Entropy spectroscopy of a tunable two-site Hubbard molecule

In the microscopic world of quantum physics, matter behaves less like solid objects and more like a fog of possibilities. When electrons are trapped in tiny, artificial cages called quantum dots, they can only occupy specific energy levels, much like a person can only stand on specific rungs of a ladder. If these electrons are allowed to move between two neighboring cages, they begin to interact, forming a simple molecule made of just two sites. Physicists call this a Hubbard molecule, a minimal model used to understand how electrons behave in complex materials. A key question in this field is knowing which of these possible electron arrangements are "active" at a given temperature. Some arrangements are so high in energy that the heat of the environment cannot reach them, leaving them frozen out. Others are low enough to be populated, contributing to the system's disorder, or entropy. Measuring this entropy is difficult because the amount of heat involved is so small that it vanishes instantly into the surrounding environment, making traditional thermometers useless.

To solve this, researchers have developed a clever workaround that uses the movement of electric charge as a proxy for heat. By observing how the number of electrons in a dot changes as the temperature shifts, they can calculate the entropy without ever measuring heat directly. However, this method hits a wall when applied to two dots instead of one. In a single dot, the relationship between charge and temperature is straightforward. In a pair of dots, the electrons can rearrange themselves in complex ways between the two sites, and a single sensor cannot easily tell which dot holds which electron. This ambiguity has prevented scientists from accurately measuring the total disorder of these two-site systems, leaving a gap in our understanding of how simple quantum molecules evolve into larger, more complex structures.

A team of researchers has now bridged this gap by demonstrating a new way to measure the entropy of a tunable two-site Hubbard molecule. They built a device using a semiconductor material where two quantum dots sit side by side, connected by a narrow tunnel that electrons can cross. The strength of this connection, or tunneling, can be adjusted with a voltage, allowing the researchers to transform the system from two separate, independent dots into a single, merged entity. To measure the entropy, they used a sensitive charge sensor, a tiny electronic gate that detects when an electron moves. The breakthrough came from how they moved the system. Instead of changing the energy of just one dot, they shifted the energy levels of both dots up and down together, keeping the difference between them constant. This specific path, which the researchers call a "delta-axis," ensures that the sensor reads the total number of electrons in the entire two-dot system, rather than getting confused by the internal shuffling between the two. By following this path, they could extract the total entropy using just one sensor, a feat that was previously thought to require multiple, complex sensors.

The team then watched how the entropy changed as they increased the tunneling strength between the dots. When the connection was weak, the two dots behaved like independent atoms. In this state, the electrons could easily access all possible arrangements, including those where the electrons were on opposite sides or had different spins, resulting in a high level of entropy. As the researchers increased the tunneling, the dots began to talk to each other more strongly, forming hybridized molecular states. The energy difference between the lowest energy state (the bonding state) and the highest energy state (the antibonding state) grew larger. Eventually, this energy gap became so wide that the heat in the environment was no longer enough to push electrons into the higher, antibonding states. These states effectively dropped out of the thermal picture, causing the total entropy of the system to drop. The measurements showed a clear, smooth transition: as the tunneling increased, the system moved from a state of high disorder, where many configurations were possible, to a state of lower disorder, where only the most stable configurations remained active.

This effect was observed in both the one-electron and two-electron regions of the device. In the one-electron case, the entropy dropped as the antibonding state became inaccessible. In the two-electron case, the situation was even more nuanced. Here, the electrons could form a singlet state (where their spins are paired) or a triplet state (where their spins are aligned). As the tunneling increased, the energy splitting between the different singlet states grew, suppressing the contribution of the higher-energy singlet state to the entropy. The experimental data matched the predictions of a theoretical model known as the two-site Hubbard model with striking precision. The researchers confirmed that their new measurement protocol correctly captured the Gibbs entropy, the standard measure of disorder in thermodynamics, across the entire range of tunneling strengths. In contrast, when they tried to measure entropy by changing only one dot at a time, the results were distorted and did not match the theoretical expectations, highlighting the necessity of their new approach.

