Signal Amplification by Sensitive Control of Bifurcation Topology
R. B. Karabalin,
1
Ron Lifshitz,
2,
*
M. C. Cross,
1
M. H. Matheny,
1
S. C. Masmanidis,
1
and M. L. Roukes
1,
†
1
Kavli Nanoscience Institute and Condensed Matter Physics, California Institute of Technology,
MC 149-33, Pasadena, California 91125, USA
2
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel
(Received 25 August 2010; published 28 February 2011)
We describe a novel amplification scheme based on inducing dynamical changes to the topology of a
bifurcation diagram of a simple nonlinear dynamical system. We have implemented a first bifurcation-
topology amplifier using a coupled pair of parametrically driven high-frequency nanoelectromechanical
systems resonators, demonstrating robust small-signal amplification. The principles that underlie
bifurcation-topology amplification are simple and generic, suggesting its applicability to a wide variety
of physical, chemical, and biological systems.
DOI:
10.1103/PhysRevLett.106.094102
PACS numbers: 05.45.
a, 62.25.
g, 85.85.+j
Interest in parametric excitation—whereby resonators
are set in motion by dynamically modulating their physi-
cal parameters—has grown over the last decade given its
many uses. These include parametric amplification, and
squeezing of classical and quantum noise [
1
], generation
of entangled pairs of photons [
2
], and effective actuation
of micro- and nanomechanical resonators [
3
–
5
]. Our
previous investigations of the parametric excitation of
coupled nanomechanical resonators [
6
,
7
] have led to
the discovery of a novel and generic amplification
mechanism. Unlike previous schemes employing a static
bifurcation [
8
], this approach is based on inducing dy-
namical changes to the topology of a simple bifurcation
diagram through the application of a small control sig-
nal. Thus, rather than positioning the device’s oscillation
state near a static bifurcation—at a point where abrupt
jumps may occur in response to small changes in the
input—the control signal induces dynamical changes in
the nature of the bifurcation itself, i.e., it changes the
way in which different branches of stable solutions are
connected to each other. The input signal does not
merely cause the system to go through a bifurcation
point, thus inducing abrupt jumps, but rather it controls
whether a high-response or a low-response stable solu-
tion is connected to the initial branch where the device is
biased. As explained below, the response thus follows,
continuously, either the high or the low branch, amplify-
ing the input signal without any abrupt jumps between
states. Here we describe a first implementation of a
bifurcation-topology amplifier (BTA) based on a coupled
pair of nonlinear high-frequency nanomechanical resona-
tors, actuated piezoelectrically [
9
] and measured by laser
interferometry at room temperature [
10
].
We consider a single resonator—in our case a nano-
mechanical doubly-clamped GaAs beam, vibrating in its
fundamental transverse mode—and assume that it under-
goes weak damping and possesses some nonlinearity both
in its potential energy and in its damping. The resonator is
driven parametrically by modulating its effective stiffness,
the equation of motion is then given by [
6
]
€
x
þ
_
x
þð
!
2
0
þ
h
cos
!
p
t
Þ
x
þ
x
3
þ
x
2
_
x
¼
0
;
(1)
where
x
represents the displacement of the resonator. The
normal frequency of the resonator is
!
0
;
is the nonlinear
spring constant, or Duffing parameter;
is the linear
damping rate; and
is the coefficient of nonlinear damp-
ing. The parameters
h
and
!
p
are the parametric driving
amplitude and driving frequency, often called the pump
frequency; these are easily varied in experimental
implementations.
We concentrate on the so-called first instability tongue,
operating near twice the resonance frequency
!
p
¼
2
!
0
þ
, with
h
sufficiently large to overcome the damp-
ing [
6
]. If we fix all other parameters, the response ampli-
tude
j
a
j
of the resonator as a function of the relative
frequency
, follows the curve shown in Fig.
1(a)
.For
>
0
, as in the case of doubly-clamped beams, the re-
sponse amplitude of the driven resonator increases with in-
creasing frequency, until it reaches a saddle-node bifurca-
tion and drops abruptly to zero. The phase of the response,
which is not plotted, is determined only to within a
phase
shift due to the fact that the drive completes two periods
whenever the resonator completes only a single period at
half the frequency. Chan
et al.
[
11
] studied noise-induced
switching between these two oscillating states, differing by
a
phase shift, and Mahboob and Yamaguchi [
12
] pro-
posed to employ them for memory storage.
Our coupled-nanoelectromechanical systems (NEMS)
BTA consists of two resonators with slightly different
resonance frequencies
!
and
!
