It takes three keys to access a certain vault in Parc de Saint-Cloud, on the outskirts of Paris. One key is held by the president of the facility, one by the director of an external international committee, and the last by the Archives de France. Inside that vault, resting in a vacuum beneath three nested glass bells, lies a cylinder of platinum and iridium commissioned in 1889. This object’s sole distinction is the curious fact that its mass is unequivocally equal to one kilogram. In fact, as the international prototype of the kilogram (IPK), it is the very definition of the kilogram. It is a societal constant, the standard by which *everyone in the world* gauges mass. The only problem? The mass of the IPK itself is changing.

Three times in the last century, the IPK has been removed from the vault for careful comparative mass measurements. Each time, researchers were alarmed to find that its mass—and thus the definition of the kilogram—had changed. The proposed remedy, a redefinition of the mass standard, is part of an overhaul that within a few years may shake the very foundations of the *Système international* (SI) units used by scientists, engineers, and governments around the globe. To look into the significance of this locked-up lump of metal and its potential successors is to go from French Revolution-era politics to fundamental constants of nature, and to peer inside the quantum mechanical machinery that drives one of the world’s most unconventional clocks right here at the University of California, Berkeley.

### A universal language

A system of units is a set of rules defining what numerical value we assign to some measured quantity, giving a standard way of comparing two objects. As any cook, chemist, baker, or builder can tell you, units matter. This author once baked an abysmal cake by using metric (Commonwealth) cups to measure out flour for a recipe written in customary (US) cups. In 1999, the Mars Climate Orbiter was destroyed as it entered the atmosphere of Mars, when a thrust calculation in US customary units (pound-force) was handed off to a program that worked in metric units (kilogram-meter per second squared). From the kitchen to outer space, failure to use a consistent unit system can have dire consequences. This is not a new idea; the need for “just balances, just weights” is expressed in the Book of Leviticus. As Barry Taylor, scientist emeritus at the National Institute of Standards and Technology (NIST) in Gaithersburg, Maryland, explains: “You have to have a universal language with which to express values… the SI units are that universal language.”

The SI units consist of seven defined base units: length, time, mass, electric current, temperature, amount of substance, and luminous intensity. A host of derived units can be formed by combining the base units in myriad ways. Many of these derived units have their own names; for example, the unit of force, a kilogram-meter per second squared, is called a Newton. Although the SI was formally adopted in 1960, its origins stretch back to the reign of King Louis XVI of France, who appointed a committee to produce a new system of units.

In 1789, however, the French Revolution interrupted his reign, his life, and his committee on standards of measurement. Originally, the mass unit, defined by the mass of one liter of water at zero degrees Celsius, had been given the name “grave.” However, the post-revolutionary government deemed this unit of mass inconvenient to ordinary people, who needed to measure comparatively small quantities of everyday goods such as flour and salt. Thus, in revolutionary spirit, one-thousandth of a grave was given its own name and christened “gram,” while the mass standard was left awkwardly prefixed as “kilogram.” In 1875, the US was one of 17 countries to sign a treaty in Paris that established the Bureau International des Poids et Mesures (BIPM) to oversee global standardization of units. Definitions of base units were laid down; for example, the second was defined as the average length of a day divided by 86,400. The kilogram and the meter were defined by the mass and length, respectively, of two platinum-iridium pieces created at the behest of the BIPM. These were carefully stored at the BIPM facilities outside of Paris, where they have remained for more than a century.

### Massive troubles

The original standards, however, were flawed. The earth (whose rotation determines the length of a day) and pieces of metal are complex systems of large numbers of interacting atoms. As such, these standards are bound to change over time. What’s wrong with a little uncertainty in the foundation of the SI? Scientists rely on precise measurements, and increasingly, so do manufacturers. Goods such as electronics, pharmaceuticals, and nanomaterials demand high-precision production—a BIPM scientist noted that engineering tolerances shrank by a factor of almost thirty over the last thirty years of the 20th century. Some communications systems demand accuracy better than one part in one trillion. Is there a more stable foundation for a system that plays such an integral role in science and commerce?

The laws of physics dictate that certain quantities (such as the speed light travels in a vacuum, or the charge of an electron) are fixtures of the universe. These constants of nature provide a natural, consistent scale for measuring other quantities. Some constants are so ubiquitous in the underlying physical theories that they are distinguished as fundamental constants; others, not cornerstones of theory but nonetheless observed in nature with unfailing consistency, are called atomic constants.

