Putting Mechanics into Quantum MechanicsKeith C. Schwab and Michael L. Roukes Citation: Phys. Today 58(7), 36 (2005); doi: 10.1063/1.2012461 View online: http://dx.doi.org/10.1063/1.2012461 View Table of Contents: http://www.physicstoday.org/resource/1/PHTOAD/v58/i7 Published by the American Institute of Physics. Additional resources for Physics TodayHomepage: http://www.physicstoday.org/ Information: http://www.physicstoday.org/about_us Daily Edition: http://www.physicstoday.org/daily_edition

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Everything moves! In a world dominated by electronicdevices and instruments it is easy to forget that all

measurements involve motion, whether it be the motion ofelectrons through a transistor, Cooper pairs or quasiparti-cles through a superconducting quantum interference de-vice (SQUID), photons through an optical interferome-ter—or the simple displacement of a mechanical element.Nanoscience today is driving a resurgence of interest inmechanical devices, which have long been used as frontends for sensitive force detectors. Among prominent his-torical examples are Coulomb’s mechanical torsion bal-ance, which allowed him in 1785 to establish the inverse-square force law of electric charges, and Cavendish’smechanical instrument that allowed him in 1798 to meas-ure the gravitational force between two lead spheres.

Today, micro- and nanoelectromechanical systems(MEMS and NEMS) are widely employed in ways similarto those early force detectors, yet with vastly greater forceand mass sensitivity—now pushing into the realm ofzeptonewtons (10⊗21 N) and zeptograms (10⊗21 g). These ul-traminiature sensors also can provide spatial resolution atthe atomic scale and vibrate at frequencies in the giga-hertz range.1 Among the breadth of applications that havebecome possible are measurements of forces between in-dividual biomolecules,2 forces arising from magnetic reso-nance of single spins,3 and perturbations that arise frommass fluctuations involving single atoms and molecules.4

The patterning of mechanical structures with nanometer-scale features is now commonplace; figure 1 and the coverdisplay examples of current devices.

The technological future for small mechanical devicesclearly seems bright, yet some of the most intriguing ap-plications of NEMS remain squarely within the realm offundamental research. Although the sensors for the appli-cations mentioned above are governed by classical physics,the imprint of quantum phenomena upon them can nowbe readily seen in the laboratory. For example, the Casimireffect, arising from the zero-point fluctuations of the elec-tromagnetic vacuum, can drive certain small mechanicaldevices with a force of hundreds of piconewtons and pro-duce discernible motion in the devices.5 But it is now pos-sible, and perhaps even more intriguing, to consider theintrinsic quantum fluctuations—those that belong to themechanical device itself. The continual progress in shrink-ing devices, and the profound increases in sensitivity

achieved to read out those devices, now bring us to therealm of quantum mechanical systems.

The quantum realmWhat conditions are required to observe the quantum prop-erties of a mechanical structure, and what can we learnwhen we encounter them? Such questions have receivedconsiderable attention from the community pursuing grav-itational-wave detection: For more than 25 years, that com-munity has understood that the quantum properties of me-chanical detectors will impose ultimate limits on their forcesensitivity.6,7 Through heroic and sustained efforts, theLaser Interferometer Gravitational Wave Observatory(LIGO) with a 10-kg test mass, and cryogenic acoustic de-tectors with test masses as large as 1000 kg, currentlyachieve displacement sensitivities only a factor of about 30from the limits set by the uncertainty principle.

But the quantum-engineering considerations for me-chanical detectors are not exclusive to the realm of gravi-tational-wave physics. In the introduction to their pio-neering book on quantum measurement, VladimirBraginsky and Farid Khalili envisage an era when quan-tum considerations become central to much of commercialengineering.7 Today, we are approaching that time—advances in the sensitivity of force detection for new typesof scanning force microscopy point to an era when me-chanical engineers will have to include \ among their listof standard engineering constants.

