed experimental scheme

In the following we describe our experimental sequence. We begin by preparing
a BEC of about $10^{4}$ 87Rb atoms in the state $|F,m_{F}\rangle=|2,2\rangle$
in a magnetic trap located around $91-96\,\mu$m $\pm 1\,\mu$m below the atom
chip surface (different experiments use different initial positions). The
harmonic frequencies of the trap are $\omega_{x}/2\pi=38$Hz and
$\omega_{y}/2\pi\approx\omega_{z}/2\pi=127$Hz. The trap is created by a copper
structure located behind the chip with the help of additional homogeneous bias
magnetic fields in the $x$, $y$ and $z$ directions. The BEC is then released
from the trap, and falls a few $\mu$m under gravity for a duration
$T_{d0}=0.9-1.4\,ms$ (see Fig. 6 for a timing diagram). During this time the
magnetic fields used to generate the trap are turned off completely. Only a
homogeneous magnetic bias field of 36.7 G in the $y$ direction is kept on to
create an effective two-level system via the non-linear Zeeman effect such
that the energy splitting between our two levels $|2,2\rangle\equiv|2\rangle$
and $|2,1\rangle\equiv|1\rangle$ is $E_{21}$ $\approx h\times$25 MHz, and
where the undesired transition is off-resonance by $E_{21}-E_{10}\approx
h\times$180 kHz. As the BEC expands, interaction becomes negligible, and the
experiment may be described by single-atom physics.

Next, we apply a radio-frequency (RF) $\pi/2$ pulse (10 $\mu$s duration) to
create an equal superposition of the two spin states, $|1\rangle$ and
$|2\rangle$, and a magnetic gradient pulse (splitting pulse) of duration
$T_{1}=4-40\,\mu$s which creates a different magnetic potential for the
different spin states $m_{F}$, thus splitting the atomic wave packet into two
wave packets with different momenta. The chip wire current is driven using
simple 12 V batteries connected in series, and is modulated using a home-made
current shutter. The SG acceleration is in the range of 59.5 - 2641 m/s2
(depending on chip current). The acceleration is measured by splitting the
wavepakcets using a single pulse, and measuring the relative distance as a
function of the time-of-flight [25].

After a delay time $T_{d1}$, we apply a stopping pulse of duration $T_{2}$
that cancels the relative momentum of the two wavepackets, and immediately
after a gradient pulse for accelerating the atoms back towards each other
($T_{3}$). After an additional delay time $T_{d2}$ we apply a final stopping
pulse ($T_{4}$) so that the two wave packets overlap in momentum and position.

A second RF $\pi/2$ pulse is applied only at the time of measurement (meaning
the two wave packets have a different spin throughout the propagation), which
completes the interferometric sequence. Without the magnetic gradient pulses
and their effect on the spatial wavefunction, the two $\pi/2$ pulses
correspond to a Ramsey sequence. Our signal is an interference pattern formed
by measuring the spin population (e.g. starting with $S_{\tilde{x}}=+1$ and
measuring $S_{\tilde{z}}$).

The visibility, or contrast, which represents spin coherence in our
experiment, is measured by changing the phase between the two interferometer
arms, and observing the resulting population oscillation between the two
states. In some measurements, we scan the magnetic gradient pulse duration,
thus adding a relative phase between the arms [32] (as in Figs. 2, 4).

In other measurements in which the gradient pulse durations are fixed, we
create a population interference pattern by shifting the phase $\phi$ of the
last RF $\pi/2$ relative to the first RF $\pi/2$ pulse. This creates
population oscillations between the states (as a function of $\phi$),
generating the oscillations shown in the insets of Fig. 5. Contrast is then
evaluated by fitting the population fringes to a function of the form
$P(\phi)=0.5C\sin(\phi+\phi_{0})+const$, where $C$ is the contrast, $\phi$ is
the applied phase shift, and $\phi_{0}$ is a constant phase term. As noted,
the resulting contrast shown in Fig. 5 is normalized to that of a pure Ramsey
sequence, i.e. a sequence without any magnetic gradients. This cancels
technical effects that reduces the contrast, such as non-zero RF detuning
(causing imperfect $\pi/2$ pulses), spin decoherence due to external magnetic
noise, etc..

As noted before, we add one or two RF $\pi$ pulses in between the two $\pi/2$
pulses, giving rise to an echo sequence which suppresses the dephasing taking
place due to magnetic noise and inhomogeneous magnetic fields in our chamber.
This allows us to increase the spin coherence time from $\sim 400$ $\mu$s up
to $\sim 4$ ms.

Relative populations in each output port are measured by applying a
homogeneous magnetic gradient that separates the spin states, and counting the
number of atoms in each output state using standard absorption imaging. The
homogeneous gradient is created by running a current in the copper structure
behind the chip for a few ms.

Finally, it is worth noting that while all gradient pulses come from the same
chip wires, the magnetic pulses may be considered as an analogy of the
original thought experiment in which there were different spatial regions with
different permanent magnets. This is so as in each pulse the current and
duration may be different and have individual jitter, and in addition the atom
position and consequently the gradient are different.