For example, in a phenomenon known as hyperfine transition, an electron in an atom can jump to a slightly lower energy state by emitting a bit of electromagnetic radiation (it will be referred to as light, though it is not necessarily visible light). The energy contained in the emitted light is an atomic constant, exactly equal to the energy difference between the two states. It doesn’t matter whether we measure this energy in our backyard, Timbuktu, or Alpha Centauri: physics tells us to expect the same answer.

How do we measure this energy? Imagine standing on an ocean buoy and counting how many wave peaks pass in a second. The number of crests per second is called the frequency. Light, too, is a wave, so we can count how many crests go by per second (if we’re very quick). It turns out this frequency tells us how much energy the light wave carries: take the frequency, and multiply by a fundamental constant called the Planck constant; the result is the energy in the wave of light. For example, low-frequency radio waves carry far less energy than high-frequency cosmic rays. The upshot: by way of a fundamental constant (the Planck constant) there is a way to talk about how quickly something happens (frequency) in terms of an atomic constant (the hyperfine transition energy).

In 1967, the second was redefined as the time it takes for the passage of 9,192,631,770 peaks of the light wave emitted from a cesium-133 atom undergoing a hyperfine transition. Importantly, this definition was tied only to constants of nature, bypassing the inherent variability of a large physical object like the earth. Why does cesium-133 get the honor of defining the unit of time? History plays a role here, too. The frequency of light emitted by the hyperfine transitions for cesium-133 is very close to a radar frequency used in World War II. When the definition was adopted, equipment for measuring light at this frequency was readily available, making the cesium transition an accessible reference constant.

Over time, other SI units were redefined in terms of fundamental and atomic constants. In 1983, the meter standard in the BIPM was retired, and the unit of length was given a modern physics makeover. A meter is now defined to be “length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second.”

However, by the beginning of the 21st century, the SI standard of mass remained stubbornly in the 19th. The kilogram is not only defined as the mass of a 120-year old piece of metal, but as the mass of that piece of metal immediately after being cleaned a particular way. Perhaps it was not surprising, then, that the mass of the IPK was found to be drifting. Five “secondary” prototypes had been created with mass identical to the IPK, and their measurements show that after the last century, they now outmass the IPK by 50 micrograms.

Perhaps the secondary standards have gained mass, or perhaps the IPK has lost mass. The platinum-iridium blend was carefully chosen to be non-reactive and non-corrosive, but metrologists (scientists who study measurement) were concerned that it can gain mass by absorbing mercury, or lose it in a process called “outgassing.” NIST scientist Barry Taylor points out that we only know how the masses of the prototypes drift relative to each other—it is possible that the masses of all the prototypes are changing ten or 100 times more than the measured differences between them. Since the kilogram shows up in the definition of three other SI units (the candela, the mole, and the ampere), the instability of this single physical object could meddle with many scientific measurements: mass, electric current, luminance, and a great many derived units. The state of affairs was alarming enough to metrologists (those who study measurement) that a resolution was passed in 1999 to tie the definition of mass to a fundamental constant.

In 2011, a proposed overhaul of the SI unit system offered a concrete alternative to the precariously drifting IPK mass standard. The “New SI,” which may be formally adopted within the next ten years, would fix exact values of seven fundamental or atomic constants, and define the base units in terms of these constants. “The current SI can be viewed as based on seven ‘invariant’ quantities or constants,” Taylor explains. This is already true for some constants—the speed of light, for example, takes an exact value in the current system. The very definition of the unit of distance implies that the speed of light is precisely 299,792,458 meters per second. Comparing the unwavering value of the speed of light to the mercurial mass of the IPK suggests that while standards all are invariant, some may be more invariant than others. As such, Taylor continues, “…the proposed change is in four of the seven constants.”

This formulation is appealing in that immutable constants can be assigned immutable, known values, which then form a solid foundation for the system of units. Taylor points out the downside: some of the definitions may be harder to understand. Mass is a perfect example. In the new framework, the unit of mass will be linked to the exact value of a number that describes the energy carried by light: Planck’s constant. To understand how this works, step inside the laboratory of Holger Müller, assistant professor of physics at the University of California, Berkeley, who measures time with an unusual device called the Compton clock.