Several laboratories worldwide are pursuing mechan-ical detection of single nuclear spins. That goal is espe-cially compelling in light of the recent success of DanRugar and colleagues at IBM in detecting a single electronspin with a MEMS device (see figure 1b and PHYSICSTODAY, October 2004, page 22).3 However, nuclear spinsgenerate mechanical forces of about 10⊗21 N, more than1000 times smaller than the forces from single electronspins. One ultimate application of this technique, struc-tural imaging of individual proteins, will involve millionsof bits of data and require measurements to be carried outnot on the current time scale of hours, but over microsec-onds. Detecting such small forces within an appropriatemeasurement time will necessitate new quantum meas-urement schemes at high frequencies, a significant chal-lenge. Yet the payoff, originally envisaged by John Sidles,will be proportionately immense: three-dimensional,chemically specific atomic imaging of individual macro-molecules.8

The direct study of quantum mechanics in micron- andsubmicron-scale mechanical structures is every bit as at-tractive as the actual applications of MEMS and NEMS.9With resonant frequencies from kilohertz to gigahertz, low

36 July 2005 Physics Today © 2005 American Institute of Physics, S-0031-9228-0507-010-9

Keith Schwab is a senior physicist at the National SecurityAgency’s Laboratory for Physical Sciences in College Park,Maryland. Michael Roukes is Director of the Kavli NanoscienceInstitute and is a professor of physics, applied physics, and bio-engineering at the California Institute of Technology in Pasadena.

Nanoelectromechanical structures are starting to approach the ultimate quantummechanical limits for detecting and exciting motion at the nanoscale. Nonclassical states ofa mechanical resonator are also on the horizon.

Keith C. Schwab and Michael L. Roukes

Putting Mechanics into Quantum Mechanics

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dissipation, and small masses (10⊗15–10⊗17 kg), these devicesare well suited to such explorations. Their dimensions notonly make them susceptible to local forces, but also make itpossible to integrate and tightly couple them to a variety ofinteresting electronic structures, such as solid-state two-level systems (quantum bits, or qubits), that exhibit quan-tum mechanical coherence. In fact, the most-studied sys-tems, nanoresonators coupled to various superconductingqubits, are closely analogous to cavity quantum electrody-namics, although they are realized in a very different pa-rameter space.

Quantized nanomechanical resonatorsThe classical and quantum descriptions of a mechanical res-onator are very similar to those of the electromagnetic fieldin a dielectric cavity: The position- and time- dependent me-chanical displacement u(r,t) is the dynamical variable anal-ogous to the vector potential A(r, t). In each case, a waveequation constrained by boundary conditions gives rise to aspectrum of discrete modes. For sufficiently low excitationamplitudes, for which nonlinearities can be ignored, the en-ergy of each mode is quadratic in both the displacement andmomentum, and the system can be described as essentiallyindependent simple harmonic oscillators.

Spatially extended mechanical devices, such as thosein figure 1, possess a total of 3A modes of oscillation, where

A is the number of atoms in the structure. Knowing theamplitude and phase of all the mechanical modes is equiv-alent to having complete knowledge of the position and mo-mentum of every atom in the device. Continuum mechan-ics, with bulk parameters such as density and Young’smodulus, provides an excellent description of the modestructure and the classical dynamics, because the wave-lengths (100 nm–10 mm) of the lowest-lying vibrationalmodes are long compared to the interatomic spacing.

It is natural to make the distinction between nano-mechanical modes and phonons: The former are low-frequency, long-wavelength modes strongly affected by theboundary conditions of the nanodevice, whereas the latterare vibrational modes with wavelengths much smallerthan typical device dimensions. Phonons are relatively un-affected by the geometry of the resonator and, except indevices such as nanotubes that approach atomic dimen-sions, are essentially identical in nature to phonons in aninfinite medium.

