Efficient computation of Casimir interactions between arbitrary 3D objects

Quantum trapping and rotational self-alignment in triangular Casimir microcavities

Casimir torque, a rotational motion driven by zero-point energy minimization, is a problem that attracts notable research interest. Recently, it has been realized using liquid crystal phases and natural anisotropic substrates. However, for natural materials, substantial torque occurs only at van der Waals distances of ~10 nm. Here, we use Casimir self-assembly with triangular gold nanostructures for rotational self-alignment at truly Casimir distances (100 to 200 nm separation). The interplay of repulsive electrostatic and attractive Casimir potentials forms a stable quantum trap, giving rise to a tunable Fabry-Pérot microcavity. This cavity self-aligns both laterally and rotationally to maximize area overlap between templated and floating flakes. The rotational self-alignment is sensitive to the equilibrium distance between the two triangles and their area, offering possibilities for active control via electrostatic screening manipulation. Our self-assembled Casimir microcavities present a versatile and tunable platform for nanophotonic, polaritonic, and optomechanical applications.

Large Casimir Flipping Torque in Quantum Trap

Casimir torque between parallel plates, a macroscopic quantum electrodynamics effect, is known to be induced by dielectric anisotropy and related to the rotational degree of freedom. We here reveal a different type of Casimir torque generated on a Au plate suspended in a quantum trap without recourse to materials anisotropy. As the Au plate deflects from the equilibrium plane with a nonzero flipping angle, the regions departing from and approaching the Teflon-coated Au substrate are subjected to attractive and repulsive Casimir forces, respectively, resulting in a restoring torque about the axis of flipping. For a quantum trap with an equilibrium separation of ∼10 nm, the stiffness per unit area of the Casimir flipping torque can be an order of magnitude larger than those of previously reported dielectric anisotropy-induced rotational torques at the same separation. The large Casimir flipping torque provides the possibility of designing a mechanical oscillator completely dominated by quantum and thermal fluctuations.

Unifying Theory for Casimir Forces: Bulk and Surface Formulations

The principles of the electromagnetic fluctuation-induced phenomena such as Casimir forces are well understood. However, recent experimental advances require universal and efficient methods to compute these forces. While several approaches have been proposed in the literature, their connection is often not entirely clear, and some of them have been introduced as purely numerical techniques. Here we present a unifying approach for the Casimir force and free energy that builds on both the Maxwell stress tensor and path integral quantization. The result is presented in terms of either bulk or surface operators that describe corresponding current fluctuations. Our surface approach yields a novel formula for the Casimir free energy. The path integral is presented both within a Lagrange and Hamiltonian formulation yielding different surface operators and expressions for the free energy that are equivalent. We compare our approaches to previously developed numerical methods and the scattering approach. The practical application of our methods is exemplified by the derivation of the Lifshitz formula.

Casimir-Induced Instabilities at Metallic Surfaces and Interfaces

Surface distortion splits surface plasmons asymmetrically in energy with a net lowering of zero-point energy. We contrast this with the symmetrical distortion of electronic energy levels. We use conformal mapping to demonstrate this splitting and find that surface corrugation always leads to a decrease in the zero-point energy of a metallic surface, but the decrease is not strong enough to drive a surface reconstruction on its own. A second metallic surface in proximity to the first gives a more significant lowering of energy, sufficient to drive the instability of a mercury thin film. This mechanism provides a fundamental length scale limit to planar nanostructures.

Strong geometry dependence of the Casimir force between interpenetrated rectangular gratings

Quantum fluctuations give rise to Casimir forces between two parallel conducting plates, the magnitude of which increases monotonically as the separation decreases. By introducing nanoscale gratings to the surfaces, recent advances have opened opportunities for controlling the Casimir force in complex geometries. Here, we measure the Casimir force between two rectangular silicon gratings. Using an on-chip detection platform, we achieve accurate alignment between the two gratings so that they interpenetrate as the separation is reduced. Just before interpenetration occurs, the measured Casimir force is found to have a geometry dependence that is much stronger than previous experiments, with deviations from the proximity force approximation reaching a factor of ~500. After the gratings interpenetrate each other, the Casimir force becomes non-zero and independent of displacement. This work shows that the presence of gratings can strongly modify the Casimir force to control the interaction between nanomechanical components.

