# A new technique to measure the gravitational mass of ultra-cold atoms in a
quadrupole magnetic trap and its implications for anti-hydrogen studies

Boaz Raz<EMAIL_ADDRESS>Gavriel Fleurov
<EMAIL_ADDRESS>Roi Holtzman  Nir Davidson  Department of
Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001,
Israel  Eli Sarid  Department of Physics, Ben-Gurion University of the Negev,
P.O.B. 653, Beer-Sheva 84105, Israel

††preprint: APS/123-QED

The supplementary material provides additional details and analysis for the
experimental and simulation procedures described in the main text. It is
ordered as follows. In Sec. I, we present the experimental setup used for this
experiment. In particular, we discuss the forced Radio-Frequency (RF)
evaporation used to cool the atoms and the effect of the Majorana spin-flips
and the lower boundary it places on the temperature of the cloud. In Sec. II,
we describe the data analysis process. In Sec. III, we present the simulation
used to analyze the experimental data. We discuss the dynamics, the initial
equilibrium distributions, and the verification of adiabatic cooling. In the
final section, Sec. IV, we present all the sources of statistical uncertainty
and systematic error in the system.

##  I Experimental Setup

In this work, we use an existing cold atoms experimental setup [1, 2],
following well-known [3, 4, 5, 6, 7] cooling and trapping techniques. As
detailed below, forced RF evaporation is used to cool the atomic cloud to the
various temperatures used in the experiment.

###  I.1 State Preparation

The atomic cloud is prepared in a quadrupole magnetic trap in a mixture of
$\ket{F=2,m_{F}=2}$ and $\ket{F=2,m_{F}=1}$ magnetic-trappable states,
followed by forced RF evaporation process [8], where atoms are transferred
into an anti-trapped state ($m_{F}<1$). The RF sweep employs a $\sim 10$ MHz
sweeping RF signal, corresponding to the Zeeman splitting of the [87]Rb
established in the magnetic trap. The atomic cloud is cooled by selectively
removing the most energetic atoms, since they reach the farthest points in the
trap and experience the largest magnetic fields, hence the largest Zeeman
splitting. The $m_{F}=2$ state has a magnetic moment twice as large as that of
$m_{F}=1$, so these atoms are lost from the trap before the $m_{F}=1$ atoms
since the sweep is from a high to a low frequency. The $m_{F}=2$ then
sympathetically cool the $m_{F}=1$ atoms, so the $m_{F}=1$ atoms are never
directly evaporated. As a result, by the end of evaporation, the relative
fraction of $m_{F}=1$ atoms is larger than it was at the beginning of the
process.

Figure 1: [87]Rb hyperfine ground state transitions. $\Delta E$ is the linear
Zeeman shift at small magnetic fields. Taken from [9]. Figure 2: [87]Rb D2
transition hyperfine structure, with frequency splittings between the
hyperfine energy levels. The approximate Landé $g_{F}$ factors for each level
are also given, with the corresponding Zeeman splittings between adjacent
magnetic sublevels. Taken from [10]

The ratio of magnetic moments is also why our measurements are limited to the
final magnetic field gradient of $\beta_{f}=2g$ (see main text for
experimental protocol and definition of $\beta_{f}$). The $m_{F}=1$ atoms
survive up to that point and distort the counting of the atoms. Scanning
$\beta_{f}$ from high to low values, we find two drops in the number of atoms
at $\beta_{f}=g$ and $\beta_{f}=2g$, as shown in Fig. 3.

###  I.2 Majorana Spin-Flips

Majorana spin-flips [11, 3, 12, 13, 14] losses occur when the relative rate of
change of the magnetic field that the atom experiences
$\absolutevalue{\ifrac{\dot{B}}{B}}$ due to its velocity is faster than the
Larmor frequency $\omega_{L}$. The atoms’ magnetic moments can not follow
these fast changes in the magnetic field, so they ”flip” their magnetic state
into an anti-trapped state. This creates a ”hole” in the trap, allowing the
atoms to escape it. The size of the ”hole” in the trap depends on the
temperature of the atomic cloud. The center of the hole is located at the
bottom of the trap, where the magnetic field is approaching zero, and the
relative change in the magnetic field is diverging. This results in losing the
coldest atoms from the trap since they spend the most time near the hole.

The loss rate due to Majorana spin-flips is proportional to $(\beta/T)^{2}$,
where $T$ is the temperature of the cloud. During the experimental protocol,
$\beta$ is lowered at a rate that allows adiabatic cooling; thus, the
temperature follows $(T\propto\beta^{2/3})$ [15]. Therefore, the overall loss
rate is proportional to $(\beta/T)^{2}=\beta^{2/3}$. The higher the magnetic
field gradient, the higher the loss rate. This means the loss rate is reduced
when the magnetic field is lowered, so the survival probability is higher.
This effect is opposite to the effect of gravity, where lowering $\beta$
causes the loss of more atoms. The probability of an atom being lost from the
trap due to Majorana spin-flips is given by
$P_{\text{Majorana}}=1-e^{-t\Gamma}$, where $\Gamma$ is the loss rate and $t$
is the time that the atoms are held in the trap. The Majorana losses become
dominant for lower initial temperatures, reducing the atoms’ lifetime in the
trap and distorting the measured loss feature. Therefore, there is an optimal
temperature for the experiment that balances Majorana’s losses with the desire
for lower temperatures, where the loss feature is narrower.

It should be noted that the exact formula for the loss rate due to Majorana
spin-flips may depend on several factors, including the magnetic field
gradient profile, the trap geometry, and the specific atomic species used.
Therefore, the formula above should be considered a general scaling relation,
and the specific parameters may need to be determined experimentally for each
system.

The effect of the Majorana spin-flips and the loss of atoms in different
magnetic states are shown in Fig. 3. When repeating the experiment with lower
magnetic field gradients, the loss feature exhibits a rise in the number of
atoms from $\beta_{f}=1.5g$ down to $\beta_{f}=1.2g$ before the drop due to
gravity. Majorana losses become substantial, roughly around $T<70\mu K$ in our
system.

Figure 3: Two drops (from right to left) in the number of atoms as a function
of $\beta_{f}$. The right drop corresponds to atoms in the $m_{F}=1$ state,
and the left drop to atoms in the $m_{F}=2$ state. The rise to the left drop -
moving from $\beta\simeq 1.5g$ to $\beta\simeq 1.2g$ is caused by the Majorana
spin-flips; see text.

##  II Data Analysis

The experimental data comprises $\sim 11,000$ data points from 16
temperatures, $76$ values of $\beta_{f}$, and 9 repetitions of each value (see
Fig. 1(c) in the main text). The system takes one data point every 35 seconds,
leading to a total experiment time of almost a week. Over this extended
period, the system is subject to several sources of long-term instabilities,
such as laser power, frequency, and polarization fluctuations, drifts in
ambient temperature and humidity in the lab, and magnetic field fluctuations.
Most of these effects are mitigated by incorporating temperature measurements
of the cloud between experiment points and monitoring the number of atoms
loaded into the MOT at the initialization stages of the system by collecting
fluorescence using a Photo-Multiplier Tube (PMT). The data is sampled wholly
randomized to ensure that averaging cancels short-term fluctuations. The
current in the coils is measured, and $\beta_{f}$ is determined during each
point of the sequence.

Furthermore, outliers more than three standard deviations from the mean of all
other points with the same $\beta_{f}$ are eliminated. The percentage of
points eliminated this way is marked in purple in Fig 5. Data points where the
PMT shows significantly low numbers due to failure in loading into the MOT are
disqualified. This step is crucial because, at lower magnetic field values,
where only a few atoms are left, a nearly empty trap cannot be distinguished
from a poorly loaded one - see Fig. 4(b-c). The data eliminated due to low PMT
values are marked in yellow in Fig. 5.

Figure 4: The data analysis process. (a) A Gaussian fit (red) to an atomic
cloud’s column density (blue) after ToF = 4 ms, horizontal axis centered
around the fitted mean. (b-c) Absorption image of a cloud of 5.35 million
atoms (b) and 0.8 million atoms (c). (d) Number of atoms left in the trap at
the end of the experiment sequence for various temperatures.

The number of atoms is determined by absorption imaging of the atomic cloud.
Four milliseconds after releasing the atoms from the trap, a resonant beam of
light is shined onto the atoms and images them onto a CCD camera. The Optical
Density (OD) is then computed [16] from each image. An example image is shown
in the inset in Fig. 4(a), and then the number of atoms is computed using the
column densities of the OD. The number of atoms is plotted as a function of
$\beta_{f}$ in Fig. 4(d). Note the stark difference between the number of
atoms at strong magnetic traps for different temperatures due to evaporation.

The atom numbers are normalized to compute the survival probability. At each
repetition for each temperature, the atom number is divided by the mean number
of atoms of the five highest $\beta_{f}$ of that repetition - three highest
for $T>169$ µK. As discussed above, the measurements were carried out over a
few days. Thus, each repetition is treated individually to avoid broadening
the error bars due to long-term drifts. After normalization, outliers of more
than three standard deviations from the mean are eliminated. These data points
appear orange in Fig. 5(b).

Figure 5: Experimental data usage for each temperature (Y-axis starts at 85%).
Data taken into the analysis (blue), data eliminated due to outliers after
normalization (red), data eliminated due to low PMT (yellow), and data
eliminated due to a mismatch in the electric current (purple).

##  III Simulation

In this section, we describe how the simulation of the evolution of the atoms
cloud is performed. We assume that the atom cloud is (i) initiated in thermal
equilibrium, namely, that the Boltzmann distribution gives the initial
distribution, and (ii) that the atoms do not interact with each other during
the protocol. These assumptions are justified since the collision rate reduces
with the trapping strength, reaching less than 1 Hz during the experiment
protocol, which takes 630 ms, so the contribution is negligible.

###  III.1 The Dynamics of the System

The Hamiltonian of a single particle is given by

$\displaystyle H(\vec{r},\vec{p},t)$
$\displaystyle=\frac{p^{2}}{2m}+U(\vec{r},t)$ (1) $\displaystyle U(\vec{r},t)$
$\displaystyle=g_{F}m_{F}\mu_{B}\beta(t)\sqrt{\frac{x^{2}+y^{2}}{4}+z^{2}}+mgz,$
(2)

where $g_{F}$ is the hyperfine Landé g-factor, $m_{F}$ is the magnetic quantum
number of the hyperfine level $F$, $\mu_{B}$ is Bohr’s magneton, $\beta$ is
the magnetic field gradient in the $z$ direction, which is chosen to point
opposite to the direction of gravity, $g$ is the gravitational acceleration
and $m$ is the atomic mass. This Hamiltonian determines the initial
distribution

$\rho(\vec{r},\vec{p},T)=\frac{\exp\left(-H(\vec{r},\vec{p})/k_{B}T\right)}{\int
d\vec{r}d\vec{p}\exp\left(-H(\vec{r},\vec{p})/k_{B}T\right)}.$ (3)

Once an initial condition for a particle is chosen, the system evolves
according to Hamilton’s equations

$\displaystyle\dot{\vec{r}}$ $\displaystyle=\frac{\partial
H}{\partial\vec{p}}=\frac{\vec{p}}{m}$ (4) $\displaystyle\dot{\vec{p}}$
$\displaystyle=-\frac{\partial
H}{\partial\vec{r}}=-g_{F}m_{F}\mu_{B}\beta(t)\frac{1}{\sqrt{\frac{x^{2}+y^{2}}{4}+z^{2}}}\left(\frac{x}{4},\frac{y}{4},z\right)-mg\hat{z}.$
(5)

###  III.2 Sampling Initial Conditions

Figure 6: Initial distributions. (a) Histogram of $10^{6}$ instances of the
sampled velocity $v_{x}$ using the Metropolis algorithm described in Sec.
III.2. The resulting histogram fits the expected Maxwell-Boltzmann
distribution in Eq. (6) with no fitting parameters. (b) Histogram of the total
energy of $10^{6}$ initial conditions sampled using the Metropolis algorithm
described in Sec. III.2. The histogram fits the expected energy distribution
in a quadrupole magnetic trap given in Eq. (7) with no fitting parameters.