It is an assumption that quantum mechanics shouldeven apply for such a large, distributed mechanical struc-ture. Setting that concern aside for the moment, one canfollow the standard quantum mechanical protocol to es-tablish that the energy of each mode is quantized:E ⊂ \w(N ⊕ 1/2), where N ⊂ 0, 1, 2, . . . is the occupation fac-tor of the mechanical mode of angular frequency w. The

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2 mm

a

c

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f

Source Drain

Gate

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dz

Figure 1. Nanoelectro-mechanical devices. (a) A 20-MHz nanome-chanical resonator ca-pacitively coupled to asingle-electron transistor(Keith Schwab, Labora-tory for Physical Sci-ences).11 (b) An ultrasen-sitive magnetic forcedetector that has beenused to detect a singleelectron spin (DanRugar, IBM).3 (c) A tor-sional resonator used tostudy Casimir forces andlook for possible correc-tions to Newtonian grav-itation at short lengthscales (Ricardo Decca,Indiana University–Purdue University Indi-anapolis). (d) A paramet-ric radio-frequency me-chanical amplifier thatprovides a thousandfoldboost of signal displace-ments at 17 MHz(Michael Roukes, Cal-tech). (e) A 116-MHznanomechanical res-onator coupled to a single-electron transistor(Andrew Cleland, Uni-versity of California,Santa Barbara).10

(f) A tunable carbonnanotube resonator op-erating at 3–300 MHz (Paul McEuen, CornellUniversity).14

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quantum ground state, N ⊂ 0, has a zero-point energy of\w/2 and is described by a Gaussian wavefunction of width∀x2¬1/2 ⊂ DxSQL ⊂ =\/(2mw). This quantity, known as thestandard quantum limit, is the root mean square ampli-tude of quantum fluctuations of the resonator position.7

The larger the zero-point fluctuations, the easier they areto detect. For example, a radio-frequency (10–30 MHz)nanomechanical resonator, with a typical mass around 10⊗15 kg, has DxSQL ⊂ 10⊗14 m, some 105–106 times larger thanDxSQL for the macroscopic test masses in the gravitational-wave detectors. Although 10⊗14 m represents a distanceonly a little larger than the size of an atomic nucleus, it isreadily detectable by today’s advanced methods, such asthose described below. At the other extreme, a carbonnanotube 1 mm long has DxSQL � 10⊗10 m, about the size ofa small atom. That relatively large value, although smallby absolute standards, makes nanotubes and nanowiresvery attractive for displaying and exploring quantum phe-nomena with mechanical systems.

A crucial consideration for reaching the quantum limitof a mechanical mode is the thermal occupation factor Nth,set by the mode frequency w and the device temperatureT. The average fluctuating energy of an individual me-chanical mode coupled to a thermal bath is expected to be

where Nth follows the Bose–Einstein distribution. For hightemperatures this expression reduces to the classicalequipartition of energy: Each mode carries kBT of energy.Figure 2a displays the deviation from classical behaviorthat occurs at low temperatures: When kBT � \w, Nth isless than 1 and the mode becomes “frozen out.” For a1-GHz resonator, this freeze-out occurs for T < 50 mK—aregime well within the range of standard dilution refrig-erators employed in low-temperature laboratories. Further-more, a 1-GHz device can be readily shielded from parasiticexternal driving forces. Nanomechanical resonators witha fundamental flexural resonance exceeding 1 GHz havebeen demonstrated by one of us (Roukes) at Caltech,1 al-

though position-detectionschemes with the band-width and sensitivity tomeasure the tiny ampli-tude of the zero-point fluc-tuations are still underdevelopment.

Fluctuations, boththermal and quantum, areof great practical and fun-damental importance:They set ultimate limitsfor sensitive force detec-tion and for the coherencetime of quantum states.These limits are deter-mined by the nanores-onator quality factor Q,which parameterizes thecoupling strength to thethermal bath. Currently,all atomic-force microscopesand experiments are in thehigh-temperature limit:kBT � \w and ∀E¬ � kBTper mode. For an acoustic-frequency mechanical res-onator typically found in

an atomic-force microscope setup, with a resonant fre-quency around 5 kHz, a mass of 10⊗12 kg, and a qualityfactor Q on the order of 104, one finds at room temperaturethat Nth ⊂ 109 and the rms displacement fluctuations,=∀xth

2 ¬ ⊂ 10⊗9 m, are more than 104 times larger than DxSQLfor this device. Thermal fluctuations limit the force sensi-tivity to 240 aN/Hz1/2 (where 1 aN ⊂ 10⊗18 N). By cooling avery thin and compliant low-frequency cantilever to near200 mK, the IBM group has been able to reduce the fluc-tuations to 6 × 10⊗13 m and achieve a record force sensi-tivity of 0.8 aN/Hz1/2.