þ
, which are excited
simultaneously using the same drive source. Given the
phase freedom in the response of each resonator, when the
resonators are uncoupled there is a 50% chance that they
will respond in phase when both are excited, and a 50%
chance that they will respond with opposite phase.
Consequently, in this case the summed response of the
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two resonators,
j
a
1
þ
a
2
j
, forms a pitchfork bifurcation, as
shown in Fig.
1(b)
, where the incoming branch—in which
only the low-frequency resonator is excited—is connected
to both the in-phase and the antiphase branches as the
second resonator becomes excited. This behavior is con-
firmed experimentally in Fig.
2(a)
, as described below.
This situation changes when coupling is introduced be-
tween the resonators. Coupling alters the topology of the
bifurcation diagram, causing a distortion of the perfect
pitchfork. To illustrate this simply, we take the coupling
to be linear, of the form
D
ð
x
1
x
2
Þ
in Eq. (
1
) for the first
resonator
x
1
, and with the interchange of indices in the
corresponding equation for
x
2
. With attractive coupling
(
D>
0
), as shown in Fig.
1(c)
, the incoming branch is
connected only to the in-phase branch. With repulsive
coupling (
D<
0
), as shown in Fig.
1(d)
, the opposite oc-
curs, and the incoming branch is connected only to the
antiphase branch. An upward quasistatic sweep of the
pump frequency that starts to the left of the bifurcation
will therefore easily distinguish between attractive and
repulsive coupling. The summed response will follow the
upper branch if the coupling is attractive—with the reso-
nators oscillating in phase—and will follow the lower
branch if the coupling is repulsive—with the resonators
oscillating in antiphase. This dependence of the bifurcation
topology on the effective coupling
D
between resonators
can be used to realize a novel method of signal amplifica-
tion. To do so one employs
D
as a control parameter, setting
it proportional to a small input signal
V
in
to be amplified.
We have realized a BTA using a pair of
6
m
500 nm
200 nm
doubly clamped GaAs beams, separated
by 400 nm, as shown in the inset of Fig.
2(a)
(see supple-
mentary information [
13
] for more details). The fundamen-
tal out-of-plane resonance frequencies of the devices are
around 13.1 MHz, differing by about 30 kHz as a result of
small fabrication variations. Resonance quality factors in
vacuum at room temperature are approximately 1700. The
GaAs beams consist of a layered structure [
9
] that develops
longitudinal strain when a voltage
V
app
is applied between
their top and bottom surfaces. When clamped at both ends,
this strain is converted into stress, which either stiffens or
softens the beams thus tuning their natural resonance fre-
quencies. Any applied voltage
V
app
also induces electric
dipoles within the layers of the beams [
14
], which causes
the beams to repel each other, thus making the coupling
D
more negative. In the absence of an applied voltage there is
residual attraction between the beams due to elastic cou-
pling through their shared elastic support. A positive dc
offset voltage
V
offset
1
:
1V
precisely compensates this
elastic attraction for these devices and nulls the intrinsic
coupling between the beams. Thus, we set the effective
coupling between the beams
V
in
¼
V
app
V
offset
, which is
proportional to
D
, as our input signal.
We measure the summed out-of-plane displacement am-
plitude of the beams
j
a
1
þ
a
2
j
by optical interferometry
using a laser beam that illuminates both resonators. The
interference signal is strongest when the beams oscillate in
FIG. 2 (color online). (a) Experimental measurement of the
parametric response of two uncoupled nanomechanical beams,
confirming the pitchfork bifurcation [cf. Figure
1(b)
]. (b) With a
constant input signal
V
ð
dc
Þ
in
larger than the noise floor, the re-
sponse to upward frequency sweeps deterministically follows the
upper or lower curves depending on the sign of
V
ð
dc
Þ
in
.
(c) Repeated upward frequency scans with no coupling,
V
ð
dc
Þ
in
¼
0
, show that each of the two possible branches are followed with
equal probability. (d) Repeated scans with
V
ð
dc
Þ
in
¼
3mV
, show
that a majority of the scans follow the upper in-phase response
curve, yet because the noise amplitude is greater than 3 mV,
some scans still follow the lower antiphase response curve.
FIG. 1 (color online). (a) Response amplitude of a single
parametric resonator given by (
1
). Stable solutions are in solid
(red); unstable in dotted (blue). (b)–(d) Summed response of two
resonators. Only the stable branches are shown, where (00) is the
zero state; in (
"
0
) only the lower-frequency resonator is excited;
in (
0
"
) only the higher-frequency resonator is excited; in (
""
)
both resonators are excited in phase, and in (
##
) both resonators
are excited in antiphase. (b) No coupling; (c) Attractive cou-
pling; (d) Repulsive coupling. See supplementary information
[
13
] for more details.