To properly sample the initial conditions from the Boltzmann distribution in
Eq. (3), we use the Metropolis algorithm [17]. The algorithm is performed as
follows. A random uniform initial condition is chosen for both
$\vec{r}=(x,y,z)$ and $\vec{p}$. First, a deviation $\delta\vec{r}$ is chosen
uniformly. The new state $\vec{r}_{\rm new}=\vec{r}+\delta\vec{r}$ is proposed
and accepted by the following condition. If the new state has lower energy
from the current state, $H(\vec{r}_{\rm new},\vec{p})=E_{\rm new}<E_{\rm
curr}=H(\vec{r},\vec{p})$, then the new state is accepted. Otherwise, it is
accepted with probability $\exp(-(E_{\rm new}-E_{\rm curr})/k_{B}T)$. The same
step is performed for a deviation in momentum $\delta\vec{p}$. These steps of
proposing new coordinates and momenta are performed repeatedly 80,000 times.
The resulting state $(\vec{r},\vec{p})$ is then a state sampled from the
Boltzmann distribution in Eq. (3), because the dynamics of this algorithm
satisfy detailed balance and it is ergodic.

We have validated that the resulting distribution of this algorithm provides
the Boltzmann distribution given in Eq. (3). The velocity distribution in each
axis fits very well to a Gaussian with the proper variance, see Fig. 6(b),
given by $k_{B}T/m$, namely

$\rho(v_{i},T)=\sqrt{\frac{m}{2\pi T}}e^{-mv_{i}^{2}/2k_{B}T},\qquad i=x,y,z.$
(6)

Moreover, the distribution of the total energy $E$ is given by [4]

$\rho(E,T)=\frac{16}{105\sqrt{\pi}(k_{B}T)^{9/2}}E^{7/2}e^{-E/k_{B}T}.$ (7)

Figure 6(a) shows the sampled initial energies and the curve in Eq. (7) with
no fitting parameters, and they match very well.

We also check that the initial conditions satisfy the virial theorem to
validate the simulation results further. The virial theorem is a relation
between ensemble averages of the kinetic and potential energies. For a linear
potential $V(r)\propto r^{1}$, it is

$\displaystyle 2\expectationvalue{E_{k}}=\expectationvalue{V},$ (8)

where $E_{k}$ is the kinetic energy. We compute the ensemble average energies
of the atoms in the simulation at each time step and follow their dynamics.
The atoms are evolved according to Eqs. (4) and (5) with
$\beta(t)=20\beta_{*}$, so their ensemble average should remain constant and
satisfy Eq. (8). See main text for the definition of
$\beta_{*}=mg/g_{F}m_{F}\mu_{B}$. In conclusion, the virial theorem is
satisfied within the simulation’s statistical uncertainties, indicating that
the simulations accurately capture the system’s thermal equilibrium behavior.

###  III.3 Adiabatic Cooling

In a quadrupole trap, a slowly changing magnetic gradient $\beta$ guarantees
adiabatic cooling [15, 18, 19], as discussed above in Sec. I.2. The
temperature will follow $\beta$ according to $T\propto\beta^{2/3}$. Note that
this process requires no collisions, and the cloud remains in thermal
equilibrium; therefore, we use the simulation to verify that the cloud’s
temperature changes adiabatically.

Figure 7: Validation of adiabatic cooling during the experimental protocol.
The temperature in the simulation is measured using a Gaussian fit to the
velocity distribution. The expected temperature according to adiabatic cooling
is also plotted with the initial temperature taken as the mean of the initial
plateau. Since atoms lost during the weakening of the trap remove heat from
the system, the data of the rise back is not expected to follow the
theoretical adiabatic heating.

Fig. 7 presents validation of adiabatic cooling based on the simulation. The
cloud is initialized at $T=220$ µK, and the simulation follows the
experimental protocol. At each time step, the temperature of the cloud is
calculated by comparing a Gaussian fit with the Maxwell-Boltzmann
distribution. The mean along the initial plateau (Time $<$ 200 ms) is
calculated and labeled as $T_{0}$, and the subsequent decrease of the
temperature is calculated according to $T=T_{0}\beta^{2/3}$. The ”measured”
temperature of the atoms in the simulation agrees with the expected
temperatures according to adiabatic cooling until atoms are lost from the trap
(green line). The temperatures are not relied upon after this point (Time ¿
370 ms).

##  IV Statistical uncertainty and systematic errors

###  IV.1 Magnetic Gradient Calibration

The system has two coils in an anti-Helmholtz configuration. Our workshop
electrical engineers designed a current source with 2 ppm stability and noise.
A real-time measurement of the current is taken during each shot, using a
voltage reading of Keysight 34470A, measured on a stable resistor (Vishay
VHP4ZT). We calibrated the current running through the coils and measured the
inherent offset of the instrument, see Fig. 8(a). The magnetic field gradient
in our system was measured immediately after the experiment. We placed a
gaussmeter (Lakeshore gaussmeter 455, $0.075\%$ accuracy) with an axial probe
(HMNA-1904-VR, DC to 20 kHz, accuracy $\pm 0.2\%$, stability $0.09$ G/K) on a
computer-controlled translation stage (repeatability $<5$ µm). The magnetic
field is measured along the gravitation axis between 5 and 9 mm above the
atoms along the coils’ symmetry axis using a custom holder to center the probe
to the geometric center of the magnetic field coils. Although reaching the
atoms’ position is impossible due to the vacuum cell’s physical constraints,
the magnetic field remains linear in the entire measured region, making the
extrapolation to the atom’s position viable. This process was repeated for
several values of currents. For example, Fig. 8(b) shows the measurement at a
set current of $1.6$ amps. A fitted linear line provides the evaluation of the
gradient and uncertainty. The value of the gradient is taken at each current,
including zero where the circuit was open, and fitted to find the linear
relation between the magnetic field gradient and the applied current to be
$\beta(I)=10.490\pm 0.005$ Gauss/(cm$\cdot$A) with a bias of $-0.134\pm 0.007$
Gauss/cm.

Figure 8: The calibration process of the magnetic gradient. (a) A histogram of
the offset voltage on the Keysight 34470A measurement. (b) An example of the
magnetic field measurement for an electric current of 1.6A. Here
$\beta=8.330\pm 0.028$ G/cm. (c) Calibration of the magnetic field gradient,
the blue line is the value at which $\beta=\beta_{*}$. Error bars are smaller
than the markers.

Microwave (MW) spectroscopy can measure magnetic field gradients with high
precision. Applying an MW field to the cold atoms can induce Rabi oscillations
between atomic states, where the frequency depends on the magnetic field
strength. An uncertainty in frequency of 1 kHz corresponds to a magnetic field
resolution of about 1.4 mG.

The magnetic field gradient in our experiment can be measured with higher
accuracy by probing the Rabi oscillation frequency of a cloud positioned with
a spatial resolution of 5 µm along the desired axis. At 300 G/cm gradients,
the magnetic field difference over 6 mm will be 180 G. The relative error in
the measured gradient will be on the order of $10^{-5}$.

###  IV.2 Temperature Measurement

The temperature of the atomic cloud is evaluated using the Time-of-Flight
(ToF) technique [20, 16], commonly used in cold atom experiments. The current
through the magnetic coils is turned off abruptly ($<1$ ms), releasing the
atoms from the trap. The atoms are then imaged after different expansion
times, and the absorption image is fitted with a Gaussian model. The width of
the Gaussian, $\sigma$, is taken to be the size of the cloud. After long
expansion times, when the cloud is much larger than the initial size in situ,
$\sigma$ is correlated with the initial momentum distribution. The temperature
in the trap is then given by $k_{B}T=m\sigma^{2}/t^{2}$ [16, 21]. Here,
$k_{B}$ is Boltzmann’s constant, $m$ is the atomic mass, and $t$ is the ToF.
The Gaussian width correlates with the trapped cloud’s most probable velocity.
In Fig. 9(a), a typical temperature measurement is plotted along with a linear
fit. The temperature and its uncertainty are extracted from the slope. A
typical absorption image of a cloud is shown for each data point.

Figure 9: The temperature measurement in the system. (a) The width of the
cloud squared is linearly fitted to the time of flight squared. A sample shot
of each time of flight is presented. The markers are larger than their
statistical errors. (b) Temperature measurements spread around their
respective mean. Horizontal labels specify the RF evaporation cut-off
frequency $\omega_{RF}$ and the combined uncertainty. Error bars represent
statistical uncertainty. Different colors represent measurements taken at a
$\sim 1$ day separation.

The measurement campaign was conducted according to the following protocol. We
chose six different RF evaporation cut-off frequencies $\omega_{RF}$ (see Fig.
9), calibrated to set the cloud at a range of temperatures between $70-230$
µK. The final magnetic gradient value $\beta_{f}$ was scanned over 76
different values in a fully randomized way to reduce biases due to fluctuating
ambient magnetic fields. Each point was repeated 9 times. Each data point
requires roughly 30 seconds, adding up to a day and a half of data collection.
At the end of this session, a temperature measurement was conducted at each of
the specified $\omega_{RF}$s. This entire session and the temperature
measurements were repeated three times. The number of atoms and the resulting
temperature of the cloud slightly drift on the scale of hours. Therefore, we
found that for measurement sessions with the same $\omega_{RF}$ but on two
different days, the resulting temperature differs on the order of 10 µK. But,
as shown in Fig. 4(d), the atom numbers and temperatures in a single
9-repetition session are very stable.

This temperature measurement has a systematic error due to the extended
duration (1 ms) of releasing atoms from the magnetic trap for imaging, leading
to a small amount of adiabatic cooling before complete release. In Fig. 10, We
compare the measured values’ results with those found by fitting them to the
simulation.

Figure 10: Comparison of Measured vs. Fitted Temperatures (blue), with error
bars representing the $1\sigma$ uncertainty for each method. The yellow line
indicates where the measured values equal the fitted values.

The results of our measurement for different temperatures are presented in
Fig. 11. The scatter of inferred values of $g$ across the 16 temperature
points suggests a variation that extends beyond the error bars, indicating
discrepancies that might not be accounted for by statistical uncertainty
alone.

Figure 11: Relative error in the measured value of $g$ as a function of the
measured temperatures with $1\sigma$ statistical error bars. The black line is
the weighted mean $(-1.9\times 10^{-4}g)$ (with statistical uncertainty
$(12\times 10^{-4}g)$). The gray shaded area is the uncertainty in the
magnetic field gradient calibration $(5\times 10^{-4}g)$. Figure 12: Relative
Uncertainty in gravitational constant measurements across temperatures: blue
for fitting, orange for magnetic field calibration, yellow for temperature
fitting, and purple for normalization coefficient.