The significance of this work extends beyond just two dots. By proving that entropy can be measured accurately in a coupled system using a single sensor, the researchers have provided a practical tool for studying larger arrays of quantum dots. These larger arrays are being developed as quantum simulators, devices designed to mimic the behavior of complex materials that are too difficult to calculate on a computer. Understanding how entropy evolves in these systems is crucial for identifying phenomena like the formation of magnetic moments or the emergence of exotic states of matter. The ability to track the thermal activity of electrons as they move from independent atoms to a unified molecular system offers a new window into the thermodynamic properties of quantum matter. The researchers have shown that by carefully controlling the path of the measurement, they can reveal the hidden thermal landscape of these tiny systems, turning a fundamental challenge in quantum physics into a solvable problem.

The device itself was fabricated on a chip made of gallium arsenide, a common material in semiconductor physics. The two dots were defined by metal gates on the surface, which could be tuned to control the energy levels and the tunneling between them. To measure the temperature dependence, the researchers used a technique called Joule heating, where they passed a small current through a nearby channel to gently warm the electrons in the dots. They verified that this heating raised the temperature of the electrons without shifting their energy levels, ensuring that the changes they observed were purely thermal. The charge sensor, a quantum point contact, acted as a highly sensitive voltmeter, detecting the tiny electrostatic changes caused by the movement of electrons. The entire experiment was conducted in a dilution refrigerator, a specialized machine that cools the device to a temperature of just 200 millikelvin, close to absolute zero, to minimize unwanted thermal noise.

The results were consistent across a wide range of conditions. When the tunneling was weak, the system behaved as two independent dots, with an entropy that reflected the spin and charge degeneracy of each dot. As the tunneling increased, the entropy decreased, reflecting the suppression of higher-energy states. The researchers were able to map out the entropy across the entire stability diagram of the device, showing how the system transitions from an atomic-like regime to a molecular regime and finally to a merged-dot regime. In the merged-dot limit, where the tunneling is so strong that the two dots act as one, the entropy followed the expected behavior for a single quantum dot with two electrons. This progression provided a complete picture of how interactions and tunneling compete to determine the thermally active states of the system.

The study also addressed a specific challenge in extending entropy measurements from single dots to multiple dots. Previous attempts to measure entropy in double dots had yielded results that disagreed with theory, largely because the measurement method did not account for the complex interplay between the two dots. By shifting both dots together, the researchers eliminated this ambiguity. Their numerical simulations confirmed that this approach reproduces the correct Gibbs entropy, whereas the traditional method of changing one dot at a time leads to apparent entropy values that deviate from the true thermodynamic entropy. This validation gives confidence that the method can be scaled up to larger systems.

In the broader context of quantum simulation, this work offers a new way to probe the thermodynamic properties of correlated electron systems. As researchers build larger arrays of quantum dots to simulate complex materials, they need tools to understand how these systems behave at finite temperatures. Entropy spectroscopy provides a direct measure of the number of accessible states, which is essential for understanding phenomena like the Mott insulator transition or the formation of magnetic order. The ability to measure this quantity in a controlled, tunable system opens the door to exploring the thermodynamics of quantum matter in ways that were previously impossible. The researchers have demonstrated that with the right protocol, the subtle thermodynamic signatures of quantum interactions can be brought into sharp focus, providing a clearer understanding of the microscopic rules that govern the behavior of electrons in solids.

The findings are not just a theoretical exercise; they are grounded in precise experimental data. The researchers measured the entropy at different temperatures and tunneling strengths, comparing their results with theoretical calculations based on the Hubbard model. The agreement between the experiment and the theory was quantitative, meaning the numbers matched closely. This level of agreement confirms that the two-site Hubbard model accurately describes the physics of the system and that the entropy measurement protocol is reliable. The researchers also noted that the method is robust against variations in the device parameters, suggesting that it can be applied to a wide range of quantum dot systems.

Ultimately, this work represents a step forward in the ability to control and measure quantum systems. By developing a method to measure entropy in a two-site molecule, the researchers have provided a tool that can be used to explore the thermodynamics of more complex systems. The ability to distinguish between states that have the same charge but different spin or orbital content is a key achievement, as it allows scientists to see how tunneling and interactions shape the thermal landscape of the system. As the field of quantum simulation continues to grow, techniques like this will be essential for unlocking the secrets of correlated electron systems and for building a deeper understanding of the quantum world. The study shows that even in the smallest of systems, the interplay between energy, temperature, and quantum mechanics creates a rich and complex tapestry of behavior that can be unraveled with careful experimentation and thoughtful analysis.