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\begin{document}

\hyphenation{Ka-pi-tul-nik}


\title{New experimental constraints on~non-Newtonian forces below $100$~$\mu$m}
\author{J.~Chiaverini, S.~J.~Smullin, A.~A.~Geraci, D.~M.~Weld, and  A.~Kapitulnik}
\affiliation{Departments of Applied Physics and of Physics, Stanford
University, Stanford, CA 94305}


\date{\today}


\begin{abstract}

We have searched for large deviations from Newtonian gravity by means
of a microcantilever-based Cavendish-style experiment.  Our data
eliminate from consideration mechanisms of deviation that posit
strengths $\sim\nobreak10^4$ times Newtonian gravity at length scales
of $20$~$\mu$m. This measurement is three orders of magnitude more
sensitive than others that provide constraints on this type of force
at similar length scales.

\end{abstract}

\pacs{04.80.Cc}
\maketitle

Physics beyond the standard model will be explored at the Large Hadron
Collider in a few years, but gravitational measurements at small length
scales are capable of investigating this region now. These
measurements become particularly important in light of recent
theoretical developments.  Moduli, massive scalar particles that could
mediate forces with strengths possibly $\sim 10^6$ times Newtonian
gravity, could have Compton wavelengths in the regime of
$10$~$\mu$m--$1$~mm. These particles would manifest themselves as
strong gravity-like forces. The hierarchy problem, the seeming
disparity ($16$~orders of magnitude) between the standard model energy
scale and the Planck scale, with nothing between these extremes,
suggests other reasons to study this regime. This problem can be
recast by asking why gravity is so weak compared to the other known
forces in nature. Recent theoretical work~\cite{add} suggests that
extra spatial dimensions, possibly as large as $1$~mm and accessible
only to gravitons, may supply a solution. Because of spreading into
the extra dimensions, gravity would be diluted compared to the other
forces, and therefore would seem quite weak. As gravity has not been
well tested below the centimeter scale until
recently~\cite{adel,review}, there are many reasons to investigate
gravity-like forces at length scales of $\lesssim 100$~$\mu$m.


Exotic massive scalar particles would modify the standard Newtonian
potential with an additional Yukawa-type term~\cite{scalar}. A similar
correction would exist for observation of gravity at length scales
roughly the size of any compactified extra dimensions~\cite{add}.  This
leads to the following equation for the gravitational potential in the
presence of such non-Newtonian effects:

\begin{equation}
\label{potential} V=- G {m_1 m_2 \over r}(1 + \alpha e^{-{r \over
\lambda}})
\end{equation}

\noindent Here $G$ is Newton's constant, $m_1$ and $m_2$ are the
masses, $r$ is the center of mass separation, $\alpha$ is the strength
(relative to Newtonian) of any new effect with a length scale of
$\lambda$.  Non-Newtonian effects are typically parameterized in terms
of $\alpha$~and $\lambda$.

In this letter we present data obtained for attracting-mass surface
separations down to $\sim\nobreak25$~$\mu$m, enabling investigation of
interaction scales below $\sim\nobreak10$~$\mu$m.


The experiment is similar to that of Cavendish~\cite{cavendish}, though
miniaturized in order that smaller mass separations may be attained.
In place of the typical torsional fiber force sensor, a
microcantilever was used. A mass attached to this cantilever was
subjected to a time varying force of a gravitational type, and the
deflection of the cantilever (measured using fiber-optic
interferometry~\cite{rugar}) provided a measure of this force. The
experiment was performed at low temperature and in cryogenic vacuum to
exploit the high quality factors attainable in microcantilevers under
these conditions.


The force sensors for these experiments were single-crystal silicon
diving-board-shaped oscillators $250$~$\mu$m in length, $50$~$\mu$m in
width, and $0.335$~$\mu$m thick.  The cantilevers were fabricated
using standard micromachining techniques~\cite{atto}, and have quality
factors in the $10^3-10^5$ range (in vacuum).  Spring constants of
these sensors were $5.0$--$5.5$~mN/m, more than two orders of magnitude
softer than standard atomic force microscopy cantilevers.  The resonant
frequency of the lowest flexural mode of the cantilever was shifted
from $\sim 7000$~Hz to $\sim 300$~Hz upon mass attachment.