###  IV.3 Variation of $g$ with $A$ and $T$

To extract the effect of temperature $T$ and normalization factor $A$ on the
final value of computed $g$, we scan these parameters in the simulation and
extract the corresponding value of $g$ for each value of the scanned parameter
by minimizing the error function $\epsilon(g,A)$ defined as:

$\displaystyle\epsilon(g,A,T)=\sum_{j}\frac{\quantity(A\times
P_{j}-f(\beta_{f,j};T,g))^{2}}{\sigma_{P,j}^{2}}$ (9)

Where $P_{j}$ is the measured survival probability, $f(\beta_{f,j};T,g)$ is
the simulated survival probability, $\beta_{f,j}$ are the final magnetic
gradient values of the experiment, and $\sigma_{P,j}$ is the one standard
deviation uncertainty in the measured value of $P_{j}$.

Figure 13: Error function $\epsilon(g,A,T=T_{exp})$ as a function of $g$
(vertical) and $A$ (horizontal) for all 16 measurement sets, at the measured
temperature $T_{exp}$. The temperature at each set is written above each
heatmap. Figure 14: Error function $\epsilon(g,A=A_{opt},T)$ as a function of
$g$ (vertical) and $T$ (horizontal) for all 16 measurement sets, at the
optimal $A_{opt}$. The measured temperature at each set is written above each
heatmap. Figure 15: . (a) The gradient field of the error function
$\epsilon(g,A)$ at a measured $T=126.4\pm 0.6$ µK. The green star is the
position of the minimum of the error function, and the yellow line is the
linear fit to the minimum of the gradient in its vicinity. The slope of the
line is $\partialderivative{g}{A}$. (b) The gradient field of the error
function $\epsilon(g,T)$ at a measured $T=126.4\pm 0.6$ µK and $A=0.8585\pm
0.0008$. The green star is the position of the minimum of the error function,
and the yellow line is the linear fit to the minimum of the gradient in its
vicinity. The slope of the line is $\partialderivative{g}{T}$.

Setting $T=T_{\text{exp}}$, where $T_{\text{exp}}$ is the experimentally
measured temperature, We scan $A$ and $g$ to calculate the error function in
the region of interest, see Fig. 13. The minimum is found as depicted in the
main text. The optimal $A$ for each measurement set was used to calculate the
error function for a relevant range of $T$ and $g$, see Fig. 14. Using $T$
from the independent temperature measurement as an input parameter, the
optimal $g_{opt}$ was taken as the one found through the heatmap of $g-A$.

To estimate the error in $g_{opt}$ as a function of $A$ and $T$, a linear line
was fitted to the direction of minimal change in $\epsilon(g,A,T)$ in a small
region around $g_{opt}$. We find it by calculating the gradient fields
$\partialderivative{g}{A}$ in Fig. 15(a) and $\partialderivative{g}{T}$ in
Fig. 15(b), and finding a linear fit to the points of minimal gradient. The
slopes of these lines are taken to be the values of the partial derivative
presented in the main text.

###  IV.4 Atom counting

Accurately counting the number of atoms in the trap is crucial for measuring
gravitational acceleration. Several sources of error can contribute to
inaccuracies in atom counting, as discussed in detail below.

The intensity of the imaging beam used for absorption imaging must be much
lower than the saturation intensity $I_{sat}$. This is achieved using a low-
power probe beam and carefully optimizing the polarization and detuning.
Additionally, any residual magnetic fields in the imaging region can cause
frequency shifts and cause an inhomogeneous broadening to the resonance
linewidth, leading to errors in atom counting. These systematic effects are
mitigated using common-mode rejection techniques, releasing the atoms from the
same trap for each experimental sequence regardless of $\beta_{f}$.

The quantum efficiency of the camera used for imaging also introduces
systematic errors in atom counting. We carefully calibrate the pixel size and
account for dark image noise to correct for these effects. Furthermore, a
reference image is acquired immediately after imaging the atoms to subtract
any background noise or imaging imperfections in each shot.

Using the repumping light before and during imaging ensures that all atoms are
in the same state before the absorption imaging. Aberrations in the imaging
system and speckles in the imaging beam can also introduce systematic errors
in atom counting. We carefully optimize the imaging system and beam parameters
to minimize these effects.

Finally, atom counting can be challenging at low atom numbers, especially when
approaching the detection limit. In this case, the relative counting
uncertainty can become significant and contribute to errors in the measurement
of gravitational acceleration. We carefully estimate the uncertainty in the
number of atoms using statistical analysis of multiple measurements and
propagate this uncertainty to the final measurement of $g$.

## References

  * Edri  _et al._ [2020] H. Edri, B. Raz, N. Matzliah, N. Davidson, and R. Ozeri, Physical Review Letters 124, 163401 (2020).
  * Shkedrov  _et al._ [2022] C. Shkedrov, M. Menashes, G. Ness, A. Vainbaum, E. Altman, and Y. Sagi, Physical Review X 12, 011041 (2022).
  * Davis  _et al._ [1995] K. B. Davis, M.-O. Mewes, M. A. Joffe, M. R. Andrews, and W. Ketterle, Physical Review Letters 74, 5202 (1995).
  * Luiten  _et al._ [1996] O. J. Luiten, M. W. Reynolds, and J. T. M. Walraven, Physical Review A 53, 381 (1996).
  * Ketterle  _et al._ [1996] W. Ketterle, M. R. Andrews, K. B. Davis, D. S. Durfee, D. M. Kurn, M. O. Mewes, and N. J. v. Druten, Physica Scripta T66, 31 (1996).
  * Müller  _et al._ [2000] J. H. Müller, D. Ciampini, O. Morsch, G. Smirne, M. Fazzi, P. Verkerk, F. Fuso, and E. Arimondo, Journal of Physics B: Atomic, Molecular and Optical Physics 33, 4095 (2000).
  * Cohen-Tannoudji and Guéry-Odelin [2011] C. Cohen-Tannoudji and D. Guéry-Odelin,  _Advances in atomic physics: an overview_ (World Scientific, Singapore ; Hackensack, NJ, 2011) pp. 529-532 and references therein.
  * DeMarco [2001] B. DeMarco,  _Quantum Behavior of an Atomic Fermi Gas_ , Ph.D. thesis, University of Colorado at Boulder (2001).
  * Daniel [2013] A. Daniel,  _Coherence Time of Rabi Oscillations of Cold Atoms_ , Ph.D. thesis, Ben-Gurion University of the Negev (2013).
  * [10] D. A. Steck, “Rubidium 87 D Line Data,” Available online at http://steck.us/alkalidata (revision 2.2.2, 9 July 2021).
  * Petrich  _et al._ [1995] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Physical Review Letters 74, 3352 (1995).
  * Sukumar and Brink [1997] C. V. Sukumar and D. M. Brink, Physical Review A 56, 2451 (1997).
  * Majorana and Bassani [2006] E. Majorana and G. F. Bassani,  _Ettore Majorana, scientific papers: on occasion of the centenary of his birth_ (Springer, Berlin ; New York, 2006) pp. 125-132: ”Oriented atoms in a variable magnetic field”.
  * Lin  _et al._ [2009] Y.-J. Lin, A. R. Perry, R. L. Compton, I. B. Spielman, and J. V. Porto, Physical Review A 79, 063631 (2009).
  * Almog  _et al._ [2018] I. Almog, J. Coslovsky, G. Loewenthal, A. Courvoisier, and N. Davidson, Applied Physics B 124, 158 (2018).
  * Ketterle  _et al._ [1999] W. Ketterle, D. S. Durfee, and D. M. Stamper-Kurn, “Making, probing and understanding Bose-Einstein condensates,”  (1999).
  * Metropolis  _et al._ [1953] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, The Journal of Chemical Physics 21, 1087 (1953).
  * Kastberg  _et al._ [1995] A. Kastberg, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and P. S. Jessen, Physical Review Letters 74, 1542 (1995).
  * Gabrielse  _et al._ [2011] G. Gabrielse, W. S. Kolthammer, R. McConnell, P. Richerme, R. Kalra, E. Novitski, D. Grzonka, W. Oelert, T. Sefzick, M. Zielinski, D. Fitzakerley, M. C. George, E. A. Hessels, C. H. Storry, M. Weel, A. Müllers, J. Walz, and ATRAP Collaboration, Physical Review Letters 106, 073002 (2011).
  * Anderson  _et al._ [1995] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995).
  * Regal and Jin [2006] C. Regal and D. S. Jin, “Experimental realization of BCS-BEC crossover physics with a Fermi gas of atoms,”  (2006).


#  $b$-Property of sublattices in vector lattices

###### Abstract

We study $b$-property of a sublattice (or an order ideal) $F$ of a vector
lattice $E$. In particular, $b$-property of $E$ in $E^{\delta}$, the Dedekind
completion of $E$, $b$-property of $E$ in $E^{u}$, the universal completion of
$E$, and $b$-property of $E$ in $\hat{E}(\hat{\tau})$, the completion of $E$.

Şafak Alpay1, Svetlana Gorokhova 2
$1$ Department of Mathematics, Middle East Technical University, 06800 Ankara,
Turkey<EMAIL_ADDRESS>
$2$ Southern Mathematical Institute of the Russian Academy of Sciences, 362027
Vladikavkaz, Russia<EMAIL_ADDRESS>
keywords: vector lattice, universal completion, Dedekind completion,
$b$-property, local solid vector lattice

MSC2020: 46A40, 46B42

##  1 Introduction and preliminaries

Vector lattices considered here are all real and Archimedean. Vector
topologies are assumed to be Hausdorff.

######  Definition 1.

A sublattice $F$ of a vector lattice $E$ is said to have $b$-property in $E$,
if $x_{\alpha}$ is a net in $F^{+}$ and $0\leq x_{\alpha}\uparrow\leq e$ for
some $e\in E$, then there exists $f\in F$ with $0\leq x_{\alpha}\uparrow\leq
f$.

Recall that a subset $F$ of $E$ is said to be majorizing in $E$ if, for each
$0<e\in E$, there exists $f\in F$ with $0\leq e\leq f$.

A subset $U$ of a vector lattice (VL) is called solid if $|u|\leq|v|$, $v\in
U$, imply $u\in U$. A linear topology $\tau$ on a VL $E$ is called locally
solid if $\tau$ has a base of zero consisting of solid sets.

A locally solid VL $E$ (LSVL) satisfies the Lebesgue property if
$x_{\alpha}\downarrow 0$ in $E$ implies
$x_{\alpha}\stackrel{{\scriptstyle\tau}}{{\to}}0$.

A LSVL $E(\tau)$ satisfies the Fatou property if $\tau$ has a base of zero
consisting of solid and order closed sets.

A sublattice $F$ in a VL $E$ is regular if $\inf A$ is the same as in $F$ and
$E$ whenever $A\subset F$ whose infimum exists in $F$. Ideals are regular in
$E$.

$E$ is called $\sigma$-laterally complete if the supremum of every disjoint
sequence exists in $E^{+}$ and laterally complete if supremum of every
disjoint subset in $E^{+}$ exists in $E$.