###  .2 Full-loop configurations

Here we describe in more detail the ’current inversion’ and ’spin inversion’
sequences shown in a timing diagram in Fig. 6, which are used to generate the
data in Fig. 5. In the first sequence, after applying a $\pi/2$ pulse and the
splitting gradient $T_{1}$ in one direction (downwards towards gravity), we
reverse the sign of the acceleration by reversing the sign of the currents in
the chip wires for the stopping and reversing gradients $T_{2}$ and $T_{3}$,
working in the opposite direction (upwards towards the chip, this is done
using two independent current shutters connected to the chip in opposite
directions). The opposite gradient causes the relative movement between the
wave packets to stop during $T_{2}$, and to change sign during $T_{3}$.
Finally the wave packets are brought to a relative stop and spatial overlap by
the second stopping pulse $T_{4}$ given in the same direction as the first
gradient. The sequence is finished by applying a $\pi$ pulse (to increase
coherence time, giving rise to an echo sequence) and a $\pi/2$ pulse, for
mixing the different spin states and enabling spin populations interference
(the $\pi/2-\pi-\pi/2$ sequence is symmetric in time). The four consecutive
gradient pulses used in the current inversion sequence are applied either
after the first $\pi/2$ pulse (i.e. only a single $\pi$ pulse is used as
described above), or in between two $\pi$ pulses to further increase the
coherence time (the $\pi/2-\pi-\pi-\pi/2$ sequence is again time symmetric).

In the second sequence, we keep the same current direction in all gradient
pulses, while reversing the spins with the help of two $\pi$ pulses. These
pulses are applied, first just before the stopping gradient pulse $T_{2}$ and
second just after the reversing gradient pulse $T_{3}$. Each $\pi$ pulse
causes the spin states to flip sign, thus changing the direction of the
applied momentum kicks in the center-of-mass frame (in lab frame, all gradient
pulses push the atoms downwards towards gravity). Both the current inversion
and spin inversion sequences are used to generate the data points for Fig. 5.

Figure 6:  Timing diagram of the full-loop experimental sequences. (A),(B) The
current inversion scheme, where the sign of the gradient $\partial B/\partial
z$ is switched during the sequence. The difference between (A) and (B) is the
additional $\pi$ pulse before the gradients, used to increase the coherence
time. (C) The spin inversion scheme, in which the sign of the gradient
$\partial B/\partial z$ is kept constant, and the relative force between the
spin states is inverted by using the $\pi$ pulses in between the gradients,
inverting the spin states.

###  .3 Minimizing the visibility loss due to momentum splitting (Fig. 4)

In Fig. 4, we validate that the loss of visibility is due to spatial
splitting, by minimizing the visibility loss due to momentum splitting for
each value of $T_{d1}$, using the following optimization procedure. We scan
for the optimal value of $T_{2}$, i.e. the value which obtains the maximal
visibility, for several delay times ($T_{d1}=100,\,200,\,300,\,400\,\mu$s), in
a similar way to what is shown in Fig. 2. We then determine the optimal
$T_{2}$ for any given value of $T_{d1}$ using a polynomial interpolation. The
blue data in Fig. 4 are taken by using the optimal values of $T_{2}$ as a
function of $T_{d1}$. This ensures that the visibility loss due to momentum
splitting is minimized for every value of $T_{d1}$, meaning we see visibility
loss mostly due to spatial splitting.

###  .4 Calculation of $l_{z}$

In the fit of the blue data of Fig. 4, $T_{d}=\tau$ represents the time at
which the contrast drops to $1/\sqrt{e}$ of its full value. Accordingly, we
estimate the spatial coherence length $l_{z}$ from the spatial splitting at
$T_{d}=\tau$. This is the sum of three contributions: 1. the splitting under
constant acceleration $a$ for a duration $T_{1}$, given by
$\frac{1}{2}aT_{1}^{2}$; 2. the splitting at constant velocity $aT_{1}$ for a
duration $T_{d}$, given by $aT_{1}T_{d}$; 3. the splitting under constant
deceleration $-a$, from initial velocity $aT_{1}$ until zero relative
velocity, for duration $T_{2}=T_{1}$, given by
$aT_{1}T_{1}-\frac{1}{2}aT_{1}^{2}=\frac{1}{2}aT_{1}^{2}$. Summing all
contributions and setting $T_{d}=\tau$, we obtain
$l_{z}=aT_{1}^{2}+aT_{1}\tau$. Using the experimental parameters $a=481.6$
m/s2, $T_{1}=5.4\,\mu$s, and the fit result of $\tau=186.8\pm 22.5\,\mu$s, we
obtain $l_{z}=0.5\pm 0.07\,\mu$m.

###  .5 Coherence scales

Here we compare the experimental results of the coherence scales to those
obtained from Eq. 1. Assuming the BEC is a minimal-uncertainty wavepacket, at
trap release time both its coherence scales are set by a single number - the
size of the BEC wavepacket. We therefore begin by calculating this number and
validating it experimentally. After some time-of-flight, however, the BEC
momentum width increases due to conversion of the mean-field potential energy
to kinetic energy [41], which we account for using the theory of Ref. [42].

The in-trap Thomas-Fermi condensate half-length $w_{0}$ in the z (gravity)
direction is given by [65] $w_{0}=\sqrt{2\mu/m}/\omega_{z}$, where the Thomas-
Fermi expression for the chemical potential $\mu$ for an harmonically confined
condensate is given by [65]
$\mu^{5/2}=15\hbar^{2}m^{1/2}N\bar{\omega}^{3}a/2^{5/2}$, and where
$\bar{\omega}=(\omega_{x}\omega_{y}\omega_{z})^{1/3}$ is the geometric mean of
the trap frequencies, $N$ is the number of atoms, and $a=5.18$ nm is the 87Rb
s-wave scattering length. For our experimental parameters of $N=10,000$ atoms,
$\omega_{x,y}/2\pi=40$ Hz and $\omega_{z}/2\pi=126$ Hz, we obtain
$w_{0}=2.88\,\mu$m.

