Table Of ContentQuantum electromechanics: Quantum tunneling near resonance and qubits from
buckling nanobars
Sergey Savel’ev(1,2), Xuedong Hu(1,3), A. Kasumov(4), and Franco Nori(1,5)
(1) Frontier Research System, TheInstituteof Physical andChemicalResearch (RIKEN),Wako-shi, Saitama, 351-0198, Japan
(2)Department of Physics, Loughborough University, Loughborough LE11 3TU, UK
6 (3) Department of Physics, University at Buffalo, SUNY, Buffalo, NY 14260-1500, USA
0 (4) Laboratoire de Physique des Solides, Associ´e au CNRS, Universit´e Paris-Sud, F-91405, Orsay, France
0 (4) Center for Theoretical Physics, Department of Physics,
2 University of Michigan, Ann Arbor, MI 48109-1040, USA
n (Dated: February 6, 2008)
a Analyzing recent experimental results1,2, we find similar behaviors and a deep analogy between
J
three-junctionsuperconductingqubitsand suspendedcarbon nanotubes. Whenthesedifferent sys-
2
temsareac-drivenneartheirresonances,theresonancesingle-peak,observedatweakdriving,splits
into two sub-peaks (Fig. 1) when the driving increases. This unusual behavior can be explained
]
l by considering quantum tunneling in a double well potential for both systems. Inspired by these
l experiments, we propose a mechanical qubit based on buckling nanobars—a NEMS so small as to
a
h be quantum coherent. To establish buckling nanobars as legitimate candidates for qubits, we cal-
- culate the effective buckling potential that produces the two-level system and identify the tunnel
s coupling between the two local states. We propose different designs of nanomechanical qubits and
e
describehowtheycanbemanipulated. Also,weoutlinepossibledecoherencechannelsanddetection
m
schemes. Acomparisonbetweennanobarsandwellstudiedsuperconductingqubitssuggestsseveral
. futureexperiments on quantumelectromechanics.
t
a
m PACSnumbers: 85.85.+j
-
d
n I. INTRODUCTION pliedtoasmallbarwithoneortwoendsfixed,thereexist
o two degenerate buckling modes, as schematically shown
c
in Fig. 2. In the quantum mechanical limit, these two
[ Micro- and nano-electromechanical systems (MEMS
modes represent the two lowest-energy states. They can
andNEMS)haveattractedwidespreadattentionbecause
1 be well separated from the higher-energy excited states,
oftheirbroadspectrumoffunctionalities,theirtiny sub-
v sothatatlowtemperaturesthebucklingnanobarcanbe
9 micron sizes, and their unique position bridging micro-
properly described as a two-level system. Furthermore,
1 electronicandmechanicalfunctions3. Sophisticatedtools
sincenanobarscanbechargedorcancarryelectricalcur-
0 ranging from mirrors and sensors, to motors and multi-
rent, the electric or magnetic field can be conveniently
1 functional devices have been fabricated4,5. As the size
0 used to manipulate their quantum states.
of these devices shrinks, experimental studies of NEMs
6
arerapidlyapproachingthequantumlimitofmechanical
0
oscillations5,6,7,8,9, where quantum coherence and super-
/
t position should result in quantum parallelism and the
a
m possibility of information processing.
The fascinating prospect of observing quantum coher-
- The emergence of quantum electromechanical devices ent phenomena in a macroscopic mechanical oscillatoris
d (see, e.g., Ref.3,4,6,7,8,9 and references therein) brings a main motivation of this study. To achieve this goal, a
n
both challenges, such as the inevitable and ubiquitous NEMS needs to possess at least two general attributes:
o
c quantumnoise,andpromises,suchasmacroscopicquan- small size and a high fundamental frequency. A carbon
: tum coherence8,10,11 or quantum teleportation12,13. In- nanotube is a naturalcandidate: itis verythin (and can
v
deed,duringthepastseveralyearsthequantummechan- be very short) while still being stiff20 (thus having high
i
X icalpropertiesofNEMSandhowthey canbe coupledto vibrationfrequency)becauseofthestrongcovalentbonds
r other quantum mechanical objects have been very ac- between carbon atoms within the graphene sheet21,22.