######  Example 1.

[1, p.198] Let $X$ be a topological space. A function $f:X\to\mathbb{R}$ is
called a step function if there exists a collection of mutually disjoint
subsets $\\{V_{i}\\}$ of $X$ such that $\bigcup_{i}V_{i}=X$, $f$ is constant
on each $V_{i}$, and $f\in C^{\infty}(X)$. Let $S^{\infty}(X)$ be the space of
step functions on an extremally disconnected topological space $X$. Then
$S^{\infty}(X)$ is a laterally complete VL.

Lateral completion $E^{\lambda}$ of a VL $E$ is defined to be the intersection
of all laterally complete vector lattices between $E$ and $E^{u}$.

Universal completion ($\sigma$-universal completion) of a VL $E$ is a
laterally ($\sigma$-laterally) and Dedekind complete (Dedekind
$\sigma$-complete) vector lattice $E^{u}$ (resp., $E^{s}$) which contains $E$
as an order dense sublattice. Every VL $E$ has a unique universal completion
[1, Theorem 7.21]

######  Example 2.

Let $X$ be an extremally disconnected topological space. $C^{\infty}(X)$, the
space of all extended continuous functions on $X$ with the usual algebraic and
lattice operations is a universally complete VL.

A net $(x_{\alpha})_{\alpha\in A}$ in a VL $E$ is order convergent to $x\in E$
if there exists a net $(x_{\beta})_{\beta\in B}$, possibly over a different
index set, such that $x_{\beta}\downarrow 0$ and, for each $\beta\in B$, there
exists $\alpha_{0}\in A$ with $|x_{\alpha}-x|\leq x_{\beta}$ for all
$\alpha\geq\alpha_{0}$. In this case we write
$x_{\alpha}\stackrel{{\scriptstyle o}}{{\to}}x$.

A net $x_{\alpha}$ in $E$ $uo$-converges to $x\in E$ if $|x_{\alpha}-x|\wedge
u\stackrel{{\scriptstyle o}}{{\to}}0$ for all $u\in E^{+}$. In this case we
write $x_{\alpha}\stackrel{{\scriptstyle uo}}{{\to}}x$.

Let $E(\tau)$ be a LSVL. A net $x_{\alpha}$ in $E$ is $u\tau$-convergent to
$x\in E$ if $|x_{\alpha}-x|\wedge u\stackrel{{\scriptstyle\tau}}{{\to}}0$ for
all $u\in E^{+}$. A net $x_{\alpha}$ in $E$ is called order Cauchy
$($$uo$-Cauchy $)$ if the doubly indexed net
$(x_{\alpha},x_{\alpha^{\prime}})_{(\alpha,\alpha^{\prime})}$ is order
convergent ($uo$-convergent) to zero. $E(\tau)$ is called $uo$-complete if
every $uo$-Cauchy net is $uo$-convergent in $E$.

The $b$-property of a VL $E$ was defined in [2] as: a VL $E$ has $b$-property
if every subset $A$ in $E$ which is order bounded in $(E^{\sim})^{\sim}$,
remains to be order bounded in $E$. Equivalently, a VL $E$ has $b$-property
iff each net $x_{\alpha}$ in $E$, with $0\leq x_{\alpha}\uparrow x$ for some
$x\in(E^{\sim})^{\sim}$, is order bounded in $E$ ([2]).

######  Example 3.

Every perfect VL, and therefore every order dual, have the $b$-property. Every
reflexive BL and every $KB$-space have $b$-property [2, 3, 4, 5]. On the other
hand, by considering the basis vectors $e_{n}$ in $c_{0}$, we see that $c_{0}$
does not have the $b$-property in $l_{\infty}$.

Let us note that Fremlin had considered subsets of a VL $E$ that are order
bounded in the universal completion $E^{u}$ of $E$. He proved that if $E$ is a
$\sigma$-Dedekind complete VL then $E$ is $\sigma$-laterally complete iff $E$
has the countable $b$-property in $E^{u}$ [1, Theorem 7.38].

######  Example 4.

Each projection band $F$ in a vector lattice $E$ has $b$-property in $E$. In
particular, every band in a Dedekind complete vector lattice has $b$-property.
An element $u$ in a VL $E$ is called an atom if whenever $v\wedge w=0$, $0\leq
v\leq u$, and $0\leq w\leq u$ imply either $v=0$ or $w=0$. If $x$ is an atom
in $E$, the principal band $B_{x}$ generated by $x$ is a projection band and
therefore has $b$-property in $E$.

######  Example 5.

Every majorizing sublattice $F$ has $b$-property in $E$. Let $0\leq
x_{\alpha}\uparrow\leq e$ for some net $x_{\alpha}\subseteq F$, $e\in E$. As
$F$ is majorizing, there exists $f\in F$ with $e\leq f$. Then $0\leq
x_{\alpha}\leq f$. Since it is well-known that $E$ is majoring in
$E^{\delta}$, $E$ has $b$-property in $E^{\delta}$.

######  Example 6.

Every order ideal $F$ in a vector lattice $E$ with $b$-property in $E$ is a
band of $E$. Indeed, let $x_{\alpha}$ be a net in $F$ such that $0\leq
x_{\alpha}\uparrow e\in E$, then $x_{\alpha}$ is $b$-bounded in $E$ and by the
$b$-property of $F$, there exists $f\in F$ with $0\leq x_{\alpha}\leq f$. As
$x_{\alpha}\uparrow e$, we have $0\leq e\leq f$ and as $F$ is an ideal, $e\in
F$.

######  Example 7.

Let $F\subseteq E$ be a sublattice of $E$ and $I(F)$ be the ideal generated by
$F$ in $E$. Then $F$ has $b$-property in $I(F)$. Having $b$-property is
transitive: if $E\subseteq F\subseteq G$ are sublattices of a VL $X$, then $E$
has $b$-property in $F$, and if $F$ has $b$-property in $G$, then $E$ has
$b$-property in $G$. If $E$ has $b$-property in $G$, then $E$ has $b$-property
in every sublattice of $G$ containing $E$ as a sublattice.

######  Example 8.

Let $(E,\|.\|)$ be a Banach lattice with order continuous norm and $F\subseteq
E$ be a norm-closed sublattice. Let $x_{n}$ be a $b$-bounded sequence in $F$
such that $0\leq x_{n}\leq e$ for some $e\in E$. Then $x_{n}$ is norm-Cauchy
and is convergent to some $x\in E$. As $F$ is norm-closed, $x\in F$ and
consequently $x_{n}\uparrow x$. That is to say $F$ has countable $b$-property
in $E$. Order continuity of the ambient space is essential in this example, if
one takes $E=l^{\infty}$ and $F=c_{0}$, Then by considering the sequence
$e_{n}$ in $c_{0}$, we see that $c_{0}$ has no $b$-property in $l^{\infty}$.

######  Example 9.

Generalizing Example 8, let $E(\tau)$ be LSVL with Lebesgue property. Then
every $\tau$-closed order ideal has $b$-property in $E(\tau)$. This is because
every $\tau$-closed ideal is a band and, as $E(\tau)$ is Dedekind complete, it
is a projection band.

######  Example 10.

Given a LSVL $E(\tau)$, let us denote by $E^{\lambda}$ its lateral completion
and $E^{u}$ its universal completion. Then the equality
$(E^{\lambda})^{\delta}=(E^{\delta})^{\lambda}=E^{u}$ (see [1, Exer.10 on
p.213]) shows that each laterally complete $E(\tau)$ has $b$-property in its
universal completion.

######  Example 11.

If $E(\tau)$ is a laterally complete LSVL, then it has the projection property
and every band on $E$ has $b$-property. Furthermore, a subset $A\subset E^{+}$
of a laterally complete VL $E$ is order bounded in $E^{u}$ iff it is order
bounded in $E$ by [1, Theorem 7.14]

Let us observe that all Lebesgue topologies on a LSVL $E(\tau)$ induce the
same topology on order bounded subsets of $E$. Therefore, if $F$ is a
sublattice of $E$ then on all subsets of $F$ with $b$-property in $E$ all
Lebesgue topologies on $E$ induce the same topology.

######  Example 12.

Let $F$ be an order dense sublattice of a vector lattice $E$. If $F$ is
laterally complete in its own right, then $F$ majorizes $E$ and therefore has
$b$-property in $E$.

We refer to [1, 10] for all undefined terms.

##  2 Main results

######  Lemma 1.

Let $F$ be a sublattice of a LSVL $E(\tau)$. Then each $b$-bounded subset $B$
of $F$ is $\tau$-bounded subset with respect to induced topology on $F$.

###### Proof.

To say that $B$ is $b$-bounded is to say that $B$ is order bounded in $E$. So,
if $U$ is a neighborhood of $0$ in $\tau$ then $B\subseteq\lambda U$ for some
$\lambda>0$. Then $B\subseteq\lambda U\cap F=\lambda(U\cap F)$. ∎

######  Lemma 2.

Let $E$ be a vector lattice and $F$ be an order dense sublattice of $E$. Then
TFAE:

  1. i$)$

$F$ has $b$-property in $E$;

  2. ii$)$

$F$ is majorizing in $E$.

###### Proof.

$i)\Longrightarrow ii)$: Let $0\leq x\in E$ be arbitrary, as $F$ is order
dense in $E$, there exists a net $x_{\alpha}$ in $F$ such that $0\leq
x_{\alpha}\uparrow x$. As $x_{\alpha}$ is $b$-bounded by assumption, there
exists $x_{0}\in F^{+}$ with $0\leq x_{\alpha}\leq x_{0}$ for all $\alpha$, as
$x_{\alpha}\uparrow x$, we have $x\leq x_{0}$ and $F$ is majorizing.

$ii)\Longrightarrow i)$: Let $x_{\alpha}$ be a net in $F$ with $0\leq
x_{\alpha}\uparrow\leq x$ for some $x\in E$. Since $F$ is assumed to be
majorizing $E$, there exists $y\in F$ with $x\leq y$. Consequently, $0\leq
x_{\alpha}\leq y$ and $F$ has $b$-property in $E$. ∎

This yields: $E$ has $b$-property in $E^{u}$ iff $E$ is majorizing in $E^{u}$.
We also have if $E(\tau)$ is a LSVL where $E$ is an ideal of
$\hat{E}(\hat{\tau})$, where $\hat{E}$ is the completion. Then $E$ has
$b$-property in $\hat{E}(\hat{\tau})$.

On the other hand, if $E(\tau)$ is a LSVL with Fatou property, then every
increasing $\tau$-bounded net of $E^{+}$ is order bounded in $E^{u}$ i.e.
every increasing $\tau$-bounded net of $E^{+}$ is $b$-bounded in $E^{u}$ by
[1, Theorem 7.51]

The following property was introduced in [8] and [9].

######  Definition 2.

A locally solid vector lattice $E(\tau)$ is called boundedly order bounded
(BOB) if every $\tau$-bounded net in $E^{+}$ is order bounded in $E$.

We show BOB is equivalent to $b$-property if the LSVL $E(\tau)$ has Fatou
property.

######  Lemma 3.

Let $E(\tau)$ be a LSVL with Fatou property. Then $E$ has $b$-property in
$E^{u}$ iff $E$ is BOB.