We also experimentally validate this value: as imaging the wave packet at
short time-of-flight $T_{d0}$ [see Fig. 6] does not give reliable results due
to refraction effects around a BEC with high optical density, we measure the
Thomas-Fermi wave packet size as a function of $T_{d0}$, and use the known
trap frequency $\omega_{z}$ to calculate $w_{0}$ according to
$w(t)=w_{0}\sqrt{1+\omega^{2}T_{d0}^{2}}$ [65]. This method gives the value
$w_{0}=3.04\pm 0.3\,\mu$m, in reasonable agreement with the theoretically
calculated value.

As the HD theory and the generalized wave-packet model [42] both assume a
Gaussian wavepacket, we express the Thomas-Fermi size in terms of Gaussian
width by fitting it to a Gaussian profile, which gives $\sigma_{z}\simeq
0.41w_{0}=1.2\,\mu$m for the in-trap size. Also taking into account the
experimental time-of-flight of $T_{d0}=$1 ms (i.e. the time difference between
trap release and the start of the SGI sequence), the wavepacket size at the
start of the SGI sequence is
$\sigma_{z}(t=0)=\sqrt{1+\omega_{z}^{2}T_{d0}^{2}}=1.5\,\mu$m.

The measured values of $l_{p}$ and $l_{z}$ can now be compared to theory (Eq.
1) using the obtained BEC wavepacket size. A momentum coherence width of
$l_{p}/m=0.12\,$mm/s corresponds to a wavepacket size of
$\sigma_{z}=6.1\,\mu$m. This means that the momentum coherence width is about
4.1 times narrower than expected for a pure condensate. The discrepancy could
originate from the contribution of the thermal fraction of the atoms, although
we evaluate the BEC fraction to be larger than 0.7, so we cannot fully explain
this effect yet.

A coherence length of $l_{z}=0.5\,\mu$m corresponds to a momentum uncertainty
of $\sigma_{p}/m=2$ mm/s, while the expected momentum uncertainty of a pure
BEC after 1 ms expansion is $\sigma_{p}/m\approx 0.6$ mm/s (including
increased momentum width due to conversion of the mean-field potential energy
to kinetic energy). This means that the coherence length is shorter than
expected for a pure condensate, a discrepancy which is only partially
explained by the thermal fraction of atoms.

###### Acknowledgements.

We thank Mark Keil for helpful discussions. We are grateful to Zina Binstock
for the electronics, and the BGU nano-fabrication facility for providing the
high-quality chip. This work is funded in part by the Israeli Science
Foundation (grants No. 1314/19 and 1381/13), the EC “MatterWave” consortium,
the DFG through the DIP program (FO 703/2-1), and the program for postdoctoral
researchers of the Israeli Council for Higher Education. A.M. is supported by
Netherlands Organization for Scientific Research (NWO) Grant number
680-91-119. SB would like to acknowledge UK EPSRC grants EP/N031105/1 and
EP/S000267/1.

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# Green Measures for a Class of non-Markov Processes

Herry Pribawanto Suryawan
Department of Mathematics, Sanata Dharma University
55281 Yogyakarta, Indonesia
Email<EMAIL_ADDRESS>José Luís da Silva
CIMA, University of Madeira, Campus da Penteada,
9020-105 Funchal, Portugal
Email<EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

In this paper, we investigate the Green measure for a class of non-Gaussian
processes in $\mathbb{R}^{d}$. These measures are associated with the family
of generalized grey Brownian motions $B_{\beta,\alpha}$, $0<\beta\leq 1$,
$0<\alpha\leq 2$. This family includes both fractional Brownian motion,
Brownian motion, and other non-Gaussian processes. We show that the perpetual
integral exists with probability $1$ for $d\alpha>2$ and $1<\alpha\leq 2$. The
Green measure then generalizes those measures of all these classes.
Keywords: Fractional Brownian motion; generalized grey Brownian motion; Green
measure; subordination.

###### Contents

  1. 1 Introduction
  2. 2 Generalized Grey Brownian Motion
    1. 2.1 Definition and Properties
    2. 2.2 Representations of Generalized Grey Brownian Motion
  3. 3 The Green Measure for Generalized Grey Brownian Motion
  4. 4 Discussion and Conclusions

##  1 Introduction

The goal of this paper (see Theorem 3.2 and Corollary 3.3 below) is to prove
the existence of the Green measure for a class of non-Gaussian processes in
$\mathbb{R}^{d}$, called generalized grey Brownian motion (ggBm for short). We
denote this family of processes by $B_{\beta,\alpha}$ with parameters
$0<\beta\leq 1$ and $0<\alpha\leq 2$. More precisely, for a Borel function
$f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$, the potential of $f$ (see [BG68,
RY99] for details) is defined as

$V_{\beta,\alpha}(f,x)=\int_{0}^{\infty}\mathbb{E}\big{[}f(x+B_{\beta,\alpha}(t))\big{]}\,\mathrm{d}t,\quad
x\in\mathbb{R}^{d}.$ (1)