a tively studied14,15,16,17,18,19. The experimental pursuit However, without strong experimental support, it might
of these studies has so far focused on cooling a device to be hard to judge the feasibility of achieving macroscopic
reachstateswherequantumfluctuationsinthelowesten- quantum coherence in NEMS. In contrast to NEMS, ex-
ergy eigenmode dominate over thermal fluctuations6,15. perimental evidence has demonstrated that supercon-
Sucheigenmodesaregenerallyharmonic-oscillatormodes ducting charge23, flux24, and phase25 qubits, using Su-
with equal energy-spacingsand follow bosonic statistics. perconducting Quantum Interference Device (SQUID),
AmongNEMStherealsoexistsystemsthatcanbewell do exhibit quantum coherence. Thus, it is important to
approximatedbytwolevelsinthequantumlimit,sothat look for possible similarities in the behavior of super-
they might be candidates for qubits. For example, when conducting qubits and NEMS. This should help future
a longitudinal strain above a certain critical value is ap- prospects to observe quantum coherence in NEMS.
2
II. SPLITTING RESONANCE AS A 4
MANIFESTATION OF QUANTUM suspended VT JJ F rf-SQUID
TUNNELLING IN THREE-JUNCTION SQUIDS E> c arbone ce, -0.5
AND NEMS < nanotube antenna on
p
s
(cid:181)n 3 Re (b)
Experiments1 on suspended single-wall carbon nan- o
otubes (with diameter of about 20 nm and length of pati 0 0.00 0.02
about 1.7 µm) excited by an electromagnetic wave show ssi P = - 15.3dB (w -w 0)/w 0
a resonance peak with an unusual shape (Fig. 1a), for di
one of the fundamental harmonics ω0. As expected, a ed 2 (a)
z
Lorentz-form resonance peak, that grows with increas- ali
ing intensity of the ac field, was observed at weak driv- m
r
ings. Surprisingly, when further increasing the ampli- o
N P=-15.6dB T=110mK
tude ofthe externallyappliedelectromagneticwave,this 1
peak splits into two sub-peaks1. When further increas- -0.001 0 (w -w )/w 0.001
0 0
ing the ac drive, these two sub-peaks gradually move
quantum U /(cid:1)w = 2
away from each other, while their heights stop growing. 0.1 resonance 0 0
It is important to stress that this phenomenon was ob-
s
servedatafrequency(ω0/2π ∼2GHz)andtemperature nit
(T 100 mK) where quantum effects start to domi- u
nate∼over thermal noise (T < ¯hω /k 100 mK, with b. 0.5 tunneling
0 B r
Btiseomilmtzwpmoaarstnaqnnuctiottenosltooabwnste(rkqvBue)a.qliuAtyalns∼fotautcmhtoeredffQiesscitps≈∼a.tNio1on5t0ie0n)tt,hhawethstiyhcshe- y <E>, a 0.00 asymmetry and( c) U0-2 0 y U0U /(cid:1)w = 20
fundamental frequency of this device can be easily in- rg discontinuity due to 0 0
e
creased (at least by an order of magnitude, well into the n nonlinearity of U(y)
e classical
operating frequency 1-15GHz of many superconducting e resonance
qubits), for experimentally available carbon nanotubes g 0.1
a
with shorter length. er
v (d)
Interestingly, a similar phenomenon has been recently A
found2 for an Al three-junction SQUID qubit coupled 0
(w -w )/w
to a Nb resonant tank circuit (Fig. 1b). It was exper- -0.02 0 0 0 0.02
imentally proven2, via the observation of quantum hys-
teresis (Landau-Zener transitions), that this circuit was FIG. 1: (a) Experimental resonance1 of the effective dissi-
operated in the quantum regime (at ω0/2π 20 MHz pation (which is proportional to the average energy hEi) as
and T > 10 mK), though for a worse rati∼o of quan- a function of the dimensionless frequency (ω −ω )/ω , for
0 0
tumtot∼hermalnoisecomparedtothecarbonnanotubes1 a suspended carbon nanotube, for different intensities P of
(i.e., ω /T one order of magnitude higher for the nan- the externally applied ac electromagnetic wave. (b): the res-
0
otube). W∼hen the magnetic flux in the SQUID was onance in the response of the three-junction SQUID probed