###### Proof.

Suppose $E$ is BOB and $x_{\alpha}$ be a net in $E$ with $0\leq
x_{\alpha}\uparrow\leq x_{0}$ for some $x_{0}\in E^{u}$. Then, by Lemma 1,
$x_{\alpha}$ is $\tau$-bounded in $E$ and, by assumption that $E$ is BOB,
$0\leq x_{\alpha}\leq x$ for some $x\in E$.

Conversely, suppose $x_{\alpha}$ is $\tau$-bounded increasing net in $E^{+}$,
then by [1, Theorem 7.50], $x_{\alpha}$ is order bounded in $E^{u}$. Thus by
$b$-property of $E$ in $E^{u}$, there exists $x\in E$ with $0\leq
x_{\alpha}\leq x$ and $E(\tau)$ is BOB. ∎

[1, Theorem 7.49] shows that, in a laterally $\sigma$-complete LSVL $E(\tau)$,
every disjoint sequence in $E^{+}$ converges to zero with respect to any LS
topology on $E$. We show a similar result. The proof is similar.

######  Proposition 1.

Let $E(\tau)$ be a LSVL which has countable $b$-property in its
$\sigma$-lateral completion. Then every disjoint sequence in $E^{+}$ converges
to zero with respect to any locally solid topology on $E$. In particular,
every locally solid topology on $E$ has the pre-Lebesgue property.

###### Proof.

Let $x_{n}$ be a disjoint sequence in $E^{+}$. Then $nx_{n}$ is also a
disjoint sequence in $E^{+}$. Then $x=\bigvee_{n=1}^{\infty}nx_{n}$ exists in
the $\sigma$-lateral completion, and we have $0\leq x_{n}\leq\frac{1}{n}x$ for
all $n$. Countable $b$-property of $E$ in its lateral completion yields a
vector $e\in E$ with $0\leq x_{n}\leq\frac{1}{n}e$ for all $n$. Thus $x_{n}$
converges to zero with respect to any locally solid topology on $E$. ∎

######  Corollary 1.

Let $E(\tau)$ be a LSVL with Lebesgue property. If $E$ has countable
$b$-property in its $\sigma$-lateral completion then the topological
completion $\hat{E}$ of $E(\tau)$ is $E^{u}$.

###### Proof.

It follows from [1, Theorem 7.51]. ∎

######  Proposition 2.

A laterally complete vector lattice $E$ has $b$-property in every vector
lattice which contains $E$ as an order dense sublattice.

###### Proof.

In this case $E$ majorizes the vector lattice that contains it. The result now
follows from [1, Theorem 7.15]. ∎

In [11, Proposition 2.22] it is proved that if $E(\tau)$ is a LSVL with
Lebesgue topology, then a sublattice $F$ of $E$ is $u\tau$-closed in $E$ iff
it is $\tau$-closed. It was asked in [11, Question 2.24] whether Lebesgue
assumption could be removed. The next result yields an answer utilizing
$b$-property.

######  Proposition 3.

Let $F$ be an order ideal of a LSVL $E(\tau)$. If $F$ has $b$-property in $E$,
then $F$ is $u\tau$-closed iff it is $\tau$-closed in $E$.

###### Proof.

As $u\tau$ is coarser than $\tau$, the forward implication is clear.

Now, suppose $F$ is $\tau$-closed and $y_{\alpha}$ is a net in $F$ with
$y_{\alpha}\stackrel{{\scriptstyle u\tau}}{{\to}}x$ for some $x\in E$. We will
show $x\in F$. The lattice operations are $u\tau$-continuous, so that
$y^{\pm}_{\alpha}\stackrel{{\scriptstyle u\tau}}{{\to}}x$. Therefore, WLOG we
may assume $0\leq y_{\alpha}$ for all $\alpha$. Let $z\in E^{+}$ be arbitrary,
then

$|y_{\alpha}\wedge z-x\wedge z|\leq|y_{\alpha}-x|\wedge
z\stackrel{{\scriptstyle\tau}}{{\to}}0.$

Since $0\leq y_{\alpha}\wedge x\leq y_{\alpha}$ for all $\alpha$, and $F$ is
an order ideal, we have $y_{\alpha}\wedge x\in F$ for all $\alpha$ and
$y_{\alpha}\wedge x\stackrel{{\scriptstyle\tau}}{{\to}}x\wedge x$.

Take $y\in F$, then $y_{\alpha}\wedge
y\stackrel{{\scriptstyle\tau}}{{\to}}x\wedge y$, since $F$ is $\tau$-closed we
have $x\wedge y\in F$ for each $y\in F^{+}$. If $z\in F^{d}$, then
$y_{\alpha}\wedge z=0$ for all $\alpha$ and we have $x\wedge z=0$. Thus $x\in
F^{dd}$. That is, $x$ is in the band generated by $F$ in $E$. Hence there
exists a net $z_{\beta}$ in $F^{+}$ such that $0\leq z_{\beta}\uparrow|x|$.
Therefore $z_{\beta}$ is $b$-bounded in $E$, by $b$-property of $F$ in $E$,
$0\leq z_{\beta}\leq x_{0}$ for some $x_{0}\in F$ and $|x|\leq x_{0}$. Hence
$x\in F$ as $F$ is an ideal. ∎

It shown in [1, Theorem 7.39] that a Dedekind complete vector lattice is
universally complete iff it is universally $\sigma$-complete and has a weak
unit. In the next result, we replace universally $\sigma$-completeness with
countable $b$-property of $E$ in $E^{u}$.

######  Proposition 4.

Let $E$ be a Dedekind complete vector lattice with countable $b$-property in
$E^{u}$ and a weak order unit. Then $E=E^{u}$.

###### Proof.

If $E=E^{u}$ then $E$ has $b$-property in $E^{u}$ and has a weak unit. Now we
prove the converse. Let $0<e$ be a weak order unit for $E$. Then $E$ is an
order ideal in $E^{u}$ by [1, Theorem 1.40]. Let $0<u\in E^{u}$ be arbitrary.
Since $e$ is also a weak unit for $E^{u}$ ($E$ is order dense in $E^{u}$), we
have $0<u\wedge ne\uparrow u$. As $u\wedge ne\in E$ for each $n$, we see that
the sequence $u\wedge ne$ is $b$-bounded in $E^{u}$. Therefore the sequence
$u\wedge ne$ has an upper bound in $E$ by assumption. Thus $0\leq u\wedge
ne\leq x$ for some $x\in E$. As $E$ is an order ideal in $E^{u}$, we have
$u\in E$. ∎

It is well known that if $E(\tau)$ is a LSVL with Levi property and
$\tau$-complete order intervals, then $E$ is complete. In the following we
reach to the same conclusion by replacing Levi property with weaker condition
that $E$ having $b$-property in $\hat{E}(\hat{\tau})$.

######  Proposition 5.

Let $E(\tau)$ be a LSVL with $\tau$-complete order intervals. If $E(\tau)$ has
$b$-property in $\hat{E}$, then $E$ is complete.

###### Proof.

The assumption on order intervals implies that $E$ is an order ideal of
$\hat{E}$ by [1, Theorem 2.42]. Let $0<\hat{x}\in\hat{E}$ be arbitrary. Since
$E$ is order dense in $\hat{E}$, there exists a net $x_{\alpha}$ such that
$0\leq x_{\alpha}\uparrow\hat{x}$. By the $b$-property of $E$ in $\hat{E}$, we
can find $x_{0}\in E$ with $0\leq x_{\alpha}\leq x_{0}$. But then since
$x_{\alpha}\uparrow x$, we have $x\leq x_{0}$ and $x\in E$. ∎

######  Proposition 6.

Let $F$ be a sublattice of an order complete vector lattice $E$. Suppose $F$
is order dense and majorizing in $E$. Then each increasing $b$-bounded net in
$F$ is $uo$-Cauchy in $F$.

###### Proof.

Let $x_{\alpha}$ be a $b$-bounded net in $F$ so that $0\leq
x_{\alpha}\uparrow\leq e$ for some $e\in E^{+}$. Since $E$ is order complete,
$x_{\alpha}\uparrow x$ for some $x\in E^{+}$. Then $x_{\alpha}$ is order
convergent in $E$ and hence $o$-Cauchy in $E$, thus $x_{\alpha}$ is
$uo$-Cauchy in $F$ by [7, Theorem 2.3] ∎

It was observed that in [7, Theorem 3.2] for a net $x_{\alpha}$ in a regular
sublattice $F$ of a vector lattice $E$, $x_{\alpha}\stackrel{{\scriptstyle
uo}}{{\to}}0$ in $F$ iff $x_{\alpha}\stackrel{{\scriptstyle uo}}{{\to}}0$ in
$E$. However this may fail for $u\tau$-convergence. $u\tau$-Convergence in a
sublattice may not imply $u\tau$-convergence in the entire space. For example,
the standard unit vectors $e_{n}$ in $l^{\infty}$ is easily seen to be a null
sequence in the unbounded norm topology of $c_{0}$ but not so in $l^{\infty}$.

######  Proposition 7.

Let $F$ be a sublattice of a LSVL $E(\tau)$. Suppose $F$ has $b$-property in
$E$. For a net $x_{\alpha}$ in $F$ for which
$x_{\alpha}\stackrel{{\scriptstyle u\tau}}{{\to}}0$ in $F$, we have
$x_{\alpha}\stackrel{{\scriptstyle u\tau}}{{\to}}0$ in $E(\tau)$.

###### Proof.

Suppose $x_{\alpha}\stackrel{{\scriptstyle u\tau}}{{\to}}0$ in $F$. WLOG we
may suppose $0\leq x_{\alpha}$ for all $\alpha$. Then $0\leq x_{\alpha}\wedge
y\stackrel{{\scriptstyle\tau}}{{\to}}0$ for each $y\in F^{+}$. On the other
hand, for each $x\in E^{+}$, $0\leq x_{\alpha}\wedge x\leq x$ and the net
$0\leq(x_{\alpha}\wedge x)$ is $b$-bounded in $F$, by the hypothesis, there
exists $y\in F^{+}$ such that $0\leq x_{\alpha}\wedge x\leq y$ for all
$\alpha$. Then

$0\leq x_{\alpha}\wedge x\leq x_{\alpha}\wedge
y\stackrel{{\scriptstyle\tau}}{{\to}}0$

from which we obtain, $x_{\alpha}\wedge
x\stackrel{{\scriptstyle\tau}}{{\to}}0$. As $x$ is arbitrary
$x_{\alpha}\stackrel{{\scriptstyle u\tau}}{{\to}}0$ in $E(\tau)$. ∎

######  Proposition 8.

Let $E(\tau)$ be a laterally complete vector lattice, then $E$ has
$b$-property in $(E^{\sim})^{\sim}_{n}$.

###### Proof.

Recall that $E$ is order dense in $(E^{\sim})^{\sim}_{n}$. Then $E$ is
majorizing in $(E^{\sim})^{\sim}_{n}$ by [1, Theorem 7.15]. Therefore $E$ has
$b$-property in $(E^{\sim})^{\sim}_{n}$ ∎

######  Proposition 9.

Let $E(\tau)$ be a LSVL with Lebesgue property. Then every order closed
sublattice $F$ of $E(\tau)$ has countable $b$-property in
$\hat{E}(\hat{\tau})$.

