Casimir problem of spherical dielectrics: numerical evaluation for general permittivities

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory
• Statistical Mechanics

Background:

• The Casimir effect describes forces between uncharged objects arising from quantum vacuum fluctuations.
• Previous work calculated the Casimir free energy for ideal metals.
• Understanding this effect for dielectric materials is crucial for various applications.

Purpose of the Study:

• To calculate the Casimir mutual free energy for a system of two dielectric concentric nonmagnetic spherical bodies at arbitrary temperatures.
• To extend previous findings for ideal metals to dielectric materials with specific refractive indices.
• To analyze the contribution of the zero-frequency TE mode for both dielectrics and real metals.

Main Methods:

• Utilizing quantum statistical mechanics as the fundamental calculational approach.
• Employing high-order Debye expansions for Riccati-Bessel functions to manage numerical precision and avoid overflow/underflow.
• Investigating the zero-frequency TE mode mathematically and incorporating physical dispersion relations, specifically the Drude model at low frequencies.

Main Results:

• The Casimir free energy for dielectric spheres was calculated for arbitrary temperatures.
• For real metals, the zero-frequency TE mode was found to be absent, resulting in half the conventional high-temperature Casimir free energy.
• The zero-frequency mode contribution was explicitly calculated for dielectrics, where the issue is absent.

Conclusions:

• The Debye expansion method provides accurate and numerically stable results for Casimir calculations.
• The absence of the zero-frequency TE mode in real metals significantly alters the Casimir free energy predictions at high temperatures.
• Further experimental validation is needed to distinguish between theoretical predictions, especially at low temperatures.