Instantons in a Double-Well are Poisson Distributed

In the quantum world, particles do not always stay where they are placed. If a particle sits in a valley between two hills, classical physics says it will remain there forever unless pushed hard enough to climb over the barrier. Quantum mechanics, however, allows the particle to slip through the hill entirely, appearing in the neighboring valley without ever crossing the top. This phenomenon, known as tunneling, is fundamental to how atoms bond and how modern electronics function. When a system has two identical valleys, the particle can exist in both at once, creating two slightly different energy states instead of one. The difference between these two states is incredibly small, shrinking exponentially as the barrier between the valleys grows higher. Physicists have long used a conceptual tool called an "instanton" to describe the rare, fleeting events where a particle tunnels from one side to the other. In this picture, the particle makes a sudden jump, and if the barrier is high enough, these jumps are so rare that they happen independently of one another, like raindrops hitting a roof.

A team of mathematicians at Princeton University has now provided a rigorous proof that this intuitive picture of independent, rare jumps is exactly correct for a specific class of quantum systems. By analyzing the heat-kernel trace—a mathematical object that describes how a system evolves over time—they demonstrated that the number of times a particle tunnels between two wells follows a precise statistical pattern known as the Poisson distribution. This distribution describes events that happen randomly but at a steady average rate, such as the arrival of phone calls at a switchboard or the decay of radioactive atoms. The researchers showed that on a specific, extremely long time scale, the tunneling events are not clustered or chaotic; they are perfectly independent, and their frequency is determined by a single, well-defined value called the hopping coefficient. This coefficient acts as a measure of how easily the particle can move between the two locations, and the team proved that it is directly linked to the energy difference between the two lowest states of the system.

The study focused on a symmetric double-well potential, a mathematical landscape with two identical dips separated by a central peak. In this setting, the particle can reside in either dip, but quantum mechanics allows it to tunnel through the peak. For decades, physicists have relied on a "dilute-gas" approximation, assuming that these tunneling events are so sparse that they do not interfere with each other. While this assumption has been useful for making predictions, it lacked a strict mathematical foundation. The new work fills this gap by decomposing the complex motion of the particle into distinct passages between the two wells. Using a localized representation of the system, the researchers tracked the particle's path as it moved between shrinking neighborhoods around the two minima. They defined a "passage" as the moment the particle crosses from one side to the other and counted how many such passages occurred over a given time interval.

The core of the discovery lies in the behavior of these passages over an exponentially long time scale. The researchers found that as the system becomes more quantum-mechanical (represented by a large parameter in their equations), the probability of observing exactly $k$ passages converges to a specific formula involving the number $k$ and the average number of passages. This formula is the hallmark of the Poisson distribution. It confirms that the tunneling events are indeed independent and random, validating the long-held physical intuition. Furthermore, the team identified the weight of a single passage with the hopping coefficient, a value that can be calculated by looking at the flow of the particle's wave function across the boundary between the two wells. This connection allowed them to derive the exact energy splitting between the two lowest quantum states directly from the statistics of the tunneling events.

The proof required overcoming significant mathematical hurdles. Unlike a simple jump, a quantum tunneling event does not happen at a single, fixed moment; it is a process that unfolds over time, and successive passages can occur very close together, potentially forming clusters. The researchers had to show that these clusters are negligible and that the vast majority of paths consist of well-separated, independent events. They achieved this by separating the possible paths into two categories: those where the passages are clearly distinct and those where they are bunched together. Through careful estimates, they demonstrated that the contribution of the clustered paths vanishes in the limit, leaving only the independent events to dominate the statistics. This rigorous separation allowed them to factorize the complex problem into manageable pieces, proving that the total number of passages behaves exactly as the Poisson law predicts.

The implications of this work extend beyond the specific model studied. By establishing a direct link between the statistical distribution of tunneling events and the energy levels of the system, the researchers provided a new way to understand the fundamental mechanism of quantum splitting. The energy difference between the two lowest states, which is crucial for understanding the stability and behavior of molecules and materials, is shown to be exactly twice the magnitude of the hopping coefficient, with a tiny correction that disappears as the system becomes more quantum. This result confirms that the familiar expansion used by physicists to calculate these energy differences emerges naturally from the factorization of the heat-kernel trace, without needing to rely on unproven approximations. The work stands as a definitive mathematical realization of the instanton picture, turning a heuristic physical argument into a proven theorem.

In the end, the paper clarifies the nature of quantum tunneling in a double-well system. It shows that the particle does not wander aimlessly or tunnel in unpredictable bursts; rather, its transitions between the two wells are governed by a simple, elegant statistical law. The number of times it crosses the barrier is random, but the randomness follows a strict pattern that depends only on the average rate of crossing. This rate is determined by the geometry of the potential and the properties of the particle's wave function. The researchers have thus bridged the gap between the intuitive, physical description of instantons and the rigorous, analytical framework of quantum mechanics, proving that the "dilute gas" of tunneling events is not just a useful metaphor, but a precise description of reality.