Despite the close analogy between the quantization ofthe motion of an extended mechanical device and the quan-tization of the electromagnetic field in a cavity, some im-portant fundamental differences exist. The total zero-pointenergy in a mechanical system is finite because the num-ber of mechanical degrees of freedom is finite. Also, evenfor a rectangular flexural resonator, the mechanical modestructure is very complex and differs from the harmonicstructure for a rectangular electromagnetic cavity res-onator. In addition, due to the resonator’s finite stretchingand consequent tensioning, there is an intrinsic mechani-cal nonlinearity that is easily observable for modest me-chanical amplitudes. The nonlinearity provides amode–mode coupling, analogous to photon–photon cou-pling in nonlinear dielectrics, and drives instabilities. Thisnonlinearity has already been put to effective use for bothparametric mechanical amplification and square-law me-chanical detection, as described below.

The challenge of motion transductionA prerequisite for attaining the ultimate potential fromnanomechanical devices is displacement sensing—that is,reading out the NEMS motional response induced by anapplied stimulus. Most typically, this requirement distillsto transduction, or conversion, between the mechanicaland electrical domains: Ultimately one wants a voltage sig-nal that provides the time record of the NEMS response.Many displacement-sensing techniques for mechanical de-vices have been proposed and demonstrated (see figure 3);perhaps not surprisingly, displacement transducers that

∀ ¬ ⊂ ⊂E N\w \wth12

1e\w /k TB 1⊗( (⊕ ,

38 July 2005 Physics Today http://www.physicstoday.org

100

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19.656 19.658 19.660 19.662 19.664FREQUENCY (MHz)

tv

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TN

= 73 mK= 75th

a b

Figure 2. Quantum limits. (a) The occupation factor Nth (black curves) of various mechan-ical resonator frequencies is a function of resonant frequency and temperature T. Shownin red is the lifetime tN of a given number state for a 10-MHz resonator with quality factorQ ⊂ 200 000 (recently demonstrated at the Laboratory for Physical Sciences).11 Also inred is the expected decoherence time tv for a superposition of two coherent states in thatresonator displaced by 100 fm. (b) The measured noise-power spectrum of the thermalmotion (black line, with a Lorentzian fit in red) atop the white noise (blue baseline) of theposition detector. The curve corresponds to the green point in panel a, with T ⊂ 73 mKand Nth ⊂ 75. These data show the closest approach to date to the uncertainty-principlelimit: The detector noise gives a displacement sensitivity a factor of 5.8 from the quantummechanical limit.

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are important for microscale devices do not prove optimalfor nanoscale devices. For example, such optical methodsas fiber-optic interferometry and reflecting off a cantilever(the so-called optical lever) have been used extensively inMEMS, especially on commercially available scanningprobe microscopes. However, because nanomechanical de-vices are small compared to the wavelength of light, dif-fraction dramatically complicates such approaches. Near-field optical methods may play an important future role.But optical adsorption and the resulting heating of the me-chanical structure prove problematic for the most sensi-tive regimes. So it appears that optimal coupling at thequantum limit may be difficult to achieve optically. Nev-ertheless, optical techniques have recently enabled the de-tection of microcantilever motion induced by a single elec-tron spin.3

Piezoresistive displacement transduction has foundwidespread applications in larger micron-scale devices.Use of this technique for nearly quantum-limited detectionof NEMS devices, however, does not appear to be straight-forward. The dissipation involved will cause appreciableheating of tiny devices, which will likely preclude opera-tion at ultralow temperatures.