We would like to investigate the class of functions $f$ for which the
potential of $f$ has the representation

$V_{\beta,\alpha}(f,x)=\int_{\mathbb{R}^{d}}f(y)\mathcal{G}_{\beta,\alpha}(x,\mathrm{d}y),$
(2)

where $\mathcal{G}(x,\cdot):=\mathcal{G}_{\beta,\alpha}(x,\cdot)$ is a Radon
measure on $\mathbb{R}^{d}$ called Green measure corresponding to the ggBm
$B_{\beta,\alpha}$, see Definition 3.1 below. First, we establish the
existence of the perpetual integral (cf. Theorem 3.2)

$\int_{0}^{\infty}f(x+B_{\beta,\alpha}(t))\,\mathrm{d}t$

with probability $1$. This leads to an explicit representation of the Green
measure for ggBm, namely (cf. Corollary 3.3)

$\mathcal{G}_{\beta,\alpha}(x,\mathrm{d}y)=\frac{D}{|x-y|^{d-2/\alpha}}\,\mathrm{d}y,\quad
d\alpha>2,\quad 1<\alpha\leq 2,$

where $D$ is a constant that depends on $\beta,\alpha$, and the dimension $d$;
see Theorem 3.2 for the explicit expression. Note that as $d\alpha>2$ and
$1<\alpha\leq 2$, the Green measure $\mathcal{G}_{\beta,\alpha}(x,\cdot)$
exists for $d\geq 2$, since $d>2/\alpha\in[1,2)$. The Brownian case
($\alpha=1)$ is covered only for $d\geq 3$.

We emphasize that the existence of the Green measure for a given process $X$
is not always guaranteed. As an example, the $d$-dimensional Brownian motion
(Bm) starting at $x\in\mathbb{R}^{d}$ has a density given by $p_{t}(x,y)=(2\pi
t)^{-d/2}\exp\big{(}-|x-y|^{2}/(2t)\big{)}$, $y\in\mathbb{R}^{d}$. It is not
difficult to see that $\int_{0}^{\infty}p_{t}(x,y)\,\mathrm{d}t$ does not
exist for $d=1,2$. This implies the non-existence of the Green measure of Bm
for $d=1,2$. On the other hand, for $d\geq 3$, the Green measure of Bm on
$\mathbb{R}^{d}$ exists and is given by
$\mathcal{G}(x,\mathrm{d}y)=C(d)|x-y|^{2-d}\,\mathrm{d}y$, where $C(d)$ is a
constant depending on the dimension $d$; see [KMdS21] and the references
therein for more details. In a two-dimensional space, the Green measure of
ggBm is determined by the parameter $\alpha$. The Green measure of ggBm for
$d=1$ requires further analysis (for Bm see [AG01], Ch. 4) which we will
postpone for a future paper.

The paper is organized as follows. In Section 2 we recall the definition and
main properties of ggBm that will be needed later. In Section 3 we show the
existence of the perpetual integral with probability $1$, which leads to the
explicit formula for the Green measure for ggBm. In Section 4, we discuss the
results obtained, connect them with other topics, and draw conclusions.

##  2 Generalized Grey Brownian Motion

We recall the class of non-Gaussian processes, called the generalized grey
Brownian motion, which we study below. This class of processes was first
introduced by Schneider [Sch90a, Sch90b], and was generalized by Mura et al.
(see [MP08, MM09]) as a stochastic model for slow/fast anomalous diffusion
described by the time fractional diffusion equation.

###  2.1 Definition and Properties

For $0<\beta\leq 1$ the Mittag-Leffler (entire) function $E_{\beta}$ is
defined by the Taylor series

$E_{\beta}(z):=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\beta n+1)},\quad
z\in\mathbb{C},$ (3)

where

$\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}\,dt,\quad
z\in\mathbb{C},\,\mathrm{Re}(z)\geq 0$

is the Euler gamma function.

The $M$-Wright function is a special case of the class of Wright functions
$W_{\lambda,\mu}$, $\lambda>-1$, $\mu\in\mathbb{C}$ via

$M_{\beta}(z):=W_{-\beta,1-\beta}(-z)=\sum_{n=0}^{\infty}\frac{(-z)^{n}}{n!\Gamma(-\beta
n+1-\beta)}.$

The special choice $\beta=1/2$ yields the Gaussian density on $[0,\infty)$

$M_{\frac{1}{2}}(z)=\frac{1}{\sqrt{\pi}}\mathrm{exp}\left(-\frac{z^{2}}{4}\right).$
(4)

The Mittag-Leffler function $E_{\beta}$ is the Laplace transform of the
$M$-Wright function, that is,

$E_{\beta}(-s)=\int_{0}^{\infty}e^{-s\tau}M_{\beta}(\tau)\,d\tau.$ (5)

The generalized moments of the density $M_{\beta}$ of order $\delta>-1$ are
finite and are given (see [MP08]) by

$\int_{0}^{\infty}\tau^{\delta}M_{\beta}(\tau)\,d\tau=\frac{\Gamma(\delta+1)}{\Gamma(\beta\delta+1)}.$
(6)

######  Definition 2.1.

Let $0<\beta\leq 1$ and $0<\alpha\leq 2$ be given. A $d$-dimensional
continuous stochastic process
$B_{\beta,\alpha}=\big{\\{}B_{\beta,\alpha}(t),\,t\geq 0\big{\\}}$ starting at
$0\in\mathbb{R}^{d}$ and defined on a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$, is a ggBm in $\mathbb{R}^{d}$ (see [MM09]
for $d=1$) if:

  1. 1.