### A rock is a clock

Recall that to measure one second, we can measure the time it takes for a specific number of peaks of a certain wave of light to go by, assuming the light was emitted by a hyperfine transition in cesium. Müller, along with graduate students Brian Estey, Pei-Chen Kuan, and collaborators, measure the frequency of cesium atoms as well. However, rather than measuring the light emitted from cesium, they measure the frequency of the atom itself. This may sound like an impossible task for a particle, but a fascinating law of physics suggests that any physical object, be it infinitesimal atom, human body, or even earth itself, can also be thought of as a wave. Recall that a wave has an associated frequency—for light waves, you can picture this as the number of peaks passing by per second. There is no such simple picture for the frequency of a wave associated with an atom, but the value of this so-called “Compton frequency” does exist for any given atom. The Compton frequency of a cesium atom is large—about a quadrillion (10^{15}) times larger than the frequency of emitted light used to define the second. As a highly-reproducible trait of atoms, it is an atomic constant that could be used to redefine the standard unit of mass. However, observing this quantity and using it to define a unit of mass poses quite a few challenges.

To explain how the Compton frequency and mass can be related, Müller draws on an analogy originally crafted by physicist Richard Feynman to explain calculations in quantum electrodynamics:

Represent each particle with a little analog clock with an hour hand that spins around and around. If the particle under consideration is an atom, its hour hand spins much faster than that of an ordinary clock. The number of complete rotations the hour hand makes per second is called the Compton frequency (in an ordinary clock, the frequency would be 1/3600, since the hour hand goes around once every 3600 seconds). The heavier the particle, the faster the hour hand spins, for the Compton frequency is equal to the particle’s energy, E, divided by the Planck constant. Since Einstein’s equation asserts that E = mc^{2}, the Compton frequency can be expressed as the particle’s mass multiplied by the speed of light squared and divided by the Planck constant.

This formula connects time and mass through two fundamental constants (which are fixed in the new SI). The trouble is that when you multiply something by the speed of light squared (a very large number), and divide by the Planck constant (a very small number), the result is an astoundingly large number. A cesium atom in Müller’s lab is represented by a clock with the hour hand whizzing by at 30,000,000,000,000,000,000,000,000 (3×10^{25}) rotations per second.

This frequency is much too large to measure directly, so one must use a spooky fact of quantum mechanics: particles often don’t have a single, known location in the world, so much as possibilities of occupying many locations. A recent *New York Times* blog post argues that the theory of quantum mechanics should be thought of as an “oddsmaker.” The equations of the theory give you probabilities for the location of a particle, not concrete predictions. To work with the theory is to have a bookie that goes around putting odds on where a particle turns up.

While we can’t control the outcome of an individual measurement, it may be possible to fix the odds, because we know the bookie’s rules. Suppose that a particle can take one of two paths, labeled A and B, in order to arrive at a detector. The rules for calculating the chance of finding the particle at the detector are as follows: take two clocks spinning at the Compton frequency of the particle. Take the first clock and send it along path A and note the position of the hour hand when the clock reaches end of the path. Do the same with the other clock, only move it along path B. If the two recorded clock hand positions point in the same direction, then the detector will find the particle. In fact, the closer the alignment of the clock hands, the better the odds of finding the particle. [Click here to view an interactive simulation.]

For example, assume that when the clock along path A arrives at the detector, its hour hand points to 9 o’clock. Now, consider two scenarios for the clock moving along path B: in one, it arrives with its hour hand pointing at 8 o’clock, very close to the position of clock A. In this case, the closely-aligned clock hands correspond to a large chance of finding the particle at the detector. In the second scenario, suppose instead that the clock along path B arrives with its hour hand at 3 o’clock. In this case, the clock hands point in opposite directions, and the calculation tells us there is no chance of measuring the particle!

In other words, the difference between the clock hands tells us everything we want to know about the probability of finding a particle. This phenomenon is known as interference, and is measured in Müller’s lab using a device called an atomic interferometer. When the clock hands are aligned, a phenomenon called constructive interference occurs, which makes the particle more likely to be seen. When the hands point in opposite directions, destructive interference occurs, and the odds of finding the particle are very slim.

The challenge, then, is getting the clocks to disagree in a precisely controlled way, which we can do by leveraging a famous result from the theory of relativity: moving clocks run more slowly. The faster the clock moves in space, the slower it ticks. Although we are accustomed to thinking about an atom as a particle, sometimes the spooky here-and-there-at-once property of quantum mechanics allows us to think of an atom as a wave. Müller’s lab uses two carefully calibrated laser beams to shoot beams of light at a cesium atom, splitting its wave in two. Half the wave travels up a long tube before it is reflected back down to meet up with the other half. [Click here to view another interactive simulation.]

Müller’s group then measures the interference between the moving part of the wave and the stationary part—effectively, the difference between the corresponding clock hands. Since the speed of the moving clock is known from the calibration of the lasers, we can calculate how much it slows down. It’s then a simple matter to solve for the rate at which the original clock ticks—the Compton frequency. One can then use the relationship between mass and frequency to calculate the mass of the cesium particle.