Microcantilever sensitivity is usually limited by thermal noise, {\it
i.e.} Brownian motion of the beam.  An expression for this noise limit,
analogous to the Johnson noise in a resistor, can be used to obtain
the minimum detectable force of a cantilever: $F_{min}=\sqrt{ 4 k_B T
k B /  \omega _0 Q }$. In this equation, $k_B$ is the Boltzmann
constant, $T$ is the temperature, $k$ is the spring constant, $\omega
_0$ is the resonant frequency (rad/s), $Q$ is the quality factor, and
$B$ is the measurement bandwidth. The typical force sensitivity for
these (thermal noise limited) measurements was $\sim\nobreak1 \times
10^{-16}$~N/$\sqrt{\rm Hz}$. With averaging times on the order of a few
hours, the present devices could reach an ultimate sensitivity below
$\sim\nobreak1\nobreak\times 10^{-17}$~N (roughly $100$~times the
expected Newtonian gravitational force).


The force measured was that between a mass placed on the end of the
cantilever (the test mass) and a larger mass oscillated a small
distance away (the drive mass).  The test masses were gold rectangular
prisms, $50\nobreak\times 50 \times 30$~$\mu$m$^3$ in size and
$\sim\nobreak1.4$~$\mu$g in mass, and were attached to the cantilevers
with a thin (typically $\lesssim\nobreak1$~$\mu$m) layer of epoxy.  The
drive mass was constructed of ten bars: five gold bars alternating
with five silicon bars. Each bar was $100$~$\mu \rm m\times 100$~$\mu
\rm m \times 1$~mm long. (See Fig.~\ref{schemmag} for schematic
representation.) This construction, when oscillated in the plane of
the drive mass and perpendicular to the direction of the long
dimension of the bars, is expected to gravitationally excite the test
mass at a harmonic of the oscillation frequency.  This shift of the
gravitational signal frequency with respect to that of the drive mass
oscillation helped prevent spurious excitation of the cantilever due
to unwanted vibration at the signal frequency.

To test the system and precisely align the drive mass with the test
mass on the cantilever, as well as to provide an {\it in situ}
equivalent measurement, a magnetic analog of the gravitational
experiment was used.  The gold bars in the drive mass were connected
electrically at alternating ends (see Fig.~\ref{schemmag}) to form a
meander. When a DC electric current is driven through the path defined
by the gold bars and their interconnects, a space-varying magnetic
field (with a spatial periodicity half that of the gravitational
response) is created above the drive mass.  This field couples to the
test mass through an evaporated layer of nickel on the test-mass
surface closest to the drive mass.  The oscillation of the drive mass
creates a time-varying magnetic field at the location of this layer. A
signal proportional to current through the meander verifies the
system's functionality.

\begin{figure}
\includegraphics[width=0.95 \columnwidth]{fig_1}
\caption{Schematic top and side views of test mass on cantilever and
drive mass below, showing alternating gold and silicon drive mass
bars.  Drive mass motion was in the $Y$-direction.} \label{schemmag}
\end{figure}

To fabricate the drive masses, $100$~$\mu$m deep trenches were
anisotropically etched in silicon wafers.  Gold was evaporated onto
the wafers to a thickness greater than the depth of the trenches, and
excess gold was removed by means of grinding and polishing of the
wafer surface.  To construct test masses, square pits were etched into
bulk silicon, and a $100$~nm layer of nickel was evaporated onto the
surface (for the magnetic force measurement). Gold was then
evaporated, and the wafers polished as for drive mass fabrication.
Finally, the bulk silicon was etched away, leaving behind rectangular
prisms of gold with a layer of nickel on one side (for fabrication
details, see~\cite{our_rsi}).


The oscillation of the drive mass beneath the test mass was facilitated
by attachment of the drive mass to the free end of a clamped
piezoelectric bimorph actuator. The actuator was $1$~mm thick and
$4$~mm wide and had a free length of $41$~mm.

The actuator's motion was characterized {\it ex situ} using a laser
beam-bounce method.  In the experimental probe, the actuator's motion
was measured capacitively using a calibration obtained from this
initial characterization.  The capacitance was measured between a thin
copper electrode attached to the side of the actuator and a similar
fixed electrode.  The actuator's drive frequency was tuned to one third
of the cantilever resonant frequency by means of magnetic excitation.
The amplitude of actuator motion was approximately 100~$\mu$m.  Finite
element calculations were performed to determine the expected
Newtonian and non-Newtonian response of the test mass.


The nonlinearity in the piezoelectric actuator caused a small amount
($\sim\nobreak4\%$) of actuator vibration at the cantilever resonance
(the third harmonic of the actuator frequency) and could lead to a
spurious signal.  Therefore, two mass-spring vibration isolation
stages separated the cantilever mount from the actuator mount. Each
stage had a resonant frequency of $\sim\nobreak2$~Hz for both vertical
and horizontal motion, and thus together the stages provided an
attenuation of $\sim\nobreak10^8$ at $300$~Hz between the mounts.


