Putting a drop of ink in a glass of water, we will see that the ink spreads, eventually mixes with the water evenly, and does not go back to the drop form again. The overall entropy of the system increases. This kind of entropy-producing irreversibility seemed to be set in stone in the second law of thermodynamics—until recently, when physicists realized that this classical understanding applies only to macroscopic systems. If the ink drop and the amount of water were made microscopic, there would be a chance that the dispersed ink molecules came back together again. The macroscopic second law of thermodynamics has been refined to reflect this new understanding. One refined version is a so-called fluctuation theorem. It stipulates, in the ink-in-water scenario, that the probability of the ink going back from a more spread state to a less spread one—an “abnormal,” entropy-consuming process—divided by the probability of it going from the less spread state to the more spread one—a “natural,” entropy-producing process—depends exponentially on the entropy difference between the less and the more spread states. The theorem has been proven correct on a number of microscopic classical systems such as a system of a micron-scale latex bead in a liquid. But does it work for small quantum systems? A concrete test has been lacking so far. In this experimental paper, we make a significant step toward an ultimate test, by quantitatively validating the fluctuation theorem by counting—in real time—electrons that pass through two tiny semiconductor quantum dots at a very low temperature of 0.3 K.

In our measurement device, two quantum dots are coupled in series, and one is also coupled to a source electrode and the other to a drain electrode. A voltage is then applied across the source and drain electrodes, and the flow of electrons from one electrode to the other through the quantum dots is then counted one electron at a time by monitoring the charge state of the dots with a sensor. The technical challenge is to ensure that the sensor works nonperturbatively—We want to count only events that come from the intrinsic thermal fluctuations of the system. To that end, we have optimized the sample design and fabrication to combine good detection efficiency of the sensor with good electronic tenability of the quantum dots. Our result is rewarding: While most of the electrons we have counted flow in the preferred direction set by the external voltage, a very small fraction of them indeed move against that direction, both movements in manners predicted by the fluctuation theorem.

Our experiment demonstrates a new way of understanding charge transport processes in small electronic devices. Equally significant is the success of our test at an entirely new energy scale that is smaller by 4 orders of magnitude than those probed in the previous experiments. Although quantum coherence is absent in our device, the step we have taken should lead to an ultimate test of the fluctuation theorem in a mesoscopic conductor where quantum coherence is present.