driven as Φ = Φ +Φ cos(ωt), the resonance in the by measuring2 the voltage for the resonant tank VT for dif-
dc ac
ferent driving amplitudes as a function of (ω−ω )/ω . For
response, probed via the tank voltage as a function 0 0
these different systems, a similar splitting of the resonance
of the dc flux, was found2 to exhibit a transformation
peak was observed1,2. Schematic diagrams for a driven sus-
from a Lorentz-form single-peak to a double-peak shape
pended carbon nanotube and three junction SQUID are also
(Fig. 1b) in striking similarity to suspended driven car-
shown at the top. The simulated resonance [average energy
bon nanotubes1. Sweeping the dc flux in the SQUID hEiversusthedimensionlessfrequency(ω−ω )/ω ]forquan-
0 0
corresponds to changing the fundamental frequency as tum (c) and classical (d) particles moving in the double-well
potential shown in the inset of (c). A quantum description
ω =ω (Φ =0)+∆ω (Φ ), (1) agreeswithexperiments1,2(Fig.1a,b): Forquantumparticles,
0 0 dc 0 dc
thestandardLorentz-form resonancepeakforweak acdrives
(c, red curve, A = 0.005) splits into two sub-peaks due to
with ∆ω (Φ Φ /2) and flux quantum Φ . Thus,
0 dc 0 0
we find th∝at the m−easured1,2 response, near resonance, tunnelling at stronger drives (blue curve, A = 0.01; green
curve, A = 0.0135). A classical description cannot describe
of both systems (carbon nanotube and three-junction
experiments1,2(Fig. 1a,b): For classical particles, the weak-
SQUID qubit) depends on the difference ω ω . This
− 0 driving Lorentz-form peak (red curve, A = 0.005) becomes
will be useful to establish below that the measurements
asymmetric and exhibit discontinuous jumps E(ω) (green
for drivencarbonnanotubes1 andthree-junctionSQUID curve,A=0.0135) dueto thenonlinearity of U(y).
qubit2 essentially probe the same effect.
3
We propose how the incorporation of quantum tun- Therefore,weconcludethatthetwo-peakresonancein-
nellingcanbe usedtounderstandthesestill-unexplained dicates that both the three-junction SQUID qubit2 and
experimentalobservationsinnanotubes1 andthreejunc- the suspended carbon nanotube1, operate in the quan-
tion SQUIDs2. In order to interpret the splitting of the tum regime. Thus, the technology for fabricating sus-
resonance peak (Fig. 1a and 1b), we employ a quasi- pended buckled carbon nanotubes, working as nanome-
classical model26 described by the equation of motion chanical qubits, already exists1. In view of the explo-
sive growth of NEMS technology, below we discuss the
y¨ + 2λy˙ + ∂U(y)/∂y = A cos(ωt), (2) prospect of such buckling charged nanobars (the clamp-
ing at the base ensures an anisotropic nanobar instead
forthedouble-wellpotentialshownintheinsetinFig.1c,
of an isotropic nanotube) as candidates of quantum bits
with small damping λ = 0.01, and different driving am-
for quantum information processing. Indeed, previous
plitudes A. We assume that the particle (which mimics
work on quantum buckling (e.g.,10,11) were not focused
the buckling mode or current in the SQUID) can tunnel
on operating these as qubits, which is our focus here29.
from one well to the other with transmission coefficient
2√2 a
D =exp √U E dy , (3) III. CONTROLLING QUANTUM STATES OF A
"− ¯h Z−a − # NANOROD IN A DOUBLE-WELL POTENTIAL.
where a is the classical turning point, and E =(y˙)2/2+
The double-well potential (Fig. 2a) corresponding to
U(y) is the energy of the particle. This minimal model
the buckling modes ysin[πl/l ] of a nanorod (Fig. 2b)
allows to qualitatively describe the resonance of a quan- max
takes the form:
tum particle in a double well potential (a more com-
plete,fullyquantummechanical,theorywillbepublished U(y)=αy2+βy4+ 2lmaxf⊥y, (4)
elsewhere27) and also to obtain the transition to a clas- π
sical description (Fig. 1c,d).
