Free-fall experiments are the stuff of scientific legend. Galileo is said to have dropped cannonballs from the Leaning Tower of Pisa to demonstrate to his students that gravitational acceleration is independent of mass. And Isaac Newton often recounted the tale of the falling apple that inspired him to think about gravitational forces over longer distances. (There’s no evidence, however, that the apple landed on his head.)
Now Erik Hebestreit, his PhD adviser Lukas Novotny, and their colleagues at ETH Zürich have shrunk the apple drop to the nanoscale regime.1 By levitating a silica nanoparticle in an optical trap, as shown in figure 1, and briefly releasing it, they can measure the effect of gravity or any other static force that acts on the particle.
An optically levitated nanoparticle, visible here as a tiny green dot on the center line of the vacuum chamber, is the basis for a nanomechanical force sensor. (Photo by Rozenn Diehl and René Reimann, ETH Zürich.)
An optically levitated nanoparticle, visible here as a tiny green dot on the center line of the vacuum chamber, is the basis for a nanomechanical force sensor. (Photo by Rozenn Diehl and René Reimann, ETH Zürich.)
The measurement is sensitive to forces as small as 10 attonewtons, or half the weight of the 136 nm particle. (It’s also the gravitational force between two 80 kg people separated by 200 km.) Although the precision remains modest, with error bars on the order of 20% of the force being measured, the technique can be used to study the largely uncharted territory of short-range, static, nanomechanical forces.
Inspired by interferometry
The nanoparticles aren’t the smallest objects to have been observed in free fall. Two decades ago, Steven Chu and colleagues studied falling atoms in an atom interferometer.2 (See Physics Today, November 1999, page 20.) By capitalizing on the cooling and trapping techniques that had won Chu a share of a Nobel Prize (see Physics Today, December 1997, page 17), they measured the acceleration due to gravity with a precision of three parts in a billion—easily enough to observe the twice-daily oscillations due to Earth’s tides. (See the article by Markus Arndt, Physics Today, May 2014, page 30.)
In 2015 Andrew Geraci and Hart Goldman presented an idea for a similar matter-wave interferometer based not on an atomic cloud but on a solid nanosphere, whose delocalized wavefunction interferes with itself.3 According to their analysis, the interferometer would be sensitive to the gravitational attraction between the nanosphere and a nearby micron-sized object, an interaction that could reveal short-range deviations from Newton’s law of universal gravitation.
It was a bold proposal. To achieve such exquisite sensitivity, experimenters would have to cool the nanosphere into the quantum ground state of an optical trap—a feat that’s still outside current capabilities.
Nevertheless, the ETH Zürich researchers were inspired by the idea. If they could implement a nanoparticle free-fall experiment, even without cooling the particle to the ground state, they could clear some of the other experimental obstacles to realizing Geraci and Goldman’s proposal. And they could create a nanomechanical sensor that would be useful in its own right. “It was the next achievable step,” recounts Martin Frimmer, an author on the new paper.
Catch and release
The principle of the scheme, illustrated in figure 2, is simple to state. A nanoparticle is held in the center of a harmonic optical trap. The trap is turned off for a fraction of a millisecond, during which the particle moves under the static force F, which may or may not be gravity. When the trap is reactivated, the particle is no longer near the center, and it oscillates with a large amplitude, which the researchers measure with high precision.
In the force-sensing scheme, the nanoparticle begins (a) in the center of a harmonic optical trap with a small but nonzero energy E0. The trap is turned off (b), and the particle moves away from the trap center under a static force F. When the trap is turned back on (c), the particle oscillates with a greater energy E. The particle’s diameter and displacement are both on the order of 100 nm; the displacement is exaggerated here for clarity. (Adapted from ref. 1.)
In the force-sensing scheme, the nanoparticle begins (a) in the center of a harmonic optical trap with a small but nonzero energy E0. The trap is turned off (b), and the particle moves away from the trap center under a static force F. When the trap is turned back on (c), the particle oscillates with a greater energy E. The particle’s diameter and displacement are both on the order of 100 nm; the displacement is exaggerated here for clarity. (Adapted from ref. 1.)
The final oscillation energy E depends on both F and the fall duration τ. It also depends on the particle’s velocity v0 at the instant the trap is turned off. Because the initial oscillation energy E0 is small but nonzero, v0 is also generally nonzero—and unpredictable.
To minimize that source of uncertainty, the researchers repeat the drop-and-catch process thousands of times for several values of τ. If F = 0 and the particle’s movement is characterized solely by v0, then the average E (essentially the average squared displacement) is proportional to τ2. A nonzero F introduces a τ4 term; by fitting the data, the researchers extract F.
The nanomechanical world holds many surprises, however, and the net force that acts on the particle isn’t always the force one hopes to measure. For example, the usual means of rapidly switching a laser beam on and off is with an acousto-optic or electro-optic modulator. Mechanical shutters are far too slow. An AOM or EOM doesn’t switch the beam all the way off but rather attenuates it by a factor of around a thousand. The residual intensity in the optical trap is enough to overwhelm the measurement of a weak static force. To reduce the trapping power to an acceptably low level, the ETH Zürich researchers used an AOM and an EOM in series, switched on and off at the same time.
Electrostatic forces posed another technical challenge. By necessity, the optical trap is positioned right next to a glass lens. And glass, being a dielectric, is prone to harboring excess surface charge. The resulting electric field turned out to be surprisingly large: several hundred volts per meter, enough to exert a force of many tens of attonewtons on a particle with a single excess elementary charge.
The researchers found no practical way to eliminate that stray field. (Says Frimmer, “There’s really no such thing as ‘no field’ in the real world.”) But they did find a way to reliably discharge the nanoparticle: by exposing it to a swarm of charged particles in tandem with an AC electric field. When the nanoparticle stops oscillating under the driving field, its charge is zero, and the experiment is ready to proceed.4 As a proof of principle that their method could also measure applied electrostatic forces, the researchers endowed the nanoparticle with an extra electron, measured the force of the residual and applied fields, measured the force of the residual field, and subtracted.
Casimir–Polder and van der Waals
Plenty of force-sensing schemes, typically based on mechanical resonators, have far better sensitivity and precision than the nano-apple drop.5 But they’re limited to measuring time-varying forces. Some forces, such as gravity, are inherently static, and others are difficult to turn on and off.
Of particular interest is the crossover between van der Waals and Casimir–Polder dispersion forces. The van der Waals force between closely spaced atoms is a nonrelativistic quantum effect mediated by the exchange of virtual photons. For larger objects and greater separations, relativistic effects come into play, as analyzed by Hendrik Casimir and Dik Polder: The nonzero propagation time for the virtual photons becomes important, and the force scales differently as a function of distance. (See the article by Steve Lamoreaux, Physics Today, February 2007, page 40.) The boundary between the nonrelativistic and relativistic regimes is expected to be found on a size scale of tens to hundreds of nanometers—a scale that, until the ETH Zürich researchers’ work, had been stubbornly difficult to explore.
The researchers plan to study those dispersion forces by dropping a nanoparticle in close proximity to a vertical glass plate. By measuring the horizontal and vertical oscillations separately, they hope to determine whether the particle falls straight down or is attracted to or repelled from the plate. That experiment, however, poses yet another technical hurdle: “We’ll need to deal with the optical reflection off the glass,” says Frimmer. “But we’re learning how to do that.”
