Casimir forces in transmission-line circuits: QED and fluctuation-dissipation formalisms

It was recently shown that transmission-line waveguides can mediate long-range fluctuation forces between neutral objects, potentially leading to novel Casimir forces in electric circuits. Here we present two approaches for the general description of these forces between electric components embedded in transmission-line circuits. The first, following ordinary quantum electrodynamics (QED), consists of the quantization and scattering theory of voltage and current waves inside transmission lines. The second approach relies on a simple circuit analysis with additional noisy current sources due to resistors in the circuit, as per the fluctuation-dissipation theorem (FDT). We apply the latter approach to derive a general formula for the Casimir force induced by circuit fluctuations between any two impedances. The application of this formula, considering the sign of the resulting force, is discussed. While both QED and FDT approaches are equivalent, we conclude that the latter is simpler to generalize and solve.

Figure 1

Transmission-line (TL) circuits and the Casimir force. (a) A TL (two parallel thick lines) is characterized by the impedance $Z_0$ of its fundamental TEM mode. Practical realizations range from a pair of parallel wires to two separated conductors in a planar geometry, such as in the widely used coplanar waveguide (see right-hand side). (b) A mirror formed by an impedance $Z$ embedded in a TL: incident waves are scattered forward ($t$) and backward ($r$) [Eq. (4)]. (c) A straightforward adaptation of the Casimir problem to TL circuits. Two circuit elements with impedances $Z_1$ and $Z_2$ are embedded in an infinite TL. Quantum fluctuations of the quantized TEM mode supported by the TL are scattered off the two circuit elements, resulting in an interaction and force between the elements (more practically, the two semi-infinite ends are terminated by resistors with resistivity $R=Z_0$; see, e.g., Sec. 3c).

Figure 2

Source of quantum fluctuations within the FDT formalism. Quantum noise emerges from resistive elements, which are modeled by their resistance $R$ connected in parallel with a current source $I_N$ with the spectrum of Eq. (9) (equivalently, in series with a voltage $V_N=R{I}_N$).

Figure 3

Equivalence between the QED and FDT approaches: noise input. (a) The voltage fluctuations inserted by a resistor $R$ into a TL with impedance $Z_0$ (calculated using the FDT approach) is identical to (b) those inserted by a semi-infinite TL with impedance $R$ (calculated using the QED approach).

Figure 4

Generalized Casimir problem in circuits. (a) Two generally complex impedances terminate the TL. This more natural formulation does not rely on the existence of semi-infinite TLs at both ends of the impedances. (b) Representation of the impedances $n=1,2$ by parallel dissipative ($R$) and reactive ($X$) parts, Eq. (23). (c) Circuit diagram of the Casimir problem. The current sources $I_{N_{1,2}}$ are those of Eq. (9) with the resistive parts $R_{1,2}$ from (b). The force is found by a simple solution of the Kirchhoff laws.

Figure 5

Force between circuit elements which terminate the TL: Examples. (a) Force between a capacitor $C$ and an inductor $L$ (example 3) as a function of their respective parameters $u_C$ and $u_L$ from Eq. (35). The force is in units of the attractive force $f_0=\pi \hslash c/\left(24l^2\right)$ between two identical perfect 1D mirrors. Both attractive ($f/f_0>0$) and repulsive ($f/f_0<0$) forces can be obtained, with the asymptotic limiting values of $f/f_0=1$ and $f/f_0=-1/2$ from examples 1 and 2, respectively. (b) Same as (a) for the first circuit element being a capacitor $C$ in series with a resistor $R$, and the second element being a short circuit ($Z_2=0$).

Figure 6

The force exerted on a circuit component (see text in Appendix pp2).