where the parameters can be estimated30 as
WhenthepotentialbarrierU iscomparablewithh¯ω ,
0 0
we find that the single resonance peak, at low driving, π2
splits into two sub-peaks (Fig. 1c) for higher drives; in α(f) = (fc f),
4l −
agreement with experimental findings1,2. The physical max
π4
origin of this effect is the following: (i) At low driv- β(f) = (4f 3f),
64l3 c−
ing the energy E of the quantum particle (whether the max
SQUID or the nanotube) is also low and the probability I Yπ2
f = . (5)
of tunnelling is negligibly small, thus, the usual Lorentz- c l2
max
formoftheresonanceoccurs;(ii)Whenthedriving(and,
thus, energy) increases, the particle starts to tunnel be- Here, we introduce the tube length l (0 l lmax), the
≤ ≤
tween the wells. A tunnelling event is equivalent to an elastic modulus Y and the moment of inertia I. To con-
inversion around the origin, i.e., changing y to y: like troltherodweneed: (i)alongitudinalcompressingforce
replacing the left well by the right one, or vic−e versa. f acting on the rod ends, and (ii) a transverse force f⊥,
As a result, right after a tunnelling event, the average which can be produced via, e.g., interacting the charged
power = y˙Fcosωt the particle receives from the ex- nanorod with an electric field.
ternal fPorceh(e.g., theipower absorption by the nanotube For zero transverse force f⊥ = 0, our qubit (Fig. 2b)
fromthe externalelectromagneticwave)changesits sign is in its degeneracy point. Indeed, this potential has two
and, hence, the energy, which increases be- minima, at y = y0(f)= α/2β that are separated
fPore→tun−nePlling,startstodecreasetemporarily. Thus,the by a potential bar±rier ±
p
average energy E also decreases when the number of
h i ∆U(f) = α2/4β. (6)
tunnelling events increases. This explains the minimum
ofhE(ω)iatω =ω0(insteadofthemaximumobservedat ThefirsttwoenergylevelsE1andE2intherightwellcan
weakdriving)andalsothesplittingoftheresonancepeak be estimated assuming a parabolic potential well shape
— because the tunnelling frequency sharply increases at U mω2(y y )2/2 with
the resonance frequency ω . ≈ 0 − 0
0
For the case of “more classical” particles, U0 ≫ ¯hω0, ω (f)= U′′(y0) 1/2 =2 α(f) 1/2 (7)
the probability of tunnelling is always very low. When 0
m m
drivingincreases,theparticle(whichisalwayslocatedin (cid:18) (cid:19) (cid:18) (cid:19)
thesamewell)beginstofeelthestrongnonlinearityofthe and the mass m of the nanorod. Thus, we obtain
potentialU(y),resultingintheenergydependenceofthe
E (f)=3h¯(α(f)/m)1/2 (8)
oscillationfrequencyω (E). Insteadofasplitresonance, 1
0
the resonance peak as a function of frequency shows the
and
standardasymmetricshapewithasharpjumpof E(ω) ,
associated with mechanical hysteresis28 (Fig. 1d)h. i E (f)=5E (f)/3. (9)
2 1
4
Due to quantum tunnelling between the left and right
potential wells, E splits into two levels
1
(a)
E1± =E1±¯h∆t. (10) E1 D U (cid:181) f ^
Thetunnellingrate∆ betweentheleftandrightbuckled
t
states is:31
(b)
2 α(f) π√2m(∆U(f) E (f)) 〈〈〈〈 1 | E^ | 0 〉〉〉〉
∆ (f) exp − 1 .
t
≈ πr m − 2h¯ α(f) !
(11)
p
It is important to stress that the longitudinal force f
phonon
allows to control the tunnelling ∆t(f) as well as the en- (c) Correlated
ergy levels itself E1(f). The higher levels are well sepa- E^ noise
∆ratt/e¯hdωfr0omt1h.e tCwhoanlogwinegsttohneest:ra(nEs1+ve−rsEe1−fo)r/c(eEf2⊥−mE1o)ve∼s 〈〈〈〈 1 | E// | 0 〉〉〉〉 (d)
the syste≪m out of the degeneracy point, allowing to ma- Q1 Q2 Q3
nipulate the proposed nano-mechanical qubit.
We now introduce another, more controllable, design
of a nanorod qubit, which is also less affected by deco-
herence coming from the relative vibrations of the rod detecting state | L 〉
S
ends (Fig. 2c). In such a design we consider a “crossed” E
electric field having a component E along the rod and (e) T
k
another E perpendicular to the rod, which is clamped
⊥
to a substrate only by one end. Changing the charge of
the top capacitor plate and the charge of the rod itself
detecting state | R 〉
can control both the electric field, E , on the rod and
k
its gradient, ∂E /∂x. By adjusting the different knobs, FIG.2: (a)Double-wellpotentialforabucklednanobar. Due
k
we can tune the relative strength between thermal acti- to tunnelling from the right potential well to the left one,
the lowest energy level is split into two levels for f = 0 as
vation and quantum tunneling, allowing the observation ⊥
shownintheleftpanel. Thelowest(blue)andthefirstexcited
of transition between these two regimes.