The geometry dependence of the Casimir force could enable applications in nanomechanical systems if the effects can be enhanced. Here, the authors demonstrate that the Casimir force between two interpenetrating nanoscale gratings can exceed the proximity force approximation by a factor of 500.

Electrodynamic Force, Casimir Effect, and Stiction Mitigation in Silicon Carbide Nanoelectromechanical Switches

Logic switches enabled by nanoelectromechanical systems (NEMS) offer abrupt on/off-state transition with zero off-state leakage and minimal subthreshold swing, making them uniquely suited for enhancing mainstream electronics in a range of applications, such as power gating, high-temperature and high-voltage logic, and ultralow-power circuits requiring zero standby leakage. As NEMS switches are scaled with genuinely nanoscale gaps and contacts, quantum mechanical electrodynamic force (EDF) takes an important role and may be the ultimate cause of the plaguing problem of stiction. Here, combined with experiments on three-terminal silicon carbide (SiC) NEMS switches, a theoretical investigation is performed to elucidate the origin of EDF and Casimir effect leading to stiction, and to develop a stiction-mitigation design. The EDF calculation with full Lifshitz formula using the actual material and device parameters is provided. Finite element modeling and analytical calculations demonstrate that EDF becomes dominant over elastic restoring force in such SiC NEMS when the switching gap shrinks to a few nanometers, leading to irreversible stiction at contact. Artificially corrugated contact surfaces are designed to reduce the contact area and the EDF, thus evading stiction. This rational surface engineering reduces the EDF down to 4% compared with the case of unengineered, flat contact surfaces.

The case for a Casimir cosmology

The cosmological constant, also known as dark energy, was believed to be caused by vacuum fluctuations, but naive calculations give results in stark disagreement with fact. In the Casimir effect, vacuum fluctuations cause forces in dielectric media, which is very well described by Lifshitz theory. Recently, using the analogy between geometries and media, a cosmological constant of the correct order of magnitude was calculated with Lifshitz theory (Leonhardt 2019 Ann. Phys. (New York) 411, 167973. (doi:10.1016/j.aop.2019.167973)). This paper discusses the empirical evidence and the ideas behind the Lifshitz theory of the cosmological constant without requiring prior knowledge of cosmology and quantum field theory. This article is part of a discussion meeting issue 'The next generation of analogue gravity experiments'.

