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Putting Mechanics Into Quantum Mechanics

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Putting Mechanics Into Quantum Mechanics Putting Mechanics into Quantum Mechanics Nanoelectromechanical structures are starting to approach the ultimate quantum mechanical limits for detecting and exciting motion at the nanoscale. Nonclassical states of a mechanical resonator are also on the horizon. Keith C. Schwab and Michael L. Roukes verything moves! In a world dominated by electronic achieved to read out those devices, now bring us to the Edevices and instruments it is easy to forget that all realm of quantum mechanical systems. measurements involve motion, whether it be the motion of electrons through a transistor, Cooper pairs or quasiparti- The quantum realm cles through a superconducting quantum interference de- What conditions are required to observe the quantum prop- vice (SQUID), photons through an optical interferome- erties of a mechanical structure, and what can we learn ter—or the simple displacement of a mechanical element. when we encounter them? Such questions have received Nanoscience today is driving a resurgence of interest in considerable attention from the community pursuing grav- mechanical devices, which have long been used as front itational-wave detection: For more than 25 years, that com- ends for sensitive force detectors. Among prominent his- munity has understood that the quantum properties of me- torical examples are Coulomb’s mechanical torsion bal- chanical detectors will impose ultimate limits on their force 6,7 ance, which allowed him in 1785 to establish the inverse- sensitivity. Through heroic and sustained efforts, the square force law of electric charges, and Cavendish’s Laser Interferometer Gravitational Wave Observatory mechanical instrument that allowed him in 1798 to meas- (LIGO) with a 10-kg test mass, and cryogenic acoustic de- ure the gravitational force between two lead spheres. tectors with test masses as large as 1000 kg, currently Today, micro- and nanoelectromechanical systems achieve displacement sensitivities only a factor of about 30 (MEMS and NEMS) are widely employed in ways similar from the limits set by the uncertainty principle. to those early force detectors, yet with vastly greater force But the quantum-engineering considerations for me- and mass sensitivity—now pushing into the realm of chanical detectors are not exclusive to the realm of gravi- zeptonewtons (10⊗21 N) and zeptograms (10⊗21 g). These ul- tational-wave physics. In the introduction to their pio- traminiature sensors also can provide spatial resolution at neering book on quantum measurement, Vladimir the atomic scale and vibrate at frequencies in the giga- Braginsky and Farid Khalili envisage an era when quan- hertz range.1 Among the breadth of applications that have tum considerations become central to much of commercial 7 become possible are measurements of forces between in- engineering. Today, we are approaching that time— dividual biomolecules,2 forces arising from magnetic reso- advances in the sensitivity of force detection for new types nance of single spins,3 and perturbations that arise from of scanning force microscopy point to an era when me- mass fluctuations involving single atoms and molecules.4 chanical engineers will have to include \ among their list The patterning of mechanical structures with nanometer- of standard engineering constants. scale features is now commonplace; figure 1 and the cover Several laboratories worldwide are pursuing mechan- display examples of current devices. ical detection of single nuclear spins. That goal is espe- The technological future for small mechanical devices cially compelling in light of the recent success of Dan clearly seems bright, yet some of the most intriguing ap- Rugar and colleagues at IBM in detecting a single electron plications of NEMS remain squarely within the realm of spin with a MEMS device (see figure 1b and PHYSICS 3 fundamental research. Although the sensors for the appli- TODAY, October 2004, page 22). However, nuclear spins ⊗21 cations mentioned above are governed by classical physics, generate mechanical forces of about 10 N, more than the imprint of quantum phenomena upon them can now 1000 times smaller than the forces from single electron be readily seen in the laboratory. For example, the Casimir spins. One ultimate application of this technique, struc- effect, arising from the zero-point fluctuations of the elec- tural imaging of individual proteins, will involve millions tromagnetic vacuum, can drive certain small mechanical of bits of data and require measurements to be carried out devices with a force of hundreds of piconewtons and pro- not on the current time scale of hours, but over microsec- duce discernible motion in the devices.5 But it is now pos- onds. Detecting such small