(red) levels correspond to the symmetric and antisymmetric
combinations of the wave functions localized in the left and
right potential wells. The energy level splitting between the
IV. MANIPULATION, DECOHERENCE, AND left andright states canbecontrolled bythetransverseforce
DETECTION OF MECHANICAL QUBITS f as shown in the top right panel. (b) A buckled rod qubit
⊥
where the compressed force applied to the rod ends controls
the potential shape [α and β in Eq. (5)] and, therefore, the
The manipulation of the mechanical qubits can be
energysplittingatthedegeneracypoint. Thetransverseforce
achieved electrically (Fig. 3). For example, in analogy
f allowstodrivethebartoadegeneracypoint. (c)Another
totheCooperpairbox23 (seetable),onecanpreparethe ⊥
proposed design for a buckled-rod qubit. In this case the
nanobar qubit in the L state by setting a transverse
| i possibledecoherenceoriginatingfromtherelativevibrationof
electric field towards the right (assuming the nanobar is
thetopandbottomrodholderscanbeavoided. Bychanging
negativelycharged). Bysuddenlyturningoffthiselectric thechargeofthetopcapacitorplateandthechargeoftherod
fieldandbringingthesystemtothedegeneracypoint,the itselfitispossibletoindependentlycontrolboththeparallel-
nanobar state is prepared in a coherent superposition of to-rod electric field E and its gradient. This design allows
k
(R L )/√2. Because of the tunnel splitting, the sys- a high level of control of both the tunnelling and the energy
t|emit±he|nistarts to oscillate coherently, with a frequency splitting. (d)Correlatednoiseproducedbysomephononscan
given by ∆ . Therefore, by detecting the nanobar posi- be eliminated using a decoherence-free subspace: qubits Q1
t
and Q2 can be associated with one logic qubit. (e) Single-
tion, as a function of ∆t (see Fig. 3), one can determine
shot measurements could be done via either single electron
the coherent oscillation frequency and the system deco-
transistor (SET) or photon reflection.
herence. Driven coherent transitions between the two
qubit states can also be similarly achieved. A sinusoidal
componentcanbeaddedtotheverticalelectricfieldthat
is used to control the tunnel coupling between the L
| i
and R states,inanalogytothemicrowavedrivingforce Universal quantum computing requires two-qubit op-
| i
on the Josephson coupling in a Cooper pair box. The erations. For charged nanobars, the inter-qubit interac-
study of free and driven coherent oscillations of a nano- tion comes naturally in terms of the electric dipole in-
bar would help clarify its quantum coherence properties teraction between the bars, quite similar to the dipole
and demonstrate its feasibility as a qubit. interaction used in other proposed qubits. In the case
5
System SC charge QB SC flux QB Nano-bar QB
| excess charge> : | current direction> : | buckling direction> :
States and and and
E
Hamiltonian
| R 〉 | L 〉
Tunneling and Gate voltage Magnetic flux: Transverse force
energy splitting (normalized: (induced, e.g., by a
| R 〉〉〉〉 - |L 〉〉〉〉 | R 〉〉〉〉 + |L 〉〉〉〉 controlled by (normalized: ) ) tfriaenldsverse) electric
| L 〉 | R 〉 Tunneling Josephson energy: Josephsonenergy: Either
controlled by •Longitudinal force
(induced by pressure or
f ^ (cid:181) E || electrical field )
•or its gradient
Longitudinal force and
its gradient control the
barrier height and its
(i) | L 〉 curvature
Coupling Electrical Magnetic Electrical(dipolar)
between qubits (e.g., capacitive or
inductive coupling)
| L 〉 | R 〉
D t
Decoherence Chargefluctuations Fluxfluctuations 1. Chargefluctuations
(ii) sources include 2. Phonon-phonon
interactions
time Read-out Electrical Magnetic(SQUID) Either electricor
(e.g., SET or JJ) optical, or mechanical
FIG. 3: A schematic diagram for controlling the quantum
state of a nanobar via coherent oscillations at the degener-
acy point. Tuning the system to its degeneracy point can Table: comparison of Josephson-junction superconducting
bedone bychanging the perpendicularelectric field E . For (JJ SC) charge, JJ SC fluxand nano-barqubits(QB).