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phase and weakest when they oscillate in antiphase. We
subsequently translate the interferometer output signal into
motional amplitude, calibrated in nanometers, using the
known value of the critical amplitude at the onset of non-
linear response for driven Duffing resonators [
6
,
15
].
Figure
2(a)
shows the summed response of the two beams
measured for zero effective coupling,
V
in
¼
0
. These data
confirm the existence of the pitchfork bifurcation, pre-
dicted in Fig.
1(b)
. Figure
2(b)
shows a sequence of upward
frequency sweeps for finite dc input signals that are
stronger than the background noise, confirming the pre-
dictions of Figs.
1(c)
and
1(d)
, that the device acts as a very
sensitive discriminator between positive and negative input
signals. Its resolution is limited only by the amplitude of
background fluctuations, which comprise noise accompa-
nying the input signal itself, thermomechanical motion of
the beams, and other sources as described in [
13
]. For
signals that are weaker than the noise, there is a finite
probability of following the ‘‘wrong’’ branch, and fluctua-
tions ultimately determine which branch is followed. At
precisely
V
in
¼
0
, the probabilities to follow the two
branches are equal, as shown in Fig.
2(c)
. This balance
tips in favor of the lower branch for
V
ð
dc
Þ
in
>
0
, and in favor
of the upper branch for
V
ð
dc
Þ
in
<
0
, as evident in Fig.
2(d)
for
V
ð
dc
Þ
in
¼
3mV
. In fact, as explained in [
13
] and demon-
strated below, for small input-signal-to-noise ratio, the
probability of following the ‘‘correct’’ branch is linear in
the signal, and the BTA indeed acts as a linear amplifier.
Our measurement setup to realize a BTA is shown in
Fig.
3(a)
[
13
]. Three waveforms with a decreasing hier-
archy of frequencies are applied to the device: (a) A fast
rf drive at frequency
!
p
, which parametrically pumps the
beams; (b) a triangular sweep waveform, at an intermediate
frequency
!
sw
, that is used to frequency modulate the rf
pump voltage and sweep it through the bifurcation point;
and (c) a slower input signal
V
in
ð
t
Þ
to be amplified. The
sweep waveform provides fast and consecutive linearly-
ramped frequency sweeps that start below the bifurcation
point and end above it. At the beginning of each upward
sweep the BTA is thereby reset, enabling it to sample anew
the sign of the input signal
V
in
ð
t
Þ
.
There are fundamental constraints on the sweep rate
2
!
ð
max
Þ
in
<!
sw
set by the Nyquist criterion and re-
quiring
!
sw
to be slower than the resonators’ relaxation
rate
. The latter ensures that the system adiabatically
follows the topology of the bifurcation diagram, which is
essential for proper BTA operation. For the experimental
resonance frequency of 13.1 MHz and quality factors of
1700
, the resonators’ response times are of order 0.1 ms.
In our measurements, we find that the output signal dimin-
ishes for sweep rates exceeding
2 kHz
, corresponding to
relaxation times around 0.5 ms, in agreement with this
estimate. We anticipate that significant increases in signal
bandwidth up to MHz scales should be readily attainable
using previously demonstrated NEMS resonators at micro-
wave frequencies.
Figure
3(b)
shows the measured spectral response of the
BTA for a positive input signal (lower curve) and for a
negative input signal (upper curve). The response of the
photodetector of the interferometer is an rf output voltage
that oscillates at the resonant frequency of the beams, with
amplitude proportional to the summed displacement of the
beams. We define the BTA output as the signal obtained by
homodyne conversion of the photodetector signal using a
local oscillator synchronized to
!
p
, but at half the pump
frequency,
!
p
=
2
. This yields a demodulated, amplified,
BTA-sampled replica of the input signal. The inset of
Fig.
3(b)
displays the demodulated output for a square-
wave input signal. The measured BTA output is the rms
summed displacements of the two resonators, averaged
over many sweeps. To within a constant shift, this is
proportional to the probability
P
"
of following the upper
branch minus the probability
P
#
of following the lower
branch. This difference in probabilities is shown [
13
]to
follow an error function.
We characterize our BTA by amplifying a simple input
signal
V
in
ð
t
Þ¼
V
ð
dc
Þ
in
þ
V
ð
ac
Þ
in
ð
t
Þ
, where
V
ð
ac
Þ
in
ð
t
Þ
is a sinusoi-
dal or a square wave with amplitude
V
ð
ac
Þ
in
and frequency
!
in
=
2
¼
167 Hz
, offset from zero by a finite dc shift
V
ð
dc
Þ
in
. A sweep rate of
!
sw
=
2
¼
557 Hz
is employed to
sample the output signal, and a lock-in amplifier is em-
ployed for taking the BTA output and extracting the am-
plitude of its modulation at the frequency of
V
ð
ac
Þ
in
ð
t
Þ
.