One technique that has been immensely successful ismagnetomotive detection, which utilizes the electromotiveforce generated by a metallic conductor (typically a metalli-zation layer) affixed to a mechanical device that moves ina large magnetic field—on the order of several tesla. Thistechnique is especially well suited for low-temperaturemeasurements for which the imposition of large magneticfields is commonplace; it has been employed to monitor thehighest-frequency devices measured to date, with funda-mental flexural modes exceeding 1 GHz.1 Unfortunately,magnetomotive motion detection becomes increasinglyawkward at higher frequencies. The effective electrical im-pedance, which arises from electromechanical coupling tothe motional part of the device, scales inversely with fre-quency and hence tends to be swamped by parasitic, staticcircuit impedances as the frequency increases into thegigahertz domain. But even at lower frequencies, at hun-dreds of megahertz, observing the thermal fluctuations ofa nanomechanical resonator by this technique has so farproven elusive—the intrinsic noise of state-of-the-art cryo-genically cooled amplifiers is too high to surmount thebugaboo of insufficient mechanical-to-electrical transduc-tion efficiency. The problem is generic for most motion-transduction methods at microwave frequencies.

Using parametric amplification, the Caltech grouphas demonstrated an attractive way to work around thisproblem. They employ the Euler instability—the nonlin-earity that causes a beam under compression to buckle—to realize a high-frequency amplifier that works entirelyon mechanical principles. Their device, depicted in figure1d, provides stable gain and up to a thousandfold dis-placement amplification in response to a weak stimulusforce at 17 MHz. With sufficient gain in the mechanical do-

main, the challenge of displacement transduction becomessubstantially easier and thermomechanical fluctuations ina cryogenically cooled device have indeed been observedwith such an amplifier.

Some of the most exciting recent transduction effortsare focused on coupling high-frequency nanomechanicalsystems to various nanoelectronic and mesoscopic devicesthat serve essentially as integrated amplifiers. Examples arequantum point contacts, quantum dots, or single-electrontransistors (SETs) used in configurations where the mo-tion of a nanomechanical device modulates the electrontransport properties. Among these readout strategies, theSET—shown by many research groups to be a very sensi-tive detector of charge—has now enabled nearly quantum-limited position detection for NEMS devices. Figures 1aand 1e each show a charged nanomechanical resonatorcapacitively coupled to an SET. The resonator’s motioninduces a change in the charge on the gate electrode of theSET; the resulting change in the SET’s conductance can bedirectly monitored. Careful consideration of noise sourcesindicates that the SET can provide motion detection withsensitivity down to the quantum limit.

Robert Knobel and Andrew Cleland at the Universityof California, Santa Barbara,10 were the first to demon-strate position detection using an SET. They used the de-vice as a narrowband mixer—with a bandwidth of only100 Hz—and detected the flexural resonance of a 100-MHzdevice. Their effort yielded continuous position detection

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B

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Figure 3. Detection techniques for nanomechanical dis-placement. (a) An optical detection scheme that is typicallyused in atomic-force microscopy but that does not workwell for NEMS. Light from an optical fiber reflects off theresonator and returns up the fiber. (b) A magnetomotivetechnique that measures the electromotive force generatedwhen the metal layer on top of the resonator moves in themagnetic field B. (c) Coupling to a mesoscopic detectorsuch as a quantum dot, a quantum point contact, or a single-electron transistor. The current IDS through the detector ismodulated by the NEMS motion.

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to within a factor of about 100 of the quantum limit, buttheir detection bandwidth was very limited.

Recently a group led by one of us (Schwab) at the Lab-oratory for Physical Sciences has shown that a radio-frequency SET can simultaneously provide both largebandwidth (75 MHz) and ultrasensitive detection.11 Thegroup directly detected the motion of a 20-MHz resonator(figure 1a), which had a Q of 50 000, and demonstratednanomechanical measurements only a factor of about 6from the quantum limit, the closest approach to date forany position measurement (see figure 2b). With such posi-tion sensitivity, they observed random vibrations driven bythe thermal energy at an effective temperature as low as60 mK. The experiment essentially constituted millikelvinnoise thermometry on a single mechanical degree of free-dom. At the experiment’s lowest temperature, the thermaloccupation factor Nth for the mechanical mode was on theorder of 58; that low value indicates that the quantumground state (Nth < 1) of NEMS devices is within reach.Necessary steps toward achieving that goal include opti-mized transduction at higher resonant frequencies and im-provements in thermalization, that is, the cooling of themechanical mode.