$\mathbb{P}(B_{\beta,\alpha}(0)=0)=1$, that is, $B_{\beta,\alpha}$ starts at
zero $\mathbb{P}$-almost surely ($\mathbb{P}$-a.s.).

  2. 2.

Any collection
$\big{\\{}B_{\beta,\alpha}(t_{1}),\ldots,B_{\beta,\alpha}(t_{n})\big{\\}}$
with $0\leq t_{1}<t_{2}<\dots<t_{n}<\infty$ has a characteristic function
given, for any $\theta=(\theta_{1},\ldots,\theta_{n})\in(\mathbb{R}^{d})^{n}$
with $\theta_{k}=(\theta_{k,1},\dots\theta_{k,d})$, $k=1,\dots,n$, by

$\mathbb{E}\left[\mathrm{exp}\left(\mathrm{i}\sum_{k=1}^{n}(\theta_{k},B_{\beta,\alpha}(t_{k}))_{\mathbb{R}^{d}}\right)\right]=E_{\beta}\left[-\frac{1}{2}\sum_{j=1}^{d}(\theta_{.,j},\gamma_{\alpha}\theta_{.,j})_{\mathbb{R}^{n}}\right],$
(7)

where $\mathbb{E}$ denotes the expectation w.r.t. $\mathbb{P}$ and

$\gamma_{\alpha}:=\gamma_{\alpha,n}:=\big{(}t_{k}^{\alpha}+t_{j}^{\alpha}-|t_{k}-t_{j}|^{\alpha}\big{)}_{k,j=1}^{n}.$

  3. 3.

The joint probability density function of
$\big{(}B_{\beta,\alpha}(t_{1}),\ldots,B_{\beta,\alpha}(t_{n})\big{)}$ is
equal to

$\rho_{\beta}(\theta,\gamma_{\alpha})=\frac{(2\pi)^{-\frac{nd}{2}}}{(\mathrm{det}\gamma_{\alpha})^{d/2}}\int_{0}^{\infty}\tau^{-\frac{nd}{2}}\mathrm{e}^{-\frac{1}{2\tau}\sum_{j=1}^{d}(\theta_{\cdot,j},\gamma_{\alpha}^{-1}\theta_{\cdot,j})_{\mathbb{R}^{n}}}M_{\beta}(\tau)\,d\tau.$
(8)

The following are the most important key properties of ggBm:

  1. (P1).

For each $t\geq 0$, the moments of any order of $B_{\beta,\alpha}(t)$ are
given by

$\begin{cases}\mathbb{E}\big{[}|B_{\beta,\alpha}(t)|^{2n+1}\big{]}&=0,\\\
\vskip
4.0pt\cr\mathbb{E}\big{[}|B_{\beta,\alpha}(t)|^{2n}\big{]}&=\frac{(2n)!}{2^{n}\Gamma(\beta
n+1)}t^{\alpha n}.\end{cases}$

  2. (P2).

The covariance function has the form

$\mathbb{E}\big{[}\big{(}B_{\beta,\alpha}(t),B_{\beta,\alpha}(s)\big{)}\big{]}=\frac{d}{2\Gamma(\beta+1)}\big{(}t^{\alpha}+s^{\alpha}-|t-s|^{\alpha}\big{)},\quad
t,s\geq 0.$ (9)

  3. (P3).

For each $t,s\geq 0$, the characteristic function of the increments is

$\mathbb{E}\big{[}e^{i(k,B_{\beta,\alpha}(t)-B_{\beta,\alpha}(s))}\big{]}=E_{\beta}\left(-\frac{|k|^{2}}{2}|t-s|^{\alpha}\right),\quad
k\in\mathbb{R}^{d}.$ (10)

  4. (P4).

The process $B_{\beta,\alpha}$ is non-Gaussian, $\alpha/2$-self-similar with
stationary increments.

  5. (P5).

The ggBm is not a semimartingale. Furthermore, $B_{\alpha,\beta}$ cannot be of
finite variation in $[0,1]$ and, by scaling and stationarity of the increment,
on any interval in $\mathbb{R}^{+}$.

  6. (P5).

For $n=1$, the density $\rho_{\beta}(x,t)$, $x\in\mathbb{R}^{d}$, $t>0$, is
the fundamental solution of the following fractional differential equation
(see [MP15])

$\mathbb{D}_{t}^{2\beta}\rho_{\beta}(x,t)=\Delta_{x}\rho_{\beta}(x,t),$

where $\Delta_{x}$ is the $d$-dimensional Laplacian in $x$ and
$\mathbb{D}_{t}^{2\beta}$ is the Caputo-Dzherbashian fractional derivative;
see [SKM93] for the definition and properties.

