Proximity force approximation for the Casimir energy as a derivative expansion

Abstract

The proximity force approximation (PFA) has been widely used as a tool to evaluate the Casimir force between smooth objects at small distances. In spite of being intuitively easy to grasp, it is generally believed to be an uncontrolled approximation. Indeed, its validity has only been tested in particular examples, by confronting its predictions with the next-to-leading-order (NTLO) correction extracted from numerical or analytical solutions obtained without using the PFA. In this article we show that the PFA and its NTLO correction may be derived within a single framework, as the first two terms in a derivative expansion. To that effect, we consider the Casimir energy for a vacuum scalar field with Dirichlet conditions on a smooth curved surface described by a function c in front of a plane. By regarding the Casimir energy as a functional of c , we show that the PFA is the leading term in a derivative expansion of this functional. We also obtain the general form of the corresponding NTLO correction, which involves two derivatives of c . We show, by evaluating this correction term for particular geometries, that it properly reproduces the known corrections to PFA obtained from exact evaluations of the energy.

Casimir interaction at liquid nitrogen temperature: Comparison between experiment and theory

“…At separations of several hundred nanometers, important for this experiment, the error introduced by the use of PFA does not exceed a fraction of a percent [43][44][45][46] , i.e., much less than the experimental error (see below). It is instructive to compare the measurement data at T = 77 K and T = 300 K. This is done in Fig.…”

“…Thus, the effective Casimir pressure between two parallel plates was determined 42 at T = 2.1 K, 4.2 K and 77 K by means of a micromachined oscillator. This was done dynamically by measuring the change in resonant frequency of an Au sphere oscillating near an Au plate with the help of the proximity force approximation (PFA) 7,8 (note that recently the applicability of PFA for the configuration of a sphere and a plate made of real materials was confirmed to a high precision [43][44][45][46] ). It was shown 42 that although the low temperature data are noisier than at room temperature, the mean measured Casimir pressures coincide at all temperatures.…”

Calculating Casimir interactions for periodic surface-relief gratings using theCmethod

“…(66) corresponds to the derivative expansion (DE) introduced in Ref. [16] for scalar fields and Refs. [17,18] for electric fields with perfect conductor or dielectric boundaries.…”

“…As the experiments have become more sensitive, it has become necessary to do the theoretical analysis outside the range of validity of the PFA. Recently a derivative expansion (DE) approach has been introduced [16][17][18], which reproduces the PFA and gives the next order correction, and which has successfully explained a number of new recent experiments [13,14]. The first theoretical calculation of the Casimir force between geometrically patterned surfaces not using the PFA was reported in 2001 [19,20], which described the normal and lateral forces between two aligned corrugated surfaces in terms of a perturbative expansion in the profile height.…”

Derivative expansion for the electromagnetic and Neumann-Casimir effects in2+1dimensions with imperfect mirrors

“…This observation can be used as the starting point for an approximation scheme, the Derivative Expansion (DE) originally proposed in Ref. [2] (for subsequent developments see Ref. [3]).…”