Cooling is less straightforward than one might ini-tially assume. Precisely in the regime where device modesbecome frozen out (and quantum effects begin to emerge),thermal conductance quantization (see PHYSICS TODAY,June 2000, page 17) imposes limits on the rate at which ananoscale device can thermalize to its environment.12

Work-arounds are few. One can perhaps attempt to engi-neer the modes mediating thermal contact—to optimizeheat transfer while preserving the quantum nature of thespecific mechanical mode under study—but it is not clearhow practical this approach will be. Perhaps a more prom-ising alternative is active cooling of the mechanical modethrough controllable external interactions.13 For example,using very low-noise temperature detection to provide op-timal feedback, it should be possible to actively cool themechanical mode close to the ground state. The prospectsare reminiscent of the strategy taken with trapped atoms:Laser-cooling of samples into a low-energy state allows forvarious quantum measurements before the atoms begin towarm up from their interaction with the environment.

Ultimately, molecular mechanical systems, assembledwith atomic precision, will subsume today’s nanomechan-ical systems patterned by top-down methods. Devicesbased on single-wall carbon nanotubes offer very excitingpossibilities because nanotubes are electronically activeand naturally form nanoelectronic devices such as single-electron transistors. Exploiting the strain dependence ofelectron transport through a carbon nanotube, PaulMcEuen’s group at Cornell University has recently de-tected the vibrational mode spectrum of a suspended nano-tube (see figure 1f).14 Although the “bottom up” fabricationis today still an art rather than a technology, nanotube and nanowire NEMS offer great promise for achievingboth very high resonant frequencies and sensitive force detection.

Uncertainty-principle limits on position detectionHow precisely can one continuously measure the position ofan object? This basic question is of central importance formechanical force detection, such as in atomic-force mi-croscopy. It is clear that quantum mechanics should placelimits on the ultimate answer, since it’s possible to constructfrom continuous measurement records both the mechanicalresonator’s position and its momentum. One should not ex-pect to be able to violate the uncertainty principle.

To continuously record the position, one must couple

the mechanical device to some form of linear detector;practically, the detector would take the form of a displace-ment transducer (such as one that linearly converts posi-tion to voltage) and an amplifying device. During the de-velopment of the maser in the 1960s, it became clear thatthe performance of any linear amplifier is limited by theuncertainty principle. In the context of position detection,the ultimate sensitivity was clarified by Carleton Cavesand colleagues6: Even for optimal engineering, the meas-urement record is obscured by an equal contribution ofquantum noise from the mechanical resonator plus itstransducer and from the subsequent linear amplifier usedas a readout. With the resonator at zero temperature, theminimum possible variance of the position record is

Two distinct forms of noise arise from the readoutprocess and affect nanomechanical measurements. Thefirst, measurement noise, is independent of the resonatorand is added to the output signal. In practice, this contri-bution may take the form of electrical or photon shot noise.The second is the somewhat more subtle back-action noise,which emanates from the readout and drives the resonatorstochastically. Back-action noise might arise from fluctuat-ing potentials or currents that are, in turn, converted by thetransducer into physical forces subsequently imposed on themechanical device. Electrical engineers are familiar withthese two forms of noise from linear amplifiers and repre-sent them as voltage and current noise, respectively. Opti-mally engineered coupling between the resonator and theamplifier is achieved when the two amplifier noise contri-butions are equal. But very few linear amplifiers, even whenoptimized in this way, can approach the quantum limit. Ac-cordingly, reaching the quantum limit for position detectioninvolves choosing an amplifier that is capable of quantum-limited detection in principle, and that can be optimally cou-pled to the mechanical resonator in practice.