###  2.2 Representations of Generalized Grey Brownian Motion

The ggBm admits different representations in terms of well-known processes. It
follows from (7) that ggBm has an elliptical distribution, see Section 2 in
[dSE15] or Section 3 in [GJ16]. On the other hand, ggBm is also given as a
product (see [MP08] for $d=1$) of two processes as follows

$\big{\\{}B_{\beta,\alpha}(t),\,t\geq
0\big{\\}}\overset{\mathcal{L}}{=}\big{\\{}\sqrt{Y_{\beta}}B^{\alpha/2}(t),\,t\geq
0\big{\\}}.$ (11)

Here, $\overset{\mathcal{L}}{=}$ means equality in law, the nonnegative random
variable $Y_{\beta}$ has density $M_{\beta}$ and $B^{\alpha/2}$ is a
$d$-dimensional fBm with Hurst parameter $\alpha/2$ and independent of
$Y_{\beta}$.

We give another representation of ggBm $B_{\beta,\alpha}$ as a subordination
of fBm (see Prop. 2.14 in [ERdS24] for $d=1$) which is used below. For
completeness, we give the short proof.

######  Proposition 2.2.

The ggBm has the following representation

$\big{\\{}B_{\beta,\alpha}(t),\,t\geq
0\big{\\}}\overset{\mathcal{L}}{=}\big{\\{}B^{\alpha/2}(tY_{\beta}^{1/\alpha}),\,t\geq
0\big{\\}}.$ (12)

###### Proof.

We must show that both representations (11) and (12) have the same finite-
dimensional distribution. For every
$\theta=(\theta_{1},\ldots,\theta_{n})\in(\mathbb{R}^{d})^{n}$, we have

$\displaystyle\mathbb{E}\left[\exp\left(\mathrm{i}\sum_{k=1}^{n}\big{(}\theta_{k},B^{\alpha/2}(t_{k}Y_{\beta}^{1/\alpha})\big{)}\right)\right]$
$\displaystyle=\int_{0}^{\infty}\mathbb{E}\left[\exp\left(\mathrm{i}\sum_{k=1}^{n}\big{(}\theta_{k},B^{\alpha/2}(t_{k}y^{1/\alpha})\big{)}\right)\right]M_{\beta}(y)\,\mathrm{d}y$
$\displaystyle=\int_{0}^{\infty}\mathbb{E}\left[\exp\left(\mathrm{i}\sum_{k=1}^{n}\big{(}\theta_{k},y^{1/2}B^{\alpha/2}(t_{k})\big{)}\right)\right]M_{\beta}(y)\,\mathrm{d}y$
$\displaystyle=\mathbb{E}\left[\exp\left(\mathrm{i}\sum_{k=1}^{n}\big{(}\theta_{k},Y_{\beta}^{1/2}B^{\alpha/2}(t_{k})\big{)}\right)\right].$

In the second equality, we used the $\alpha/2$-self-similarity of fBm. This
completes the proof. ∎

##  3 The Green Measure for Generalized Grey Brownian Motion

In this section we show the existence of the Green measure for the ggBm, see
(1) and (2). Let us begin by discussing the existence of the Green measure for
a general stochastic process $X$.

Let $X=\\{X(t),\;t\geq 0\\}$ be a stochastic process in $\mathbb{R}^{d}$
starting from $x\in\mathbb{R}^{d}$. If $X(t)$, $t\geq 0$, has a probability
distribution $\rho_{X(t)}(x,\cdot)$, then Eq. (1) becomes

$V_{X}(x,f)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(y)\,\rho_{X(t)}(x,\mathrm{d}y)\,\mathrm{d}t.$
(13)

Then, applying the Fubini theorem, the Green measure
$\mathcal{G}_{X}(x,\cdot)$ of $X$ is given by

$\mathcal{G}_{X}(x,\mathrm{d}y)=\int_{0}^{\infty}\rho_{X(t)}(x,\mathrm{d}y)\,\mathrm{d}t,$

assuming the existence of $\mathcal{G}_{X}(x,\cdot)$ as a Radon measure on
$\mathbb{R}^{d}$. That is, for every bounded Borel set
$B\in\mathcal{B}_{b}(\mathbb{R}^{d})$ we have

$\mathcal{G}_{X}(x,B)=\int_{0}^{\infty}\rho_{X(t)}(x,B)\,\mathrm{d}t<\infty.$

If the probability distribution $\rho_{X(t)}(x,\cdot)$ is also absolutely
continuous with respect to the Lebesgue measure, say
$\rho_{X(t)}(x,\mathrm{d}y)=\rho_{t}(x,y)\,\mathrm{d}y$, then the function

$g_{X}(x,y):=\int_{0}^{\infty}\rho_{t}(x,y)\,\mathrm{d}t,\quad\forall
y\in\mathbb{R}^{d},$ (14)

is called the Green function of the stochastic process $X$. Moreover, the
Green measure in this case is given by
$\mathcal{G}_{X}(x,\mathrm{d}y)=g_{X}(x,y)\,\mathrm{d}y$.

This leads us to the following definition of the Green measure of a stochastic
process $X.$

######  Definition 3.1.

Let $X=\\{X(t),\;t\geq 0\\}$ be a stochastic process on $\mathbb{R}^{d}$
starting from $x\in\mathbb{R}^{d}$ and $\rho_{X(t)}(x,\cdot)$ be the
probability distribution of $X(t)$, $t\geq 0$. The Green measure of $X$ is
defined as a Radon measure on $\mathbb{R}^{d}$ by

$\mathcal{G}_{X}(x,B):=\int_{0}^{\infty}\rho_{X(t)}(x,B)\,\mathrm{d}t,\;\;B\in\mathcal{B}_{b}(\mathbb{R}^{d}),$

or

$\int_{\mathbb{R}^{d}}f(y)\mathcal{G}_{X}(x,\mathrm{d}y)=\int_{\mathbb{R}^{d}}f(y)\int_{0}^{\infty}\rho_{X(t)}(x,dy)\,\mathrm{d}t,\;\;f\in
C_{0}(\mathbb{R}^{d})$

whenever these integrals exist.