forces within an appropriate sible, and perhaps even more intriguing, to consider the measurement time will necessitate new quantum meas- intrinsic quantum fluctuations—those that belong to the urement schemes at high frequencies, a significant chal- mechanical device itself. The continual progress in shrink- lenge. Yet the payoff, originally envisaged by John Sidles, ing devices, and the profound increases in sensitivity will be proportionately immense: three-dimensional, chemically specific atomic imaging of individual macro- Keith Schwab is a senior physicist at the National Security molecules.8 Agency’s Laboratory for Physical Sciences in College Park, The direct study of quantum mechanics in micron- and Maryland. Michael Roukes is Director of the Kavli Nanoscience submicron-scale mechanical structures is every bit as at- Institute and is a professor of physics, applied physics, and bio- tractive as the actual applications of MEMS and NEMS.9 engineering at the California Institute of Technology in Pasadena. With resonant frequencies from kilohertz to gigahertz, low 36 July 2005 Physics Today © 2005 American Institute of Physics, S-0031-9228-0507-010-9 a b Figure 1. Nanoelectro- mechanical devices. (a) A 20-MHz nanome- chanical resonator ca- pacitively coupled to a single-electron transistor (Keith Schwab, Labora- 2mm tory for Physical Sci- ences).11 (b) An ultrasen- sitive magnetic force detector that has been used to detect a single c d electron spin (Dan Rugar, IBM).3 (c) A tor- sional resonator used to study Casimir forces and look for possible correc- tions to Newtonian grav- itation at short length scales (Ricardo Decca, Indiana University– Purdue University Indi- anapolis). (d) A paramet- ric radio-frequency me- chanical amplifier that provides a thousandfold boost of signal displace- ments at 17 MHz e f (Michael Roukes, Cal- tech). (e) A 116-MHz nanomechanical res- onator coupled to a single-electron transistor (Andrew Cleland, Uni- versity of California, W 10 Source Drain Santa Barbara). dz (f) A tunable carbon nanotube resonator op- erating at 3–300 MHz (Paul McEuen, Cornell Gate University).14 dissipation, and small masses (10⊗15–10⊗17 kg), these devices A is the number of atoms in the structure. Knowing the are well suited to such explorations. Their dimensions not amplitude and phase of all the mechanical modes is equiv- only make them susceptible to local forces, but also make it alent to having complete knowledge of the position and mo- possible to integrate and tightly couple them to a variety of mentum of every atom in the device. Continuum mechan- interesting electronic structures, such as solid-state two- ics, with bulk parameters such as density and Young’s level systems (quantum bits, or qubits), that exhibit quan- modulus, provides an excellent description of the mode tum mechanical coherence. In fact, the most-studied sys- structure and the classical dynamics, because the wave- tems, nanoresonators coupled to various superconducting lengths (100 nm–10 mm) of the lowest-lying vibrational qubits, are closely analogous to cavity quantum electrody- modes are long compared to the interatomic spacing. namics, although they are realized in a very different pa- It is natural to make the distinction between nano- rameter space. mechanical modes and phonons: The former are low- frequency, long-wavelength modes strongly affected by the Quantized nanomechanical resonators boundary conditions of the nanodevice, whereas the latter The classical and quantum descriptions of a mechanical res- are vibrational modes with wavelengths much smaller onator are very similar to those of the electromagnetic field than typical device dimensions. Phonons are relatively un- in a dielectric cavity: The position- and time- dependent me- affected by the geometry of the resonator and, except in chanical displacement u(r,t) is the dynamical variable anal- devices such as nanotubes that approach atomic dimen- ogous to the vector potential A(r, t). In each case, a wave sions, are essentially identical in nature to phonons in an equation constrained by boundary conditions gives rise to a infinite medium. spectrum of discrete modes. For sufficiently low excitation It is an assumption that quantum mechanics should amplitudes, for which nonlinearities can be ignored, the en- even apply for such a large, distributed mechanical struc- ergy of each mode is quadratic in both the displacement and ture. Setting that concern aside for the moment, one can momentum, and the system can be described as essentially follow the standard quantum mechanical protocol to es- independent simple harmonic oscillators. tablish that the energy of each mode is quantized: 1 Spatially extended mechanical devices, such as those E ⊂ \w(N ⊕ /2), where N ⊂ 0, 1, 2, . is the occupation fac- in figure 1, possess a total
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