Coupling nanomechanics to quantum systemsA large effort in condensed matter physics today is focusedon investigating the coherent quantum mechanical be-havior of individual solid-state devices. That researchbegan with mesoscopic physics in the 1980s and continuestoday, motivated by interest in quantum information andcomputation. Beautiful and convincing quantum behav-ior—quantized energy, coherent evolution, superpositionstates, and entanglement—has been observed with vari-ous single-electron devices, SQUIDs, and quantum dots. Agrowing number of researchers now believe that we havethe necessary tools to observe similar behavior in smallmechanical structures.

To explore quantum effects in NEMS, it is essential tomove beyond continuous measurement and linear cou-pling: Continuous linear measurements cannot distin-guish between classical and quantum oscillators. Thequantum properties of the most heavily studied simpleharmonic oscillator—the modes of the electromagneticfield—have been revealed by coupling to a nonlinear de-tector such as a photon counter (for energy detection) or toa coherent two-level system, as in a single atom in an elec-tromagnetic cavity.

Analogous measurement concepts in nanomechanicsare starting to take shape. Reviewing the experimentalstatus of quantum nondemolition measurements in 1996,Braginsky and Khalili wrote that “no experimentalscheme had been proposed so far” for measuring the en-ergy of a mechanical resonator.7 The Caltech group has set

\mw ln 3

.DxQL⊂ 1.35 xSQLD= �

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about to change that; recently they demonstrated how tobuild a mechanical square-law detector—whose output isproportional to the square of the position coordinate—analogous to Braginsky and Khalili’s energy detector. Ex-perimental work on this front is under way, with theoret-ical investigation led by researchers at Caltech and theUniversity of Queensland.15

The detection of a single electron spin by Rugar’s IBMgroup is the first realization of a single quantum two-levelsystem coupled to a mechanical mode.3 This success demon-strates both that the coupling between the cantilever andspin can be strong enough to produce measurable displace-ments and that the coherence of the spin is sufficient to per-form many coherent manipulations, even while the spin iscoupled to the resonator. A positive-feedback scheme ap-pears possible: The coherent back action of the cantileveronto the spins can directly stimulate the transition of morespins and give rise to amplified cantilever oscillations, a me-chanical lasing scheme dubbed the cantilaser.16 This pro-posed technique has the advantage that the dynamics of thecantilever–spin system is not limited to the mechanical re-laxation time tm ⊂ Q/w0, and it may prove useful in the fastmechanical detection of spin ensembles.

The quantum system formed by capacitively coupling aNEMS resonator to a solid-state qubit has garnered perhapsthe most attention from theorists. Andrew Armour, MilesBlencowe, and one of us (Schwab) first described one suchsystem, a NEMS coupled to a Cooper-pair box.17 The CPB isa superconducting device in which both the electrostaticcharging energy of a small conducting island and theJosephson coupling energy are important in the coherentquantum dynamics of the device’s charge states. For certainbias configurations, the CPB is well described by a simpletwo-level qubit Hamiltonian in which the island’s capaci-tance determines the spacing between the levels +0¬ and +1¬,which have zero and one extra Cooper pair, respectively; theJosephson energy determines the transition rate betweenthe two charge states. Experiments by a number of re-searchers have demonstrated CPB eigenstates with excited-state lifetimes of up to 7 ms and coherence times of super-position states as long as 0.5 ms.

The CPB and the mechani-cal resonator interact simplythrough the electrostatic forcemediated by the charge on thebox. The coupling strength canbe comparable to the resonatorquantization energy for reason-able parameters. Adding in thesingle-mode harmonic-oscillatorHamiltonian that describes themechanical resonator, the sys-tem is directly analogous to cav-ity quantum electrodynamics. Infact, for certain CPB biases, thesystem is described by a Hamil-tonian (the so-called Jaynes–Cumming Hamiltonian) widelyemployed throughout atomicphysics. Various groups explor-ing this analogy have concludedthat the NEMS–CPB systemshould allow for experimentssuch as number-state detection,quantum nondemolition meas-urements, and a resonator cool-ing process analogous to lasercooling of atoms.