A stiff metallized shield was placed between the drive and test masses
to minimize the effects of electrostatic and Casimir forces.  These
forces exist between the drive mass and the shield and between the
shield and the cantilever, but a stiff shield prevents direct AC
coupling between the drive and test mass.  The shield was a  $2.2$~mm
by $4.2$~mm silicon-nitride membrane $3$~$\mu$m thick onto both sides
of which was evaporated $100$~nm of gold.  Supported by a 1~cm$^2$
silicon wafer die, the shield was attached to the cantilever wafer
$15$~$\mu$m from the cantilever.


The vertical $(Z)$ and in-plane $(X, Y)$ relative displacements of the
test mass and drive mass were determined by means of a capacitive
position sensor (CPS) similar to that described in~\cite{field_cps}.
Two simpler capacitors, in conjunction with the CPS, provided
measurement of relative tilt. After cooling, the position was adjusted
until the original capacitive readings were regained, using a
three-axis translation stage with a vacuum feedthrough.   Because of
the drive mass geometry, the position in $X$ (the direction along the
drive mass bars) need only be determined to an accuracy of
$\sim\nobreak100$~$\mu$m. In the $Y$-direction, greater precision was
required as the gravitational response was expected to vary by
$\sim\nobreak100$\% over 50~$\mu$m in $Y$. The CPS enabled
determination of position in this direction to $\pm\nobreak2$~$\mu$m.
The tilt was adjusted by differential heating of the upper vibration
isolation stage support springs.

After lateral re-alignment, an absolute value of the vertical
separation between the masses was determined by means of direct
mechanical contact (sensed by the cantilever) between the drive mass
and the shield. Subsequent motion away from the shield was measured
with the CPS, and the separation between the masses was thus known to
an accuracy of $2$~$\mu$m.


The cantilever, actuator, and vibration isolation stages were
suspended in a vacuum can at the end of a probe that was inserted into
a liquid helium research dewar. An exchange gas space separated the
inner vacuum can from the liquid helium in order to reduce external
vibrations due to helium boil-off.  The entire system was supported by
$2$~m long $\sim\nobreak2$~Hz springs for additional isolation. The
experiments were performed at $T=9$--$11$~K, and at gas pressures of
less than $10^{-4}$~torr (room temperature reading) with further
cryopumping while cold.


The sinusoidal voltage supplied to the actuator and the cantilever
displacement signal were sampled and stored simultaneously at $10$~kHz
using an analog-to-digital converter. The displacement data stream was
then sorted into bins using the phase of the actuator drive voltage and
averaged by actuator phase. As the actuator voltage phase is a measure
of the lateral position of the drive mass, this technique is similar to
a lock-in technique in that signals that are not phase-coherent with
the actuator drive signal are rejected.


\begin{figure}
\includegraphics[width=0.9 \columnwidth]{fig_2}
\caption{(a) Actuator voltage (left axis), raw cantilever displacement
(right axis), and averaged cantilever displacement (right axis,
$10$~min averaging time) as a function of time over one period of
actuator motion. (b) Averaged data for one period of actuator motion
for four values of DC current through the drive mass meander. }
\label{rawdata}
\end{figure}

The third harmonic of the Fourier transform of the averaged data is
the motion of the cantilever that is phase-correlated with the third
harmonic of the drive mass motion, {\it i.e.} the gravitational-like
response of the test mass (see Fig.~\ref{rawdata}(a)).  Using the
spring constant and the measured quality factor of the cantilever, the
force on the test mass due to the drive mass can be determined.


Data were collected as a function of drive mass meander current and
$Y$-position in order to align the drive mass to the test mass.
Fig.~\ref{rawdata}(b) shows data for one period of the drive mass
actuator motion as a function of drive mass meander current.  The
phase of the magnetic response changes by $\pi$ as the current is
changed from negative to positive. Also present is a small asymmetry
in the magnitude of the magnetic signal for positive and negative
values of the nominally same current magnitude.  The observed
zero-current signal adds phasewise to the signal for a particular
current, producing this asymmetry.