Plasmonic Surface Lattice Resonances: Theory and Computation

Plasmonic surface lattice resonances (SLRs) are mixed light-matter states emergent in a system of periodically arranged metallic nanoparticles (NPs) under the constraint that the array spacing is able to support a standing wave of optical-frequency light. The properties of SLRs derive from two separate physical effects; the electromagnetic (plasmonic) response of metal NPs and the electromagnetic states (photonic cavity modes) associated with the array of NPs. Metal NPs, especially free-electron metals such as silver, gold, aluminum, and alkali metals, support optical-frequency electron density oscillations known as localized surface plasmons (LSPs). The high density of conduction-band electrons in these metals gives rise to plasmon excitations that strongly couple to light even for particles that are several orders of magnitude smaller than the wavelength of the excitation source. In this sense, LSPs have the remarkable ability to squeeze far-field light into intensely localized electric near-fields that can enhance the intensity of light by factors of ∼10³ or more. Moreover, as a result of advances in the synthesis and fabrication of NPs, the intrinsic dependence of LSPs on the NP geometry, composition, and size can readily be exploited to design NPs with a wide range of optical properties. One drawback in using LSPs to enhance optical, electronic, or chemical processes is the losses introduced into the system by dephasing and Ohmic damping-an effect that must either be tolerated or mitigated. Plasmonic SLRs enable the mitigation of loss effects through the coupling of LSPs to diffractive states that arise from arrays satisfying Bragg scattering conditions, also known as Rayleigh anomalies. Bragg modes are well-known for arrays of dielectric NPs, where they funnel and trap incoming light into the plane of the lattice, defining a photonic cavity. The low losses and narrow linewidths associated with dielectric NPs produce Bragg modes that oscillate for ∼10³-10⁴ cycles before decaying. These modes are of great interest to the metamaterials community but have relatively weak electric fields associated with dielectric NPs and therefore are not used for applications where local field enhancements are needed. Plasmonic lattices, i.e., photonic crystals composed of metallic NPs, combine the characteristics of both LSPs and diffractive states, enabling both enhanced local fields and narrow-linewidth excitations, in many respects providing the best advantages of both materials. Thus, by control of the periodicity and global symmetry of the lattice in addition to the material composition and shape of the constituent NPs, SLRs can be designed to simultaneously survive for up to 10³ cycles while maintaining the electric field enhancements near the NP surface that have made the use of LSPs ubiquitous in nanoscience. Modern fabrication methods allow for square-centimeter-scale patches of two-dimensional arrays that are composed of approximately one trillion NPs, making them effectively infinite at the nanoscale. Because of these advances, it is now possible to experimentally realize SLRs with properties that approach those predicted by idealized theoretical models. In this Account, we introduce the fundamental theory of both SLRs and SLR-mediated lasing, where the latter is one of the most important applications of plasmonic SLRs that has emerged to date. The focus of this Account is on theoretical concepts for describing plasmonic SLRs and computational methods used for their study, but throughout we emphasize physical insights provided by the theory that aid in making applications.

Nonadditivity of van der Waals forces on liquid surfaces

We present an approach for modeling nanoscale wetting and dewetting of textured solid surfaces that exploits recently developed, sophisticated techniques for computing exact long-range dispersive van der Waals (vdW) or (more generally) Casimir forces in arbitrary geometries. We apply these techniques to solve the variational formulation of the Young-Laplace equation and predict the equilibrium shapes of liquid-vacuum interfaces near solid gratings. We show that commonly employed methods of computing vdW interactions based on additive Hamaker or Derjaguin approximations, which neglect important electromagnetic boundary effects, can result in large discrepancies in the shapes and behaviors of liquid surfaces compared to exact methods.

Influence of surface roughness on dispersion forces

Surface roughness occurs in a wide variety of processes where it is both difficult to avoid and control. When two bodies are separated by a small distance the roughness starts to play an important role in the interaction between the bodies, their adhesion, and friction. Control of this short-distance interaction is crucial for micro and nanoelectromechanical devices, microfluidics, and for micro and nanotechnology. An important short-distance interaction is the dispersion forces, which are omnipresent due to their quantum origin. These forces between flat bodies can be described by the Lifshitz theory that takes into account the actual optical properties of interacting materials. However, this theory cannot describe rough bodies. The problem is complicated by the nonadditivity of the dispersion forces. Evaluation of the roughness effect becomes extremely difficult when roughness is comparable with the distance between bodies. In this paper we review the current state of the problem. Introduction for non-experts to physical origin of the dispersion forces is given in the paper. Critical experiments demonstrating the nonadditivity of the forces and strong influence of roughness on the interaction between bodies are reviewed. We also describe existing theoretical approaches to the problem. Recent advances in understanding the role of high asperities on the forces at distances close to contact are emphasized. Finally, some opinions about currently unsolved problems are also presented.

Quantum vacuum photon modes and superhydrophobicity

Nanostructures are commonly used for developing superhydrophobic surfaces. However, available wetting theoretical models ignore the effect of vacuum photon-mode alteration on van der Waals forces and thus on hydrophobicity. Using first-principles calculations, we show that superhydrophibicity of nanostructured surfaces is dramatically enhanced by vacuum photon-mode tuning. As a case study, wetting contact angles of a water droplet above a polyethylene nanostructured surface are obtained from the interaction potential energy calculated as a function of the droplet-surface separation distance. This new approach could pave the way for the design of novel superhydrophobic coatings.