In other words, $\mathcal{G}_{X}(x,B)$ is the expected length of time the
process remains in $B$.

In order to state the main theorem which establishes the existence of the
Green measure for ggBm, first, we introduce a proper Banach space of functions
$f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$ such that

$\int_{0}^{\infty}f\big{(}x+B_{\beta,\alpha}(t)\big{)}\,\mathrm{d}t$

is finite $\mathbb{P}$-a.s. Without loss of generality, we may assume that
$f\geq 0$ above. We define the space $CL(\mathbb{R}^{d})$ of continuous real-
valued on $\mathbb{R}^{d}$ by

$CL(\mathbb{R}^{d}):=\big{\\{}f:\mathbb{R}^{d}\rightarrow\mathbb{R}\mid
f\;\text{is continuous, bounded and}\;f\in L^{1}(\mathbb{R}^{d})\big{\\}}.$

The space $CL(\mathbb{R}^{d})$ becomes a Banach space with the norm

$\|f\|_{CL}:=\|f\|_{\infty}+\|f\|_{1},\quad\forall f\in CL(\mathbb{R}^{d}),$

where $\|\cdot\|_{\infty}$ denotes the $\sup$-norm and $\|\cdot\|_{1}$ is the
norm in $L^{1}(\mathbb{R}^{d}$). The choice of $CL(\mathbb{R}^{d})$ allows us
to show that the family of random variables (also known as perpetual integral
functionals)

$\int_{0}^{\infty}f\big{(}x+B_{\beta,\alpha}(t)\big{)}\,\mathrm{d}t,\quad f\in
CL(\mathbb{R}^{d})$

have finite expectations $\mathbb{P}$-a.s.

######  Theorem 3.2.

Let $f\in CL(\mathbb{R}^{d})$ and $x\in\mathbb{R}^{d}$ be given and consider
ggBm $B_{\beta,\alpha}$ with $d\alpha>2$ and $1<\alpha\leq 2$. Then, the
perpetual integral functional $\int_{0}^{\infty}f(x+B(t))\,\mathrm{d}t$ is
finite $\mathbb{P}$-a.s. and its expectation equals

$\begin{gathered}\mathbb{E}\left[\int_{0}^{\infty}f(x+B_{\beta,\alpha}(t))\,\mathrm{d}t\right]=D\int_{\mathbb{R}^{d}}\frac{f(x+y)}{|y|^{d-2/\alpha}}\,\mathrm{d}y,\end{gathered}$
(15)

where
$D=D(\beta,\alpha,d)=\frac{1}{\alpha}2^{-1/\alpha}\pi^{-\frac{d}{2}}\Gamma\left(\frac{d}{2}-\frac{1}{\alpha}\right)\frac{\Gamma(1-\frac{1}{\alpha})}{\Gamma(1-\frac{\beta}{\alpha})}$.

###### Proof.

Given $x\in\mathbb{R}^{d}$ and $f\in CL(\mathbb{R}^{d})$ non-negative, let
$\rho_{\beta}(\cdot,t^{\alpha})$ denote the density of $B_{\beta,\alpha}(t)$,
$t\geq 0$, given by (see (8) with $n=1$)

$\rho_{\beta}(y,t^{\alpha})=\frac{1}{(2\pi
t^{\alpha})^{d/2}}\int_{0}^{\infty}\tau^{-d/2}\mathrm{e}^{-\frac{|y|^{2}}{2t^{\alpha}\tau}}M_{\beta}(\tau)\,\mathrm{d}\tau,\quad
y\in\mathbb{R}^{d}.$

First, we show the equality (15). It follows from the above considerations
that

$\mathbb{E}\left[\int_{0}^{\infty}f(x+B_{\beta,\alpha}(t))\,\mathrm{d}t\right]=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(x+y)\rho_{t}^{\beta,\alpha}(y)\,\mathrm{d}y\,\mathrm{d}t.\\\
=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(x+y)\frac{1}{(2\pi
t^{\alpha})^{d/2}}\int_{0}^{\infty}\tau^{-d/2}M_{\beta}(\tau)e^{-\frac{|y|^{2}}{2t^{\alpha}\tau}}\,\mathrm{d}\tau\,\mathrm{d}y\,\mathrm{d}t.$

Using Fubini’s Theorem, we first compute the $t$-integral and use the
assumption $d\alpha>2$. We obtain

$\int_{0}^{\infty}\frac{1}{(2\pi
t^{\alpha}\tau)^{d/2}}e^{-\frac{|y|^{2}}{2t^{\alpha}\tau}}\,\mathrm{d}t=C(\alpha,d)\frac{\tau^{-\frac{1}{\alpha}}}{|y|^{d-2/\alpha}},$

where

$C(\alpha,d):=\frac{1}{\alpha}2^{-1/\alpha}\pi^{-\frac{d}{2}}\Gamma\left(\frac{d}{2}-\frac{1}{\alpha}\right).$