These simple descriptions ofcoupled quantum systems suggest an intriguing possibility,first described by Armour and colleagues.17 The interactiondisplaces the resonator according to the charge state of theCPB: The device moves to the left if the CPB is in the +0¬charge state, and moves to the right if the CPB is in the +1¬charge state. This coupling raises the simple question,Which way does the mechanics move if one prepares theCPB into the superposition state (+0¬ ⊕ +1¬)/=2+? If the me-chanics truly is quantum, then naively one expects the res-onator to displace in both directions and ultimately form anentangled state with the qubit (see figure 4).

To design and fabricate experimental devices that allowfor such observations, at least two criteria need to be satis-fied. First, the coupling between the qubit and the resonatormust be sufficient so that the difference in displacementgenerated by the two qubit states should be greater thanDxSQL. Meeting this condition will ensure that an entangledstate is formed with two nearly orthogonal positions of theresonator. Second, the decoherence time of the resonatormust be larger than the measurement time. The decoher-ence time of a state formed by a superposition of two co-herent states displaced by a distance Dx and coupled to athermal bath modeled by simple harmonic oscillators wascalculated by Wojciech Zurek and colleagues at Los AlamosNational Laboratory in 1992. Their result,

is plotted in figure 2a. For a very small displacement of thetwo superposition states, Dx ⊂ 3 DxSQL ⊂ 100 fm, for whichthe states are still quantum mechanically distinct, tv is ex-pected to be surprisingly long—greater than 1 ms at 100 mK.For an entangled state with Dx comparable to the width ofa typical resonator, 100 nm, tv is estimated to be 10⊗18 s!Thus Schrödinger-cat states formed by a mechanical res-onator coupled to a qubit appear to be feasible, although thedistance between these states would be subatomic.

Illuminating the quantum worldNanomechanical resonators will be increasingly useful for ex-plorations of quantum mechanics, whether as ultrasensitive

\k TB

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P x( )

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TIME

Figure 4. Quantum superpositions may be possible with a quantum two-level sys-tem (TLS) coupled to a nanomechanical resonator. In this schematic, the flexure ofthe cantilever depends on whether the TLS is in the down state +A¬ (red) or the upstate +R¬ (blue). The plot shows the probability P of finding the resonator center-of-mass at a given position x for different times after coupling the resonator to the TLS.If the TLS is prepared in a superposition state, an entangled state is formed and theresonator will be in both locations.

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probes of quantum and mesoscopic forces, as detectors ofsingle quantum systems, as “buses” in quantum informa-tion devices,18 or as devices to allow the study of quantumbehavior in ordinary bits of matter. The construction of ascanning microscope that can detect single nuclear spinsremains a grand challenge of nanomechanics. Success inthese areas and others will require highly engineered me-chanical structures that routinely operate near quantummechanical limits.

The generation and detection of the uniquely quan-tum states of a small mechanical device, such as energyeigenstates (so-called Fock states), superposition states, orentangled states, are particularly interesting because themechanical structures may be considered “bare systems”:There is no macroscopic quantum condensate to protectthe device from excitation or decoherence. In supercon-ductors, superfluids, or Bose–Einstein condensates, thenumber of quantum states of a macroscopic sample is dras-tically reduced to only a few degrees of freedom. For in-stance, in a CPB containing billions of electrons, super-conductivity suppresses almost all the normal-statedegrees of freedom down to just two: number and phase.

The mechanical devices shown in figure 1 are com-posed of ordinary matter, full of defects and imperfections,and help answer the question: What does it take to observethe quantum nature of an ordinary system? The motiva-tion is similar to that behind the recent beautiful work ofAnton Zeilinger and colleagues at the University of Vi-enna, who have shown matter-wave interference with mol-ecules of increasing complexity and have studied decoher-ence by directly controlled interactions. In this vein, wehope experiments with nanomechanical devices at thequantum limit may further illuminate the boundary be-

tween the microscopic realm, governed by quantum me-chanics, and our macroscopic world, governed by classicalmechanics.

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