In other words, the mass of any physical object could be related to time via the experiment above (though it is much simpler with atoms). As Müller likes to summarize: “a rock is a clock!” He borrowed the phrase from a skeptic who had sternly stated that “a rock is not a clock.” The Compton clock makes it possible to express the mass of a cesium atom in terms of a time measurement and the Planck constant—the faster the “clock” ticks, the more mass the particle has. With this definition in hand, one can redefine the standard unit of mass by counting out lots and lots of atoms and adding up their masses until you get a whole kilogram.

### Building blocks: The kilogram as a sum of silicon spheres

Unfortunately, the number of atoms that would sum to one kilogram is impossibly large. How could a scientist count so many things in a reasonable amount of time? This final step is being tackled by the Avogadro Project, an international collaboration that produced two spheres of silicon crystal in which the number of atoms can be counted very accurately.

Imagine that you are building a skyscraper out of sugar cubes. If you compute volume of the finished structure, and you know the volume of each individual sugar cube, then dividing the first number by the second lets you figure out the number of sugar cubes in the structure without actually taking the trouble to count them one by one. This is basic idea behind the Avogadro Project. Eight silicon atoms form a crystal in the shape of a cube with a known, constant volume. Scientists can arrange many of these cubes so that they form a sphere large enough to measure directly. Now, by precisely measuring the volume of the sphere (a much easier task), they then must simply divide this by the volume of each cube to compute the total number of cubes (and thus the total number of atoms).

### Weighing in on mass standards

The Avogadro project is an incredibly clever use of basic geometry to connect the vastly different mass of a microscopic object (an atom too small to notice without special equipment) to that of a macroscopic object (a sphere large enough to hurt if dropped on your toes). However, it does involve an extra step of effort: building the Avogadro sphere and measuring its volume. A competitor of the Compton clock, called a watt balance, attempts to cut out the extra step. The watt balance uses an electric current to lift a macroscopic object, relying on two rather complicated quantum effects to determine its exact mass in terms of the Planck constant. In Barry Taylor’s view, the future of mass standards lies in watt balances rather than the Compton clock. Estey views the Compton clock as more a “proof of concept” experiment rather than a feasible way to measure mass. “We like the Compton clock because it demonstrates how fundamental it [the relationship between mass and the Planck constant] is, ” he says.

Müller, however, is optimistic that the Compton clock will be used to realize the mass standard. Although the uncertainty in the kilogram determination is comparable between the Compton clock and the watt balance, the world has only completed two watt balance experiments. They do not, at present, agree with each other. The watt balance is finicky—so sensitive that “if you run it now and run it an hour later, you’ll get a different result because the moon has moved,” he avers. Furthermore, the Compton clock offers less uncertainty on atomic scales—precisely where, Müller argues, you need it most. “Macroscopic masses are not all that well defined. They change their mass all the time,” he points out. The IPK, shedding at least 50 micrograms a century even in a carefully sealed vault in Paris, corroborates.

### Not your grandfather’s clock

Regardless of which experimental method comes to define the new kilogram, the IPK’s 125-year reign may soon come to a close as the redefinition of mass pushes experimentalists to the frontiers of modern physics. The Compton clock, Müller points out, not only offers a new definition of mass but a new concept of what it means to keep time. He picks up a metal rod lying on his desk and explains, “what you usually do when you measure time is take something that oscillates [*Müller swings the rod back and forth*] and you count the oscillations.” But, “it’s really two objects: the rod, and the earth that attracts it.”

To make a conventional clock, he says, “you need at least two objects and you need them to interact to get an oscillation that you can count. Maybe you can only build clocks by having two or more bodies that interact, which would mean that in a universe where particles don’t interact there would be no time.” In other words, the usual way of measuring time only works if things can exert come kind of influence on each other, like the gravitational pull between the earth and the metal rod. If they couldn’t influence each other, we couldn’t measure the passage of time. In building a clock from a rock, Müller, Estey, and their colleagues have broken this convention by measuring time with a single body.

In a world where the quantum bookie states the rules, redefining the kilogram has massive implications. The Müller lab’s clock lays the foundation for probing subtle properties of all kinds of particles. Moreover, tying the standard of mass to an atomic constant will be a final step in bringing standards for measurement into the 21^{st} century. As an integral part of the new SI, the realization of a new mass standard will not only be more stable, but more widely available. Someday you may not need the keys to a high-security vault in Paris to find the exact mass of a kilogram. All you will need is a couple of cesium atoms, some high-powered lasers, and the laws of physics.