###### Proof.

Let $x_{n}$ be a $b$-bounded sequence in $E$. Then there exists
$\hat{x}\in\hat{E}$ with $0\leq x_{n}\uparrow\hat{x}$. Since $E(\tau)$ is
assumed to have Lebesgue property, it has the $\sigma$-Lebesgue property as
well as the Fatou property by [1, Theorem 4.8]. Since the topology
$\hat{\tau}$ of $\hat{E}$ is also Lebesgue, the sequence $x_{n}$ is
$\hat{\tau}$-Cauchy in $\hat{E}$. Then
$x_{n}\stackrel{{\scriptstyle\hat{\tau}}}{{\to}}x$ for some $x\in\hat{E}$.
Since $\tau$ is Fatou and $F$ being order closed is $\tau$-closed by [1,
Theorem 4.20]. Thus $x\in F$. As $x_{n}\uparrow$,
$x_{n}\stackrel{{\scriptstyle\tau}}{{\to}}x$, hence $x=\sup x_{n}$, and $F$
has $b$-property in $\hat{E}$. ∎

######  Proposition 10.

Let $F$ be a $uo$-closed sublattice of a Dedekind complete vector lattice $E$.
Then $F$ has $b$-property in $E$.

###### Proof.

Let $x_{\alpha}$ be a net in $F$ with $0\leq x_{\alpha}\uparrow x$ for some
$x\in E$. As $E$ is Dedekind complete, $x_{\alpha}\uparrow\hat{x}$ for some
$\hat{x}\in E$. Then $x_{\alpha}\stackrel{{\scriptstyle o}}{{\to}}\hat{x}$,
consequently $x_{\alpha}\stackrel{{\scriptstyle uo}}{{\to}}\hat{x}$ in $E$ as
$F$ is $uo$-complete, $\hat{x}\in F$. ∎

######  Proposition 11.

Let $E$ be a vector lattice admitting a minimal topology $\tau$. Let $x_{n}$
be a $b$-bounded sequence in $E^{u}$. Then $x_{n}$ is $\tau$-Cauchy in $E$.

###### Proof.

Let $x_{n}$ be such that $0\leq x_{n}\uparrow x^{u}$ for some $x^{u}\in
E^{u}$. Since $E^{u}$ is Dedekind complete, $x_{n}$ being order bounded in
$E^{u}$, has a supremum in $E^{u}$, let it be $x$. Therefore
$x_{n}\stackrel{{\scriptstyle o}}{{\to}}x$, it follows that $x_{n}$ is
$uo$-Cauchy in $E^{u}$. Since $E$ is order dense in $E^{u}$, and order dense
sublattices are regular, $E$ is regular in $E^{u}$ and by [7, Theorem 3.2],
$x_{n}$ is $uo$-Cauchy in $E$. As every minimal topology is Lebesgue, $\tau$
is Lebesgue and $x_{n}$ is $u\tau$-Cauchy. As $\tau$ is unbounded, it follows
that $x_{n}$ is $\tau$-Cauchy on $E$. ∎

######  Definition 3.

A locally solid vector lattice $E(\tau)$ is called boundedly $uo$-complete if
every $\tau$-bounded $uo$-Cauchy net in $E$ is $uo$-convergent.

######  Proposition 12.

A boundedly $uo$-complete LSVL $E(\tau)$ has $b$-property in $E^{u}$.

###### Proof.

Let $0\leq x_{\alpha}\uparrow x^{u}$, where $x^{u}\in E^{u}$, be a net in $E$.
As $x_{\alpha}$ is a $b$-bounded subset of $E$, it is $\tau$-bounded by Lemma
1. We show $x_{\alpha}$ has an upper bound in $E$. As $E^{u}$ is Dedekind
complete, $\sup x_{\alpha}$ exists in $E^{u}$. Let this supremum be $x$. Then
$0\leq x_{\alpha}\uparrow x$ in $E^{u}$. Thus
$x_{\alpha}\stackrel{{\scriptstyle o}}{{\to}}x$. It follows that $x_{\alpha}$
is $uo$-Cauchy in $E$ as $E$ is order dense and a regular sublattice of
$E^{u}$. Thus $x_{\alpha}$ being $uo$-Cauchy and $\tau$-bounded converges to
some $x^{\prime}\in E$. But as $x_{\alpha}\stackrel{{\scriptstyle o}}{{\to}}x$
we must have $x=x^{\prime}$ ∎

######  Definition 4.

A Banach lattice is monotonically complete (the Levy property) if every norm
bounded increasing net in $E^{+}$ has supremum.

We now show that every boundedly $uo$-complete Banach lattice $E$ has
$b$-property in $(E^{\sim}_{n})^{\sim}_{n}$. The proof uses an idea of [6] in
that $(E^{\sim}_{n})^{\sim}_{n}$ is monotonically complete and the canonical
map $J:E\to(E^{\sim}_{n})^{\sim}_{n}$ maps a bounded increasing net in $E^{+}$
to a net in $(E^{\sim}_{n})^{\sim}_{n}$ with similar properties.

######  Proposition 13.

Let $E$ be a boundedly $uo$-complete Banach lattice with $E^{\sim}_{n}$
separating points of $E$. If $0\leq x_{\alpha}\uparrow$ is an increasing net
in $E^{+}$ which is order bounded in $(E^{\sim}_{n})^{\sim}_{n}$, then
$x_{\alpha}$ has an upper bound in $E$.

###### Proof.

Since the net $x_{\alpha}$ is order bounded in $(E^{\sim}_{n})^{\sim}_{n}$, it
is norm bounded in $(E^{\sim}_{n})^{\sim}_{n}$ and hence norm bounded in $E$
by Lemma 1.

Let $J:E\to(E^{\sim}_{n})^{\sim}_{n}$ be the natural embedding, where
$J(x)(f)=f(x)$ for each $x\in E$ and $f\in E^{\sim}_{n}$. The map $J$ is a
vector lattice isomorphism and the range $J(E)$ in $(E^{\sim}_{n})^{\sim}_{n}$
is order dense in $(E^{\sim}_{n})^{\sim}_{n}$ by [1, Theorem 1.43]. Therefore,
$J(E)$ is a regular sublattice of $(E^{\sim}_{n})^{\sim}_{n}$.

By [10, 2.4.19], $(E^{\sim}_{n})^{\sim}_{n}$ is a monotonically complete
Banach lattice. Thus, the increasing net $J(x_{\alpha})$ has a supremum in
$(E^{\sim}_{n})^{\sim}_{n}$ say $x$.

So $J(x_{\alpha})\uparrow x$ and $J(x_{\alpha})$ is order Cauchy in
$(E^{\sim}_{n})^{\sim}_{n}$. It follows that $J(x_{\alpha})$ is $uo$-Cauchy in
$(E^{\sim}_{n})^{\sim}_{n}$ and in the regular sublattice $J(E)$. As $J$ is
$1$-$1$ and onto $J(E)$ is lattice isomorphism, $x_{\alpha}$ is $uo$-Cauchy in
$E$. $E$ is boundedly $uo$-complete, $x_{\alpha}\stackrel{{\scriptstyle
uo}}{{\to}}x_{1}$ for some $x_{1}\in E$. On the other hand $0\leq
x_{\alpha}\uparrow$, thus $x_{\alpha}\uparrow x_{1}$ and $x_{\alpha}$ is order
bounded in $E$. ∎

## References

  * [1] Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, 2nd edition. American Mathematical Society, Providence, RI (2003)
  * [2] Alpay, Ş., Altın, B., Tonyali, C.: On property $b$ of vector lattices, Positivity, 7, 135-139 (2003),
  * [3] Alpay, Ş., Altın, B., Tonyali, C.: A note on Riesz spaces with property $b$, Czechoslovak Math. J. 56 (131), 765-772 (2006)
  * [4] Alpay, Ş., Emelyanov, E., Gorokhova, S.: Bibounded $uo$-convergence and $bbuo$-duals of vector lattices, https://arxiv.org/abs/2009.07401v1 (2020)
  * [5] Alpay, Ş., Ercan, Z.: Characterizations of Riesz spaces with $b$-property, Positivity, 13 no. 1, 21-30 (2009)
  * [6] Gao, N., Leung, D., Xanthos, F.: Dual representation of risk measures on Orlicz spaces. Preprint: arXiv (2018)
  * [7] Gao, N., Troitsky, V., Xanthos, F.: $Uo$-convergence and its applications to Cesáro means in Banach lattices. Isr. J. Math. 220, 649-689 (2017)
  * [8] Labuda, I.: Completeness type properties of locally solid Riesz spaces, Studia Math. 77, 349-372 (1984)
  * [9] Labuda, I.: On boundedly order-complete locally solid Riesz spaces, Studia Math. 81, 245-258 (1985)
  * [10] Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer-Verlag, Berlin (1991)
  * [11] Taylor, M.A.: Unbounded Convergences in Vector Lattices, Master’s thesis, University of Alberta, (2018)


# LLM-Oracle Machines

Jie Wang Richard Miner School of Computer and Information Sciences, University
of Massachusetts, Lowell, MA 01854, USA. Copyright ©Jie Wang, 2024.

###### Abstract

Contemporary AI applications leverage large language models (LLMs) to harness
their knowledge and reasoning abilities for natural language processing tasks.
This approach shares similarities with the concept of oracle Turing machines
(OTMs). To capture the broader potential of these computations, including
those not yet realized, we propose an extension to OTMs: the LLM-oracle
machine (LLM-OM), by employing a cluster of LLMs as the oracle. Each LLM acts
as a black box, capable of answering queries within its expertise, albeit with
a delay. We introduce four variants of the LLM-OM: basic, augmented, fault-
avoidance, and $\epsilon$-fault. The first two are commonly observed in
existing AI applications. The latter two are specifically designed to address
the challenges of LLM hallucinations, biases, and inconsistencies, aiming to
ensure reliable outcomes.

##  1 Introduction

In an oracle Turing machine, the oracle embodies a decision problem. It acts
as a hypothetical, all-powerful entity that can instantly determine whether a
query, an instance generated during the computation, is a positive instance of
the decision problem. OTMs have played a significant role in the development
of both computation theory and computational complexity theory.

Drawing inspiration from the concept of using external knowledge to assist
with computing tasks, and motivated by the recent advancements in LLMs with
their powerful knowledge bases and inference capabilities, we use a cluster of
LLMs in place of the oracle in OTMs.

We treat LLMs as black boxes, capable of answering queries within their
domains of expertise, albeit with a delay111The assumption of delay may be
omitted if we are focusing on generating trustworthy results.. Both queries
and responses are exchanged in natural language, encompassing formatting.
Since each LLM is trained on diverse datasets using different technologies,
their capabilities vary. Therefore, we use a cluster of LLMs as the oracle,
selecting an appropriate LLM to respond to a query during an LLM-OM
computation.

Unlike in an OTM, where the oracle represents a decision problem and a query
asks whether an instance is positive, a query generated in the computation of
an LLM-OM consists of a task and a sequence of specifications, with the
response being a solution to complete the task.

In a nutshell, the computation of an LLM-OM takes an input representing a task
to accomplish, generates queries, acquires answers to each query from the
appropriate LLM in the LLM-oracle, and continues this process until the LLM-OM
reaches a halting state with the final answer.

