The entropy of an electronic system offers important insights into the nature of its quantum mechanical ground state. This is particularly valuable in cases where the state is difficult to identify by conventional experimental probes, such as conductance. Traditionally, entropy measurements are based on bulk properties, such as heat capacity, that are easily observed in macroscopic samples but are unmeasurably small in systems that consist of only a few particles [1, 2]. In this work, we develop a mesoscopic circuit to directly measure the entropy of just a few electrons, and demonstrate its efficacy using the well understood spin statistics of the first, second, and third electron ground states in a GaAs quantum dot [3][4][5][6][7][8]. The precision of this technique, quantifying the entropy of a single spin-1 2 to within 5% of the expected value of k B ln 2, shows its potential for probing more exotic systems. For example, entangled states or those with non-Abelian statistics could be clearly distinguished by their low-temperature entropy [9][10][11][12][13].Our approach is analogous to the milestone of spin-tocharge conversion achieved over a decade ago, in which the infinitesimal magnetic moments of a single spin were detected by transforming them into the presence or absence of an electron charge [14,15]. Following this example, we perform an entropy-to-charge conversion, making use of the Maxwell relationthat connects changes in entropy, particle number, and temperature (S, N , and T , respectively) to changes in the chemical potential, µ, a quantity that is simple to measure and control. The Maxwell relation in Eq. 1 forms the basis of two theoretical proposals to measure non-Abelian exchange of Moore-Read quasiparticles in the ν = 5 2 state via their entropy [9,10]. Reference 10 proposes a strategy by which quasiparticle entropy could be deduced from a V m id V p G sens N − 1 N ∂S/∂N = 0 b V m id V m id V p N − 1 N ∂S/∂N > 0 c Vp δGsens Vp δGsens I heat I sens V sens V p G sens δG sens AC DC 500nm FIG. 1.Measurement protocol (a) Scanning electron micrograph of a device similar to the one measured. Electrostatic gates (gold) define the circuit in a 2D electron gas (2DEG), with grey gates grounded. Squares indicate ohmic contacts to the 2DEG. The temperature of the electron reservoir in the middle (red) is oscillated using AC current, I heat , at frequency f heat through the quantum point contact (QPC) on the left. A portion of the 5 µm-wide reservoir has been removed here for clarity. The occupation of the quantum dot, tunnel coupled to the right side the reservoir, is tuned by Vp and monitored by Isens through the charge sensor QPC. Isens is split into DC and AC components, the latter being measured by a lock-in amplifier at 2f heat . (b) and (c) Simulated DC charge sensor signal, Gsens, for a transition from N − 1 → N electrons at two temperatures (T Red > T Blue ), showing two possible cases for ∂S ∂N . Insets show the corresponding difference, δGsens, between hot and cold curves.the temperature-depende...

Multiple vertebrate embryonic structures such as organ primordia are composed of confluent cells. Although mechanisms that shape tissue sheets are increasingly understood, those which shape a volume of cells remain obscure. Here we show that 3D mesenchymal cell intercalations are essential to shape the mandibular arch of the mouse embryo. Using a genetically encoded vinculin tension sensor that we knock-in to the mouse genome, we show that cortical force oscillations promote these intercalations. Genetic loss- and gain-of-function approaches show that Wnt5a functions as a spatial cue to coordinate cell polarity and cytoskeletal oscillation. These processes diminish tissue rigidity and help cells to overcome the energy barrier to intercalation. YAP/TAZ and PIEZO1 serve as downstream effectors of Wnt5a -mediated actomyosin polarity and cytosolic calcium transients that orient and drive mesenchymal cell intercalations. These findings advance our understanding of how developmental pathways regulate biophysical properties and forces to shape a solid organ primordium.

Numerous hypotheses invoke tissue stiffness as a key parameter that regulates morphogenesis and disease progression. However, current methods are insufficient to test hypotheses that concern physical properties deep in living tissues. Here we introduce, validate, and apply a magnetic device that generates a uniform magnetic field gradient within a space that is sufficient to accommodate an organ-stage mouse embryo under live conditions. The method allows rapid, nontoxic measurement of the three-dimensional (3D) spatial distribution of viscoelastic properties within mesenchyme and epithelia. Using the device, we identify an anteriorly biased mesodermal stiffness gradient along which cells move to shape the early limb bud. The stiffness gradient corresponds to aWnt5a-dependent domain of fibronectin expression, raising the possibility that durotaxis underlies cell movements. Three-dimensional stiffness mapping enables the generation of hypotheses and potentially the rigorous testing of mechanisms of development and disease.