The third harmonic of two sets of data acquired over $1000$~sec and
averaged as a function of measurement time is plotted in
Fig.~\ref{datavtime}. Also plotted is a line representing the
theoretical thermal noise of the cantilever.  Data taken while the
actuator was oscillating far from the test mass match the theoretical
curve within error and imply that the dominant experimental noise is
thermal in nature. (Note that the theoretical curve depends on
experimental parameters.) Data taken with the drive mass oscillating
and in close proximity to the test mass show a signal that clearly is
not diminished with averaging and is patently above the thermal noise.
This anomalous signal is too large (by a factor of $\sim\nobreak1000$)
to be Newtonian in origin. The magnitude and phase of the signal
(relative to the actuator drive) as a function of $Y$-direction offset
was not consistent with a mass-dependent force.

\begin{figure}
\includegraphics[width=0.9 \columnwidth]{fig_3}
\caption{Background and experimental signal, as well as theoretical
thermal noise, as a function of averaging time.  The experimental
signal levels off at a force of $8.9\times 10^{-17}$~N.}
\label{datavtime}
\end{figure}


These data lead to the conclusion that the experiment's force
resolution is presently limited by a background effect, most likely an
electrostatic interaction between the shield and the cantilever as
discussed below. While the shield prevents direct coupling of this sort
between the masses, a secondary effect due to shield motion could
spuriously drive the cantilever. Other possible background sources are
too small to appear at this level. For instance, randomly quenched
magnetic impurities in the gold of the drive mass produce a moment
that would result in a force below the level of Newtonian gravity, and
coordinated pressure variations in the ballistic, residual gas in the
probe are similarly small.


Shield motion was characterized (using the fiber interferometer) under
standard experimental conditions. Data were taken as a function of
relative position of the drive and test masses as well as actuator
amplitude. The response was not sensitive to the current through the
drive mass meander or relative mass position. With the actuator
swinging its full amplitude, the shield motion at the first harmonic
was $\sim\nobreak10$~pm.  The signal at the third harmonic was an order
of magnitude less.


Motion of the membrane could drive the cantilever by creating an AC
Casimir force. An upper bound can be calculated assuming the
cantilever is a conductor and using the Casimir result~\cite{casimirp}
for two conducting planes (force per unit area), $F_A\nobreak=-\pi ^2
\hbar c / 240 z^4$. For $10$~pm of motion of the shield, the
oscillatory force is on the order of $1\nobreak\times 10^{-21}$~N,
much less than the thermal noise of the cantilever for experimental
measurement times.


The motion of the membrane could also induce cantilever motion if
there were a significant potential difference between the metallized
top of the shield and the cantilever. Making a worst-case assumption
of two parallel plates with an area the size of the cantilever, a
voltage difference on the order of $0.3$~V would produce a force on
the cantilever of $1\nobreak\times 10^{-16}$~N. Potentials of this
size have been observed in other experiments~\cite{stipe}, and we
cannot yet exclude the possibility that such a force currently limits
our experimental sensitivity. Shield motion was most likely due to
inhomogeneities in the drive mass surface caused by the polishing of
dissimilar materials.  The design to be used in upcoming experiments
contains insulating and conducting layers above the drive mass bars;
the conducting layer will shield the effect of any drive mass surface
variations.


\begin{figure}
\includegraphics[width=0.9 \columnwidth]{fig_4}
\caption{Strength versus length scale parameter space (see
Eq.~(\ref{potential})) for non-Newtonian effects showing area excluded
by present (darker shaded region bounded by signal plus 1$\sigma$-level
error) and previous (lighter shaded region) experiments. Lines labeled
Lamoreaux, UWashington, UColorado, and Irvine are
from~\cite{lamoreaux}, \cite{adel}, \cite{price}, and~\cite{irvine},
respectively. Theoretical predictions (dashed lines) are adapted
from~\cite{scalar}~(dilaton and moduli) and from~\cite{add}
and~\cite{savas}~(gauge bosons).} \label{phasespace}
\end{figure}

The existence of a spurious force of average magnitude $8.9\times
10^{-17}$~N allows calculation of an upper limit on the magnitude of
any gravity-like force. Fig.~\ref{phasespace} displays the region of
non-Newtonian interaction strength versus interaction distance
parameter space that can be excluded by these results, as well as
other current experimental bounds.  For a length scale of $15$~$\mu$m,
the present data improve constraints on the strength of non-Newtonian
effects by $\sim\nobreak3$~orders of magnitude. The region eliminated
is an important one, as it contains most of the allowable,
previously-unexplored space for scalar moduli particles.  Moduli that
mediate mass-dependent forces are a very general result of many types
of theories in which supersymmetry is broken at a relatively low energy
scale.


We have presented results from a search for gravity-like deviations
from Newtonian theory at distance scales below $100$~$\mu$m using a
microcantilever approach, with masses of size on the order of these
scales. Our data offer a new constraint on non-Newtonian effects in the
range of $10$~$\mu$m that nearly eliminates the possible observation of
scalar moduli which couple gravitationally.



We thank Thomas Kenny and Savas Dimopoulos for useful discussions. JC
and DMW thank DoD for fellowship support. SJS thanks NSF for fellowship
support. This work made use of the National Nanofabrication Users
Network facilities supported by the National Science Foundation under
Award .


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\end{document}