Unlike in an OTM where the oracle reliably provides an answer to a query, LLMs
can generate fabricated or misleading information, resulting in incorrect or
inadequate answers (e.g., see [1, 2]). LLMs may also provide answers with
different meanings to the same query at different times. These issues of
information hallucination (or better phrased as “information nonsense” because
LLMs cannot distinguish between truth and lies), inadequacy, and inconsistency
are common in LLMs.

While advancing technologies aim to mitigate these issues, complete
elimination remains challenging. Therefore, we assume that there exists a
probability that an LLM may provide an unacceptable answer to a query.
Depending on the context of the application, an unacceptable answer could be
one that is outright incorrect or one that, while not incorrect, fails to meet
the required level of adequacy.

##  2 LLM-OM Basics and Variants

An LLM-OM can be viewed as a deterministic algorithm with access to an LLM
oracle. Similar to an OTM, each computation step in an LLM-OM represents a
transition. We denote an LLM-OM as $M$, the input representing the task
inquiry (including optional specifications) as $Q$, and the output as the
answer $A$ to $Q$. This output $A$ can take various forms, including human-
like text, code snippets, or other representations. Both inputs $Q$ and
outputs $A$ can be encoded as binary strings for compatibility with
traditional computing models.

During the computation, $M$ generates a query in the form of
$(x;y_{1},y_{2},\ldots,y_{k})$, denoted as $q_{x}$, where $x$ is an
intermediate task for completing $Q$ and each $y_{i}$ is an attribute. These
attributes can be

  * •

Selected text to provide relevant context for $x$.

  * •

Specific requirement to detail the desired outcome of $x$.

  * •

Solution format to instruct $M$ on how to express the answer (e.g., text,
code).

  * •

Verification method to specify how to validate the answer’s correctness.

  * •

Self-critique instruction to guide $M$ to evaluate its own response.

Other specifications with any additional details relevant to completing $x$
may also be attributes. Collectively, these attributes form a prompt that
instructs the LLMs on how to approach the intermediate task $x$ effectively.

The computation begins with $M$ interpreting the input task $Q$. If possible,
it decomposes $Q$ into a sequence of smaller, more manageable subtasks,
denoted as $Q_{1},Q_{2},\ldots,Q_{m}$. However, if $Q$ cannot be further
broken down, it remains the sole subtask.

###  2.1 Basic variants: adaptive and non-adaptive

The basic variant of the LLM-OM utilizes two query types: adaptive and non-
adaptive.

In an adaptive LLM-OM: subtasks are interdependent. This means some subtasks
require answers from previous subtasks before generating all queries. For a
subtask $Q_{i}$, the LLM-OM generates a query $q_{i,x}$ specifying the task
within the query. It then retrieves an answer $a_{x}$ from a chosen LLM in the
oracle and uses $a_{x}$ to determine the next step. The final answer is
derived by combining answers from subtasks, potentially involving the LLM-
oracle again.

In a non-adaptive LLM-OM, subtasks are independent. Each subtask $Q_{i}$ is
completed by sending a set of independent queries generated during the
computation to the LLM-oracle. The final answer is produced solely from these
answers, without further interaction with the LLM-oracle.

Ideally, the final answer $A$ should directly address the original inquiry $Q$
and remain relevant to the topic. Let $\text{toc}(X)$ denote the collection of
topics within text $X$.

  * •

We say that $A$ is relevant to $Q$, denoted by $A\lesssim Q$, if
$\mbox{toc}(Q)\subseteq\mbox{toc}(A)$.

Remark 1. The basic functionality of many LLM web applications, like free
versions of ChatGPT and Gemini, exemplifies the non-adaptive LLM-OM variant.
In these applications, users submit queries, which directly translate to the
subtasks in the LLM-OM. The system then independently generates a set of
queries based on the user’s input and sends them to the underlying LLM-oracle.
Finally, the application presents the user with the final answer derived
solely from these responses, without further interaction with the LLM.

###  2.2 Augmented LLM-OM

The augmented LLM-OM takes a pair $(T,Q)$ as input. Here, $T$ represents an
“augmented text” in natural language, acting as verified background knowledge
or ground truth. $Q$ remains the task inquiry, specifying what information to
extract or infer from $T$. This information can include specific sentences,
topics, summaries, entities, relationships between entities, events, event
relationships, logical consequences, or numerical consequences. Similar to the
basic variant, queries generated during the computation can be either adaptive
or non-adaptive.

Ideally, the answer $A$ to the inquiry $Q$ should align with the augmented
text $T$. Here’s a formal definition of this alignment. Let $\mbox{mng}(X)$
denote the set of meanings of text $X$.

  * •

We say that $A$ complies with $T$ with respect to $Q$, written as
$A\lesssim_{Q}T$, if the following conditions hold:

    1. 1.

Relevance: $A$ is relevant to $Q$. Namely, $Q\lesssim A$.

    2. 2.

Inclusiveness: $A$ is inclusive in $T$. Namely,
$\mbox{mng}(A)\subseteq\mbox{mng}(T)$.

Note that even if two texts $X$ and $Y$ have the same set of meanings (i.e.
$\mbox{mng}(X)=\mbox{mng}(Y)$), it doesn’t necessarily mean they are
identical. There are many ways to express the same idea with different
wording.

To complete the task $x$ represented by query $q_{x}$, the LLM-OM $M$ first
identifies relevant content, denoted as $C_{x}$, within the augmented text
$T$. It then leverages this content $C_{x}$ along with an appropriate LLM from
the LLM-oracle to generate an answer to $x$, denoted as $a_{x}$, ideally
complying with $C_{x}$ with respect to $q_{x}$, namely,
$a_{x}\lesssim_{q_{x}}C_{x}$.

Remark 2. Web applications like ChatGPT 4o that employ Retrieval-Augmented
Generation (RAG) techniques with an LLM can be seen as a practical application
of the augmented LLM-OM framework. In RAG-based applications, the retrieved
information acts as the augmented text that confines the LLM’s response. This
retrieved information helps the LLM generate answers that are more grounded in
factual evidence and more likely to comply with the user’s query.

###  2.3 Fault-avoidance LLM-OM

The basic and augmented LLM-OM variants do not inherently guarantee
consistency, correctness, or adequacy in the final answer $A$ for several
reasons:

  1. 1.

Limitations of LLMs: Even the most advanced LLMs can be susceptible to biases,
factual inaccuracies, and hallucinations in their responses. These issues can
directly translate into inconsistencies or errors in the final answer
generated by the LLM-OM.

  2. 2.

Incomplete or unclear input: If the initial user inquiry $Q$ or the augmented
text $T$ in the augmented variant is ambiguous, incomplete, or misleading, it
can lead the LLM-OM down an incorrect path and ultimately result in an
inadequate or incorrect answer.

  3. 3.

Dependence on LLM selection: The choice of LLM from the LLM-oracle can also
influence the outcome. Different LLMs have varying strengths and weaknesses,
and an inappropriate selection might hinder the generation of a consistent or
accurate answer.

  4. 4.

Query design challenges: Crafting effective intermediate queries $q_{x}$ is
crucial. Poorly designed queries can lead the LLM-OM to misunderstand the
intent or miss key aspects of the overall task, compromising the final
answer’s adequacy.

For the purpose of illustration, we assume that both the user inquiry $Q$ and
the augmented text $T$ in the augmented variant are well-defined and free from
ambiguity, incompleteness, and misleading information. This idealized scenario
allows us to focus on the limitations inherent to the LLM-oracle itself,
independent of potential issues with the input data. We define these terms
(consistency, correctness, and adequacy) as follows:

  * •

$M$ is consistent if, for any user inquiries $Q$ and $Q^{\prime}$ posed at
different points in time, if $\mbox{mng}(Q^{\prime})=\mbox{mng}(Q)$, then $M$
outputs an answer $A$ to $Q$ and an answer $A^{\prime}$ to $Q^{\prime}$ such
that $\mbox{mng}(A)=\mbox{mng}(A^{\prime})$.

  * •

Suppose $M$ is augmented with input and output being $(T,Q)$ and $A$. We say
that $A$ is correct for $Q$ with respect to $T$ if the following conditions
hold:

    1. 1.

Compliance: $A$ complies with the ground truth $T$ with respect to $Q$.
Namely, $A\lesssim_{Q}T$.

    2. 2.

Exclusiveness: $A$ doesn’t include unnecessary information outside the scope
of $Q$. In other words, if we let $\mbox{intent}(Q)$ denote the set of
intended meanings of $Q$, then $\mbox{mng}(A)\subseteq\mbox{intent}(Q)$.

    3. 3.

Completeness: $A$ doesn’t miss any information contained in $T$ that are
relevant to $Q$. In other words,
$(\mbox{mng}(T)-\mbox{mng}(A))\cap\mbox{intent}(Q)=\emptyset$.

    4. 4.

Distribution agreement: If $A$ has more than one meaning, then
$\mbox{dist}(A)=\mbox{dist}(T_{A})$, where $\mbox{dist}(A)$ denotes the
distribution of meanings within answer $A$ and $\mbox{dist}(T_{A})$ denotes
the distribution of meanings in the corresponding text for $A$ in the
augmented text $T$.

We say that $M$ is correct with respect to $T$ if for any input $(T,Q)$, $M$
returns an answer $A$ that is correct for $Q$ with respect to $T$.

  * •

Suppose $M$ is an augmented LLM-OM with input $(T,Q)$ that generates an answer
$A$. We say that answer $A$ is adequate for $Q$ with respect to $T$ if $A$
complies with $T$ with respect to $Q$. However, unlike correctness, adequacy
allows for some flexibility in the answer:

    1. 1.

Non-exclusiveness: $A$ may contain information that extend beyond the specific
information requested in the user inquiry $Q$. Namely,
$\mbox{mng}(A)-\mbox{intent}(Q)\not=\emptyset$.

    2. 2.

Incompleteness: $A$ may miss information from $T$ that pertains to $Q$.
Namely, $(\mbox{mng}(T)-\mbox{mng}(A))\cap\mbox{intent}(Q)\not=\emptyset$.

    3. 3.

Distribution discrepancy: $\mbox{dist}(A)\not=\mbox{dist}(T_{A})$.

Essentially, adequacy acknowledges that the answer $A$ may not be perfect, but
it provides valuable information to the query from the augmented text.

We say that $M$ is adequate if for all inputs $(T,Q)$, $M$ returns an answer
$A$ that is adequate for $Q$ with respect to $T$.

  * •

If $M$ is not augmented, assume the existence of a set of texts, denoted by
$U$, that represents the true knowledge and information for the areas of
interest. We say that $M$ is correct if $M$ is correct with respect to $U$,
$M$ is adequate if $M$ is adequate with respect to $U$. Set $U$ is referred to
as the absolute truth.

Note that if $A$ is correct for $Q$ with respect to $T$, then $A$ is adequate
for $Q$ with respect to $T$, but not vice versa.

A fault-avoidance LLM-OM is an augmented LLM-OM that is consistent and
correct. A weak fault-avoidance LLM-OM is an augmented LLM-OM that is
consistent and adequate. It is necessary for $M$ to identify the best-matched
content $C_{x}$ from $T$ for each query $q_{x}$, and the chosen LLM from the
LLM-oracle must comply with $C_{x}$ when generating an answer $a_{x}$ to
$q_{x}$.

