Stable Levitation and Alignment of Compact Objects by Casimir Spring Forces

Abstract

We investigate a stable Casimir force configuration consisting of an object contained inside a spherical or spheroidal cavity filled with a dielectric medium. The spring constant for displacements from the center of the cavity and the dependence of the energy on the relative orientations of the inner object and the cavity walls are computed. We find that the stability of the force equilibrium can be predicted based on the sign of the force, but the torque cannot be.The Casimir force between atoms or macroscopic objects arises from quantum fluctuations of the electrodynamic field [1,2]. In all known examples it is found that the force is attractive, as long as the space between the objects is empty, and the magnetic susceptibility of the objects is negligible compared to their electric susceptibilities. But when space is filled with a medium with electric permittivity M intermediate between that of two objects, 1 < M < 2 , the force between the two becomes repulsive [3]. This effect has recently been verified experimentally [4]. But while repulsive forces are nothing new, they become interesting for applications when they produce stable equilibria, which, for example, the Coulomb force cannot.For an infinite cylinder enclosed in another, the Casimir force has recently been shown to have a stable equilibrium in the two directions perpendicular to the cylinder axes, when the material properties are chosen so that the force between two slabs of the same materials would be repulsive [5]. If the inner and outer cylinders have square cross sections, the direction of the torque exerted by one cylinder on the other is found to agree qualitatively with the predictions of the pairwise summation or proximity force approximations (PSA or PFA). Orientation dependence of the Casimir energy has also been studied recently for a small spheroid facing another spheroid or an infinite plate [6], or for a small spheroid located inside a spherical metal cavity [7]. (The smallness of the spheroid refers to keeping only the first term in the series expansion of the Casimir energy in the largest length scale of the spheroid.) The preferred orientation of the small spheroid flips when it is moved between the inside and the outside of the spherical shell; there is no torque if the spheroid faces an infinite metal plate, which is the limit of an infinitely large shell.In this paper we investigate the first configuration, depicted in Fig. 1, in which a compact object levitates stably due to the Casimir force alone. We characterize the Casimir energyby the spring constant k and the coefficients k n in a series expansion in a/R, where a is the magnitude of the displacement from the center of the cavity and R the radius of the (undeformed) spherical cavity. (k has units of energy here.) Unlike the case of infinite cylinders, where one object can be displaced along the cylinder axes without changing the energy, our case exhibits, for appropriately chosen materials, true stability in all directions and applies to realistic situations. For example...

Casimir potential of a compact object enclosed by a spherical cavity

“…The Casimir energy of dielectric spheres immersed in a dielectric medium is derived using the preceding equations in Ref. [3]. We present results for conducting boundary conditions in the following section.…”

“…[8], and stable three-dimensional configurations of objects in spheroidal drops of liquid or metallic shells were presented in Ref. [3].…”

“…In this article, we provide results for a metallic sphere inside an otherwise empty metallic spherical shell, while dielectric objects immersed in a dielectric medium are treated in Ref. [3].…”

Casimir quantum levitation tuned by means of material properties and geometries

“…On the other hand, such phenomena may also be of significant interest when designing new nanodevices, such as the Casimir anharmonic oscillator [6,7] or the ultrasensitive sensor [8]. Several interesting works have appeared regarding the use of repulsive Casimir force to achieve stable levitation of nano-objects, such as cylinder in cylinder [9], sphere on surface [10], or rotated hockey pucks [11]. Most of them are, however, achieved in sophisticated geometries that are difficult to fabricate using well-established micro-and nanofabrication technologies.…”

Mimicking Boyer’s Casimir repulsion with a nanowire material

“…The Casimir-Lifshitz forces [1][2][3] and especially the socalled Casimir repulsion [4][5][6][7][8][9][10] or "quantum levitation" [11][12][13][14] phenomena in structures of different geometries have been topics of continuous research for many decades. In simple geometries involving planar dielectric or metallic plates in a vacuum, the electromagnetic Casimir force is always attractive, while when the plates are immersed in a dielectric fluid, one can obtain a repulsive force if the permittivities of the plates and the fluid are chosen appropriately [15].…”