⊥
thecase when E and E is zero at t=0, theperpendicular
⊥ k states ofthe nanobar,so thatthe Nyquist-Johnsonnoise
electricalfieldE hastobeappliedfirstandthenthelongitu-
⊥ is also suppressed. The main source of quantum me-
dinalfieldE (e.g., atthemomentmarkedbythesolid circle
k chanicaldecoherencemightbeinternaldissipationcaused
on the E (time) curve). This initializes the system during
⊥
by phonon-phonon interactions. Since the nanobars are
the stage (i). Switching off the perpendicular electric field
E brings the system back to its degeneracy point, and the clamped onto a much larger substrate, in which lower
⊥
nanorodstartstooscillateatime∆tduringstage(ii). Toob- energy phonon modes exist, coupling at the base of the
serve coherent oscillations, measurements (e.g., optically or nanobarscouldleadtorelaxation/excitationofthequbit
electrically) must be done for many values of ∆t. states. However, the long-wave-length phonons produce
a correlated noise, which can be reduced by working
within a properly chosen decoherence-free subspace (see
schematics in Fig. 2d and, e.g.,27,34). Another source
of nanobarqubits, one can againuse the transverseelec- of decoherence would be charge fluctuations. Charged
tric field E to tune selectedqubits into resonance,then nanobars allow increased maneuverability of the NEMS;
T
apply microwaves to perform conditional rotations and but the surrounding and internal charge fluctuators can
other operations. produce charge noises that affect the relaxation and de-
For NEMS, and for more conventional applications phasing of the qubit.
suchasresonators,majorsourcesofnoise32,33 (thatlimit Single-wall nanotubes are a natural candidate for the
thequalityfactorQ)areinternalthermomechanicalnoise mechanical qubit we consider here. However, our pro-
(suchasheatflowsdueto inhomogeneousdistributionof posalisnotlimitedtothem. Indeed,amultiple-wallnan-
strain),Nyquist-Johnsonnoisefromthedrivingelectrical otube couldprovidea higher-frequencymechanicaloscil-
circuit,adsorption-desorptionnoise whenaresonantoris lator and more favorable condition for observing macro-
movinginanon-vacuum,noisefrommovingdefects,etc. scopic quantum tunnelling and the coherentevolutionof
Inthequantummechanicallimit,someofthesenoisesbe- mechanical motion. Silicon-based systems provide an-
comeunimportant. Forexample,thereshouldbenoheat otherenticingalternative,especiallyfromtheperspective
flow: the nanobar should not be driven beyond its first offabrication. Whileitmightbedifficulttoclampseveral
excited state. The NEMS can be placed in a vacuum to single-wall nanotubes to make identical nanobars (with
reduce adsorption-desorption noise. The whole system, controlledinter-bardistance,samelength,samebuckling
including its electrical components, has to be cooled in orientation,and same response to externalstress), fabri-
a dilution refrigerator in order to reach the two lowest cating a lithographically-patterned silicon-based nanos-
6
tructure would be much more reliable. SiC nanobars pended nanobarshave already been tested38 as veryfast
have shown higher stiffness22 compared to Si. Unless and very low-power-consumption storage memory de-
nano-assemblednanotubes canbe made withsufficiently vices. Still, many challenges lie ahead on the road to
highprecision,materialsthatcanbelithographicallyfab- practical quantum electromechanics. We hope that our
ricatedseemmorepromisingcandidatesforalarger-scale proposal here stimulate more research towards the ulti-
mechanical quantum information processor. This would mate quantum limit of NEMS.
beanironicturnofevents,giventhatthefirstcomputers
(by C. Babbage) were mechanical.
The detection of the nanobar state can be done either
electrically6,18,35 or optically36,37. It has been shown in ACKNOWLEDGMENT
recent experiments6 that single electron transistors are
verysensitiveto smallchargedisplacements. Opticalde- This work was supported in part by the National Se-
tectorscanalsobeused,wherelightscatteringcandetect curity Agency (NSA) and Advanced Research and De-
the state of the bent nanorod (Fig. 2e). velopment Activity (ARDA) under Air Force Office of
Very recently, Ref.38 fabricated suspended nanobars Research (AFOSR) contract number F49620-02-1-0334;
(driven by a 25 MHz current through an attached elec- and also supported by the US National Science Founda-
trode) switching between two distinct states. These sus- tion grant No. EIA-0130383.
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