Next we compute the $\tau$-integral using (6) so that

$\int_{0}^{\infty}\tau^{-1/\alpha}M_{\beta}(\tau)\,\mathrm{d}\tau=\frac{\Gamma(1-\frac{1}{\alpha})}{\Gamma(1-\frac{\beta}{\alpha})},\quad\alpha>1.$

Combining gives

$\mathbb{E}\left[\int_{0}^{\infty}f(x+B_{\beta,\alpha}(t))\,\mathrm{d}t\right]=D\int_{\mathbb{R}^{d}}\frac{f(x+y)}{|y|^{d-2/\alpha}}\,\mathrm{d}y,$

where

$D=D(\beta,\alpha,d)=C(\alpha,d)\frac{\Gamma(1-\frac{1}{\alpha})}{\Gamma(1-\frac{\beta}{\alpha})}.$

Therefore, the equality (15) is shown.
Now we show that the right-hand side of (15) is finite for every non-negative
$f\in CL(\mathbb{R}^{d})$. To see this, we may use the local integrability of
$|y|^{d-2/\alpha}$ in $y$ and obtain

$\displaystyle\int_{\mathbb{R}^{d}}\frac{f(x+y)}{|y|^{d-2/\alpha}}\,\mathrm{d}y$
$\displaystyle=$ $\displaystyle\int_{\\{|y|\leq
1\\}}\frac{f(x+y)}{|y|^{d-2/\alpha}}\,\mathrm{d}y+\int_{\\{|y|>1\\}}\frac{f(x+y)}{|y|^{d-2/\alpha}}\,\mathrm{d}y$
$\displaystyle\leq$ $\displaystyle C_{1}\|f\|_{\infty}+C_{2}\|f\|_{1}\leq
C\|f\|_{CL}.$

Therefore, the integral in (15) is, in fact, well defined. In other words, the
integral $\int_{0}^{\infty}f\big{(}x+B_{\beta,\alpha}(t)\big{)}\,\mathrm{d}t$
exists with probability 1. This completes the proof. ∎

As a consequence of the above theorem, we immediately obtain the Green measure
of ggBm $B_{\beta,\alpha}$, that is, comparing (2) and (15).

######  Corollary 3.3.

The Green measure of ggBm $B_{\beta,\alpha}$ for $d\alpha>2$ is given by

$\mathcal{G}_{\beta,\alpha}(x,\mathrm{d}y)=\frac{D}{|x-y|^{d-2/\alpha}}\,\mathrm{d}y.$

######  Remark 3.4.

  1. 1.

It is possible to show that given $f\neq 0$, the perpetual integral
$\int_{0}^{\infty}f(x+B_{\beta,\alpha}(t))\,\mathrm{d}t$ is a non-constant
random variable. As a consequence, for $f\geq 0$ the variance of
$\int_{0}^{\infty}f(x+B_{\beta,\alpha}(t))\,\mathrm{d}t$ is strictly positive.
The proof uses the notion of conditional full support of ggBm. We will not
provide a detailed explanation of this result that closely follows the ideas
of Theorem 2.2 in [KMdS21] to which we address the interested readers.

  2. 2.

Note also that the functional in (1)

$V_{\beta,\alpha}(\cdot,x):CL(\mathbb{R}^{d})\longrightarrow\mathbb{R}$

is continuous. In fact, from the proof of Theorem 3.2 for any $f\in
CL(\mathbb{R}^{d})$ yields

$|V_{\beta,\alpha}(f,x)|\leq K\|f\|_{CL},$

where $K$ is a constant depending on the parameters $\beta,\alpha$, and $d$.

##  4 Discussion and Conclusions

We have derived the Green measure for the class of stochastic processes called
the generalized grey Brownian motion in Euclidean space $\mathbb{R}^{d}$ for
$d\geq 2$. This class includes, in particular, fractional Brownian motion and
other non-Gaussian processes. To address the case where $d=1$, a
renormalization process is needed. However, this will be postponed to future
work. For $\beta=\alpha=1$ ggBm $B_{1,1}$ is nothing but a Brownian motion. In
this case, the Green measure exists for $d\geq 3$.

The relationship between the Green measure and the local time of the ggBm can
be described as follows. For any $T>0$ and a continuous function
$f:\mathbb{R}^{d}\longrightarrow\mathbb{R}$, the integral functional

$\int_{0}^{T}f(B_{\beta,\alpha}(t))\,\mathrm{d}t$ (16)

is well-defined. For $d=1$ the integral (16) with $f\in L^{1}(\mathbb{R})$ is
represented as

$\int_{0}^{T}f(B_{\beta,\alpha}(t))\,\mathrm{d}t=\int_{\mathbb{R}}f(x)L_{\beta,\alpha}(T,x)\,\mathrm{d}x,$

where $L_{\beta,\alpha}(T,x)$ is the local time of ggBm up to time $T$ at the
point $x$, see [dSE15, GJ16]. The Green measure corresponds to the asymptotic
behaviour in $T$ of the expectation of local time $L_{\beta,\alpha}(T,x)$. The
existence of this asymptotic depends on the dimension $d$ and the transient or
recurrent properties of the process.

### Acknowledgments

This research was funded by FCT-Fundação para a Ciência e a Tecnologia,
Portugal grant number UIDB/MAT/04674/2020,
https://doi.org/10.54499/UIDB/04674/2020 through the Center for Research in
Mathematics and Applications (CIMA) related to the Statistics, Stochastic
Processes and Applications (SSPA) group.

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