Casimir forces are of fundamental interest because they originate from quantum fluctuations of the electromagnetic field 1 . Apart from controlling the Casimir force via the optical properties of the materials 2-11 , a number of novel geometries have been proposed to generate repulsive and/or non-monotonic Casimir forces between bodies separated by vacuum gaps 12-14 . Experimental realization of these geometries, however, is hindered by the difficulties in alignment when the bodies are brought into close proximity. Here, using an on-chip platform with integrated force sensors and actuators 15 , we circumvent the alignment problem and measure the Casimir force between two surfaces with nanoscale protrusions. We demonstrate that the Casimir force depends non-monotonically on the displacement. At some displacements, the Casimir force leads to an effective stiffening of the nanomechanical spring. Our findings pave the way for exploiting the Casimir force in nanomechanical systems using structures of complex and non-conventional shapes.The prediction of the attraction between two neutral perfect conductors by Casimir 1 was obtained by considering the boundary conditions imposed by two planar surfaces on the quantum fluctuations of the electromagnetic field. Alternatively, the Casimir force is sometimes considered as an extension of the van der Waals force between fluctuating dipoles to solid bodies and in the regime of retardation. This attractive force increases monotonically when the distance between the two planes decreases. As the Casimir force becomes the dominant interaction between electrically neutral surfaces separated by nanoscale gaps, they are of practical importance in nanomechanical devices 16,17 . In experiments, one of the flat surfaces is often replaced by a spherical body 4-11 due to the difficulty in maintaining parallelism at small separations 18 . By introducing corrugations to one of the surfaces, recent experiments 19,20 have demonstrated the non-trivial geometry dependence of the Casimir force. Measuring the Casimir Device fabrication Supplementary Figure S1 | The fabrication procedure of the device (not to scale). a, A cross-sectional view of the silicon-on-insulator wafer. The silicon device layer, the buried oxide layer and the substrate are shown in blue, yellow and grey respectively. b, The silicon oxide etch mask (red) is created using the resist pattern from lithography. c, Silicon in the regions not protected by silicon oxide is removed by DRIE. d, HF selectively etches the silicon oxide isotropically, undercutting of the top silicon structure by ~ 2.7 µm. The middle silicon piece is thin enough to be suspended. The other two pieces have oxide underneath and therefore are anchored to the substrate.

Thermal properties of materials are often determined by measuring thermalization processes; however, such measurements at the nanoscale are challenging because they require high sensitivity concurrently with high temporal and spatial resolutions. Here, we develop an optomechanical cantilever probe and customize an atomic force microscope with low detection noise ≈1 fm/Hz 1/2 over a wide (>100 MHz) bandwidth that measures thermalization dynamics with ≈10 ns temporal resolution, ≈35 nm spatial resolution, and high sensitivity. This setup enables fast nanoimaging of thermal conductivity (η) and interfacial thermal conductance (G) with measurement throughputs ≈6000× faster than conventional macroscale-resolution timedomain thermoreflectance acquiring the full sample thermalization. As a proof-of-principle demonstration, 100 × 100 pixel maps of η and G of a polymer particle are obtained in 200 s with a small relative uncertainty (<10%). This work paves the way to study fast thermal dynamics in materials and devices at the nanoscale.

We study the Casimir torque between two metallic one-dimensional gratings rotated by an angle θ with respect to each other. We find that, for infinitely extended gratings, the Casimir energy is anomalously discontinuous at θ = 0, due to a critical zero-order geometric transition between a 2Dand a 1D-periodic system. This transition is a peculiarity of the grating geometry and does not exist for intrinsically anisotropic materials. As a remarkable practical consequence, for finite-size gratings, the torque per area can reach extremely large values, increasing without bounds with the size of the system. We show that for finite gratings with only 10 period repetitions, the maximum torque is already 60 times larger than the one predicted in the case of infinite gratings. These findings pave the way to the design of a contactless quantum vacuum torsional spring, with possible relevance to micro-and nano-mechanical devices.