Entropy of a double quantum dot

We use charge sensing to detect entropy changes in a double quantum dot defined by electrostatic gating of a GaAs/AlGaAs heterostructure. This system can be tuned to be two separate systems, like two independent, artificial atoms, or a single coherent system, like a molecule. We study entropy changes in both regimes due to changes in the occupation of the system. First we recover the single-dot result for each dot, that the occupation of the dot by a single electron corresponds to an increase in the entropy of $k_{\mathrm{B}} \log 2$. Next we examine two different charge transitions in the “molecular” regime, and how it reveals itself in terms of the measured entropy. We also uncover a realization of Pauli blockade that clutters the entropy signal. By applying a rate equation model, we demonstrate the effect’s nonequilibrium origins and exclude it from the analysis of the system’s entropy. Understanding these experiments in this simplest coupled system enables the study of the entropy in other, more complicated coupled quantum systems, such as ones with topological or highly entangled ground states.

💡 Research Summary

In this work the authors extend a recently developed charge‑sensing technique for measuring entropy to a double quantum dot (DQD) defined in a GaAs/AlGaAs heterostructure. The DQD can be tuned continuously from two isolated “artificial atoms” to a single coherent “artificial molecule” by adjusting the inter‑dot tunnel coupling. The experimental platform consists of a two‑dimensional electron gas patterned with metallic gates that define the two dots, source and drain leads that act as thermal reservoirs, and a nearby quantum point contact (QPC) that serves as a non‑invasive charge detector. By applying an AC heating current to one of the leads, the temperature of that lead oscillates at twice the heating frequency (2 ω). This temperature modulation changes the occupation probabilities of the dot states, which in turn modulates the QPC current at 2 ω.

Two complementary routes are used to extract the entropy change ΔS associated with a charge transition. The first exploits a Maxwell relation, ∂V_mid/∂θ = –ΔS/k_B, where V_mid is the gate voltage at which the dot and lead are equally likely to be occupied and θ is the temperature expressed in gate‑voltage units. By measuring the derivative of V_mid with respect to θ (obtained from the 2 ω QPC signal) the authors directly obtain ΔS. The second route integrates the measured 2 ω current over gate voltage, using the same Maxwell relation in integral form. Both methods give consistent results.

In the regime of large detuning (Γ_int ≈ 0) each dot behaves as an independent single‑electron island. For the transitions (0,0) → (0,1) and (0,0) → (1,0) the measured entropy increase is ΔS ≈ k_B ln 2, reflecting the twofold spin degeneracy of a singly occupied level. Fits to the 2 ω signal yield temperatures of ~55 mK and ΔS values within 10 % of the theoretical expectation, confirming the reliability of the technique.

When the inter‑dot tunnel coupling is increased, the system enters a “molecular” regime. Near a triple point where three charge configurations are nearly degenerate, the authors observe a pronounced peak in ΔS that grows from k_B ln 3 up to k_B ln 4 as the temperature of the heated lead is raised. This behavior is interpreted as the activation of both bonding and antibonding molecular orbitals: each orbital is spin‑degenerate, giving a total fourfold degeneracy (log 4) when k_B T becomes comparable to the tunnel splitting (~5.8 µeV). The width and height of the ΔS peak scale with the lead temperature, confirming that the observed entropy reflects the full many‑body degeneracy of the coupled system rather than a simple single‑dot effect.

A striking, unexpected feature appears in the 2 ω maps near the triple point: triangular regions of positive signal that are not associated with any change in charge configuration. The authors identify these as signatures of Pauli blockade combined with non‑equilibrium population dynamics. To understand them they construct a rate‑equation model that includes (i) a slow relaxation rate Γ_r from an excited to a ground state (≈1 neV, i.e., ~1.5 MHz) and (ii) the fact that the QPC has different sensitivities to charge on the left versus the right dot. Simulations reproduce the triangular features when only one lead is heated, and show that heating both leads eliminates the triangles, exactly as observed experimentally. Because these signals arise from kinetic bottlenecks rather than thermodynamic state counting, the authors exclude them from the entropy analysis.

The paper demonstrates that entropy can be measured reliably in a minimal coupled system, that the technique can resolve changes in degeneracy from 2 to 4, and that non‑equilibrium effects such as Pauli blockade must be carefully modeled and removed. The authors argue that this methodology provides a blueprint for probing entropy in more exotic platforms—e.g., quantum dots coupled to Majorana zero modes, hybrid semiconductor–superconductor devices, or engineered lattices with topological order—where entropy is a key diagnostic of fractionalization, non‑Abelian statistics, or many‑body entanglement. By establishing a clear experimental protocol and a theoretical framework for disentangling equilibrium entropy from dynamical artifacts, the work paves the way for thermodynamic studies of quantum information carriers in nanoscale devices.