Casimir forces on a silicon micromechanical chip

Quantum fluctuations give rise to van der Waals and Casimir forces that dominate the interaction between electrically neutral objects at sub-micron separations. Under the trend of miniaturization, such quantum electrodynamical effects are expected to play an important role in micro- and nano-mechanical devices. Nevertheless, utilization of Casimir forces on the chip level remains a major challenge because all experiments so far require an external object to be manually positioned close to the mechanical element. Here by integrating a force-sensing micromechanical beam and an electrostatic actuator on a single chip, we demonstrate the Casimir effect between two micromachined silicon components on the same substrate. A high degree of parallelism between the two near-planar interacting surfaces can be achieved because they are defined in a single lithographic step. Apart from providing a compact platform for Casimir force measurements, this scheme also opens the possibility of tailoring the Casimir force using lithographically defined components of non-conventional shapes.

Non-equilibrium Casimir force between vibrating plates

We study the fluctuation-induced, time-dependent force between two plates confining a correlated fluid which is driven out of equilibrium mechanically by harmonic vibrations of one of the plates. For a purely relaxational dynamics of the fluid we calculate the fluctuation-induced force generated by the vibrating plate on the plate at rest. The time-dependence of this force is characterized by a positive lag time with respect to the driving. We obtain two distinctive contributions to the force, one generated by diffusion of stress in the fluid and another related to resonant dissipation in the cavity. The relation to the dynamic Casimir effect of the electromagnetic field and possible experiments to measure the time-dependent Casimir force are discussed.

Nanoplasmonic lattices for ultracold atoms

We propose to use subwavelength confinement of light associated with the near field of plasmonic systems to create nanoscale optical lattices for ultracold atoms. Our approach combines the unique coherence properties of isolated atoms with the subwavelength manipulation and strong light-matter interaction associated with nanoplasmonic systems. It allows one to considerably increase the energy scales in the realization of Hubbard models and to engineer effective long-range interactions in coherent and dissipative many-body dynamics. Realistic imperfections and potential applications are discussed.

Casimir force on a surface with shallow nanoscale corrugations: geometry and finite conductivity effects

We measure the Casimir force between a gold sphere and a silicon plate with nanoscale, rectangular corrugations with a depth comparable to the separation between the surfaces. In the proximity force approximation (PFA), both the top and bottom surfaces of the corrugations contribute to the force, leading to a distance dependence that is distinct from a flat surface. The measured Casimir force is found to deviate from the PFA by up to 10%, in good agreement with calculations based on scattering theory that includes both geometry effects and the optical properties of the material.

Casimir repulsion between metallic objects in vacuum

We give an example of a geometry in which two metallic objects in vacuum experience a repulsive Casimir force. The geometry consists of an elongated metal particle centered above a metal plate with a hole. We prove that this geometry has a repulsive regime using a symmetry argument and confirm it with numerical calculations for both perfect and realistic metals. The system does not support stable levitation, as the particle is unstable to displacements away from the symmetry axis.

Theoretical ingredients of a Casimir analog computer

We derive a correspondence between the contour integration of the Casimir stress tensor in the complex-frequency plane and the electromagnetic response of a physical dissipative medium in a finite real-frequency bandwidth. The consequences of this correspondence are at least threefold: First, the correspondence makes it easier to understand Casimir systems from the perspective of conventional classical electromagnetism, based on real-frequency responses, in contrast to the standard imaginary-frequency point of view based on Wick rotations. Second, it forms the starting point of finite-difference time-domain numerical techniques for calculation of Casimir forces in arbitrary geometries. Finally, this correspondence is also key to a technique for computing quantum Casimir forces at micrometer scales using antenna measurements at tabletop (e.g., centimeter) scales, forming a type of analog computer for the Casimir force. Superficially, relationships between the Casimir force and the classical electromagnetic Green's function are well known, so one might expect that any experimental measurement of the Green's function would suffice to calculate the Casimir force. However, we show that the standard forms of this relationship lead to infeasible experiments involving infinite bandwidth or exponentially growing fields, and a fundamentally different formulation is therefore required.