Verifying the correctness or adequacy of the answer $A$ for a given query $Q$
with respect to $T$ calls for concrete implementations of $\mbox{toc}(X)$,
$\mbox{mng}(X)$, $\mbox{intent}(Q)$, and $\mbox{dist}(X)$ using techniques in
natural language processing, machine learning, deep learning, and other
methods.

Consistence, however, is harder to verify. We may aim to develop a method that
provides a certain guarantee that $M$ is consistent with a desired high
probability.

###  2.4 $\epsilon$-fault LLM-OM

An $\epsilon$-fault LLM-OM is a non-augmented LLM-OM that is consistent and
correct with probability of $1-\epsilon$ with respect to the absolute truth
for the areas of interest, where $\epsilon$ is some small positive parameter.
Likewise, a weak $\epsilon$-fault LLM-OM is similarly defined by replacing
correctness with adequacy.

Let $L_{1},L_{2},\ldots,L_{k}$ be the LLMs in the LLM-oracle, where each
$L_{i}$ has a small probability $p_{i}$ of generating hallucinated answers to
queries. Different LLMs may hallucinate on different queries. However, $M$
doesn’t know which LLM will hallucinate an answer to a given query, and $M$
alone cannot verify if an answer is incorrect, as the absolute truth is not
provided as an augmented input.

We aim to investigate whether it is possible to utilize these LLMs to obtain
an answer to a query with a certain level of guarantee that the answer is
correct or adequate with a desired probability $1-\epsilon$ for some
$\epsilon$.

Making certain reasonable assumptions may be useful in the quest to obtain
such a result. For example, we may assume that for a given query, there is
always an $L_{i}$ that produces a correct and adequate answer; we just don’t
know which one.

## References

  * [1] J. Li, J. Chen, R. Ren, X. Cheng, W. X. Zhao, J.-Y. Nie, and J.-R. Wen, “The dawn after the dark: An empirical study on factuality hallucination in large language models,” 2024. arXiv 2401.03205.
  * [2] R. Stureborg, D. Alikaniotis, and Y. Suhara, “Large language models are inconsistent and biased evaluators,” 2024. arXiv 2405.01724.


# Poster: No safety in numbers: traffic analysis of sealed-sender groups in
Signal
††thanks: Supported by NSF grant 1814753

Eric Brigham  Department of Computer Science
University of Minnesota
Minneapolis, United States
<EMAIL_ADDRESS>Nicholas Hopper  Department of Computer Science
University of Minnesota
Minneapolis, United States
<EMAIL_ADDRESS>
###### Index Terms:

Signal, end-to-end encrypted messaging applications, statistical disclosure
attacks

##  I Introduction

Secure messaging applications often offer privacy to users by protecting their
messages from would be observers through end-to-end encryption techniques [8]
[9] [11]. However, the metadata of who communicates with whom cannot be
concealed by encryption alone. Signal’s Sealed Sender mechanism attempts to
enhance its protection of this data by obfuscating the sender of any message
sent with the protocol [5]. However, it was shown by Martiny et al. [6] that
due to the message delivery protocols in Signal, the record of who receives
messages can be enough to recover this metadata. In this work we extend the
attack presented in [6] from deanonymizing communicating pairs to
deanonymizing entire group conversations.

##  II Background

Statistical disclosure attacks, or SDA, are a known attack vector which can be
used to link senders and receivers of messages in anonymous mix networks [3]
[4] [7] [10]. The traditional attack methods rely on two key components that
seemingly make them ineffective in an environment like Signal: identities of
senders are known and there is a mix entity present [1]. Under Signal’s Sealed
Sender mechanism, the identity of a message’s sender would be hidden, and
there is no mix present. The authors of [6] argue that in the presence of
immediate responses, both of these apparent shortcomings can be overcome and a
modified SDA attack can be developed and leveled against the users by the
server in order to deanonymize communicating pairs. Crucially, they observe
that immediate responses are guaranteed in Signal due to the fact that
delivered receipts cannot be disabled by the user. The authors show both
convincing theoretical analysis as well as simulations that the attack does
achieve its goal. In this work we extend their proposed attack from the
setting of communicating pairs to that of groups, where there are three or
more users communicating through a single channel.

##  III The Attack

We consider an attack setting where a single user Bob is being targeted in the
hope of discovering his group $G$ of $k$ associates who are communicating
together through a single channel. In Signal, when Bob sends a message through
a group channel, within a short period of time which we refer to as the epoch,
all members of the group can be seen to have received a message, and shortly
thereafter, Bob will receive delivered receipts from all members; we refer to
this sequence of delivered receipts as the flurry. Contrasting this with a
random epoch of the same length where Bob has not sent a group message, the
group members may or may not be receiving messages, but Bob will not receive a
quick succession of messages. We observe then that by monitoring who receives
messages most often before a flurry, we can identify the likely members of the
group.

This behavior was confirmed by examining output logs generated by group
messaging with the Android emulator Genymotion similarly to [2]. While the
proposed attack generalizes to a group of any size, we use a group of three
members for simplicity of exposition. We assume that all members of the group
are online at the time of the attack, otherwise delivered receipts cannot be
sent. Similarly to [6] we define random and target epochs, but the foundation
of our attack is the flurry. The following definition/terms will be used
throughout:

  * •

Flurry a string of ‘To Bob’ messages which appear to the server as being sent
in succession, or very close to it. This occurs after Bob sent a message
through the group channel, after which we would necessarily observe delivered
receipts to Bob from all members of the group. Please refer to Fig. 1 for a
visual representation.

  * •

Attack Window : the time-frame in which the attack takes place, necessarily
spanning multiple messages sent to and from Bob

  * •

Target Epoch: an epoch during the attack window preceding a flurry of “To Bob”
delivered receipts. This contains some number of messages sent by Bob using
sealed sender with all recipients(group members) observable.

  * •

Random Epoch: an epoch taken from the total set of messages, chosen uniformly
at random independent from Bob’s activities.

Figure 1: The left box corresponds to a target epoch due to the existence of
the flurry, while the right box could correspond to some random epoch

We define our attack on the assumption that a flurry exists. The attack
follows the basic functioning presented in [6], but is modified for a group
setting and predicated on the existence of a flurry. It is as follows:

  1. 1.

Create an empty table of counts, initializing all values to zero.

  2. 2.

Sample a target epoch. For each user that received a message during the target
epoch, increment their count in the table.

  3. 3.

Sample a random epoch. For each user that received a message during the random
epoch, decrement their count in the table.

  4. 4.

Repeat steps two and three for some number $n$ target and random epochs (same
amount each).

  5. 5.

The users in the table with the highest counts are most likely to be in a
group together. Note that because a flurry is always produced when Bob sends
to the group, but is unlikely to occur in pairwise communication patterns, the
process will produce the highest scores for group members communicating with
Bob.

Note it possible to define an attack on the absence of a flurry, for instance
if Bob were a passive observer of the group, but this is left to a full
version of the paper.

##  IV Theoretical Analysis

The subsequent analysis will rely on the following assumptions, of which one,
two, and four are identical to those found in [6], the third and fifth are
unique to this setting:

  1. 1.

The probability of receiving a message during a particular epoch is
independent of receiving a message during any other epoch

  2. 2.

Each user has a fixed probability of receiving a message during a random
epoch, call it $r_{u}$

  3. 3.

Any group member $u$ has a fixed probability $t_{u}$ of receiving a message
during a target epoch, with $t_{u}>r_{u}$

  4. 4.

All users not in the group have the same probability of receiving a message
during a target or random epoch, $t_{u}=r_{u}$

  5. 5.

Bob is in one group (Note that in this case, assuming no false positive
flurries, we have $t_{u}=1$ for all $u\in G$.)

In this section we present theoretical analysis of attack success. It follows
much in the same way as the analysis presented in [6], but for a group
setting.
Theorem: Given $m$ total users in a messaging system. Let each group member
$u\in G$ have probabilities $r_{u},t_{u}$ of appearing in random or target
epochs respectively. Then under the stated probability assumptions, the
probability that all $k$ group members are ranked higher than all non-
associates after $n$ random and target epochs is at least:

$1-\frac{m|G|}{C^{n}}$

Where the parameter $C=\min_{u\in G}exp((t_{u}-r_{u})^{2}/4)>1$, depend solely
on group member probabilities $r_{u},t_{u}$. The proof of this theorem is left
to a full version of the paper.
The consequences of this are similar to those found in [6], chief among them
being that the number of epochs needed to de-anonymize group members with high
probability scales logarithmically with the total number of users – meaning
that the attack is efficient.

##  V Future work

While all that has been included here are theoretical results, simulations are
in progress and will be included for a full paper. We also hope to explore the
dynamics of the target user being in multiple groups simultaneously. Finally,
we hope to develop and test defense mechanisms that do not require Signal to
modify their own implementation.

## References

  * [1] David L Chaum.  Untraceable electronic mail, return addresses, and digital pseudonyms.  Communications of the ACM, 24(2):84–90, 1981.
  * [2] Cas Cremers, Charlie Jacomme, and Aurora Naska.  Formal analysis of session-handling in secure messaging: Lifting security from sessions to conversations.  In Usenix Security, 2023.
  * [3] George Danezis.  Statistical disclosure attacks: Traffic confirmation in open environments.  In Security and Privacy in the Age of Uncertainty: IFIP TC11 18 th International Conference on Information Security (SEC2003) May 26–28, 2003, Athens, Greece 18, pages 421–426. Springer, 2003.
  * [4] George Danezis, Claudia Diaz, and Carmela Troncoso.  Two-sided statistical disclosure attack.  In Privacy Enhancing Technologies: 7th International Symposium, PET 2007 Ottawa, Canada, June 20-22, 2007 Revised Selected Papers 7, pages 30–44. Springer, 2007.
  * [5] J. Lund.  Technology preview: Sealed sender for signal.  Signal Blog: https://signal.org/blog/sealed-sender/https://signal.org/blog/sealed-sender/, 2018\.
  * [6] Ian Martiny, Gabriel Kaptchuk, Adam J Aviv, Daniel S Roche, and Eric Wustrow.  Improving signal’s sealed sender.  In NDSS, 2021.
  * [7] Nick Mathewson and Roger Dingledine.  Practical traffic analysis: Extending and resisting statistical disclosure.  In Privacy Enhancing Technologies, volume 3424, pages 17–34. Springer, 2004.
  * [8] Trevor Perrin Moxie Marlinspike.  The x3dh key agreement protocol.  Signal Blog: https://signal.org/docs/specifications/x3dh/, 2016\.
  * [9] Trevor Perrin Moxie Marlinspike.  The sesame algorithm: Session management for asynchronous message encryption.  Signal Blog: https://signal.org/docs/specifications/sesame/, 2017\.
  * [10] Carmela Troncoso, Benedikt Gierlichs, Bart Preneel, and Ingrid Verbauwhede.  Perfect matching disclosure attacks.  In Privacy Enhancing Technologies: 8th International Symposium, PETS 2008 Leuven, Belgium, July 23-25, 2008 Proceedings 8, pages 2–23. Springer, 2008.
  * [11] Nik Unger, Sergej Dechand, Joseph Bonneau, Sascha Fahl, Henning Perl, Ian Goldberg, and Matthew Smith.  Sok: secure messaging.  In 2015 IEEE Symposium on Security and Privacy, pages 232–249. IEEE, 2015.