Figure
4(a)
displays a family of curves showing the modu-
lation amplitude of the BTA output, measured as a function
of
V
ð
dc
Þ
in
, for a set of fixed input amplitudes
V
ð
ac
Þ
in
. For large
j
V
ð
dc
Þ
in
j
relative to
V
ð
ac
Þ
in
the BTA output is not at all modu-
lated, giving rise to the widths of the observed peaks in
Fig.
4(a)
, while the sharpness of the rise and fall of the
curves depends on the noise amplitude. The maximum
peak height, obtained for
V
ð
dc
Þ
in
¼
0
, which is proportional
to
P
"
P
#
, is plotted in Fig.
4(b)
as a function of the input
amplitude
V
ð
ac
Þ
in
, showing the expected error function
dependence. For input amplitudes exceeding the noise,
V
ð
ac
Þ
in
>
20 mV
rms, the maximum peak heights saturate
FIG. 3 (color online). (a) Schematics of the experimental
setup. (b) The output response measured by a spectrum analyzer.
The upper (red) spectral response curve is obtained for a negative
input signal, and the lower (black) spectral response curve for a
positive input signal. Inset: Demodulated output signal for a
square-wave input signal.
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because the BTA deterministically switches between the
two branches in every period of the input signal. For small
input amplitudes the average output signal is linear, with a
slope that depends on noise. An increase in the noise
amplitude will decrease the slope, thereby delaying the
saturation of the BTA response and extending the dynamic
range but at the cost of suppressed gain.
The utility of the BTA clearly depends on its particular
implementation. Here, with our coupled-NEMS BTA as an
example, we can estimate its sensitivity as a charge detec-
tor. We easily resolve an input signal
V
ð
ac
Þ
in
¼
1mV
corre-
sponding to
70
e
of charge on the resonators’ electrodes
(
e
¼
1
:
6
10
19
C
). Figure
4(c)
shows that a 0.056 Hz
integration bandwidth yields a signal-to-noise ratio of
100
. A detailed analysis of the noise [
13
], shows that
this demonstrates a charge noise spectral density for this
BTA of
ffiffiffiffiffi
S
q
p
30
e=
ffiffiffiffiffiffi
Hz
p
. The supplementary information
[
13
] also contains analyses of the principal noise sources
limiting coupled-NEMS BTA performance. These indicate
that very significant enhancement, ultimately limited by
fundamental noise sources, should be attainable [
16
]. In
this limit, we estimate that charge sensitivity will improve
to a level of order
0
:
5
e=
ffiffiffiffiffiffi
Hz
p
for our current device at
room temperature, and
10
5
e=
ffiffiffiffiffiffi
Hz
p
for an optimized
device with GHz frequency and reduced capacitance, op-
erating at cryogenic temperatures.
The first implementation of BTA principles, described
here, is based on coupled-NEMS resonators. However, the
simple and generic principles underlying the BTA suggest
its applicability to a wide variety of physical systems, such
as laser cavities, superconducting resonators, coupled
Josephson junctions, and possibly even to oscillating
chemical and biological systems.
This research is partially supported by the National
Science Foundation under Grant No. DMR-0314069, and
the US-Israel Binational Science Foundation (BSF) under
Grant No. 2004339. We thank Philip Feng and Inna
Kozinsky for useful discussions, and Iwijn De Vlaminck
and Gustaaf Borghs from IMEC (Leuven, Belgium) for
providing us with the GaAs material.
*
ronlif@tau.ac.il
†
roukes@caltech.edu
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FIG. 4 (color online). (a) BTA output for different values of
the signal amplitude
V
ð
ac
Þ
in
, as a function of its offset
V
ð
dc
Þ
in
from
zero. (b) BTA output as a function of the input amplitude
V
ð
ac
Þ
in
,
for zero offset voltage. The response closely matches the theo-
retically predicted error function [
13
], shown as a solid (red)
line. (c) The smallest measured signal in (a), with
V
ð
ac
Þ
in
¼
1mV
corresponding to about
70
e
applied to each resonator’s conduct-
ing layer, averaged for a longer time corresponding to an
integration bandwidth of 0.056 Hz.
PRL
106,
094102 (2011)
PHYSICAL REVIEW LETTERS
week ending
4 MARCH 2011
094102-4