Circuit-Induced Casimir-Lifshitz Force

• Circuit-induced Casimir-Lifshitz force is a fluctuation-mediated interaction arising from electromagnetic scattering in circuit components with well-defined response functions.
• It is computed using a scattering-T operator framework that incorporates impedance, capacitance, inductance, and geometry to determine the force’s magnitude and direction.
• This phenomenon enables experimental tunability in NEMS/MEMS and quantum circuits by engineering material, topological, and geometric parameters to achieve precise force control.

The circuit-induced Casimir-Lifshitz force is the fluctuation-mediated force that arises between circuit elements due to their electromagnetic response to vacuum or thermal fluctuations, generalizing the traditional Casimir-Lifshitz effect from macroscopic bodies to integrated electronic systems such as transmission lines, nanocircuits, and micro/nanoelectromechanical devices. This interaction, rooted in the scattering of electromagnetic fluctuations by circuit components with well-defined response functions, is governed by both material parameters (resistivity, capacitance, inductance, dielectric function) and circuit geometry (separation, connectivity, layout). The circuit-induced Casimir-Lifshitz force serves as a unifying framework in analyzing, predicting, and engineering mesoscopic quantum forces in the context of contemporary quantum and hybrid technologies.

1. Scattering-T Operator Framework for Fluctuation Forces

The foundational theoretical approach to compute circuit-induced Casimir-Lifshitz forces is an adaptation of general scattering and statistical field theory methods, previously applied to macroscopic dielectric or conducting bodies, to circuits. The core procedure involves:

• Representing each circuit element as an object with a defined electromagnetic response, such as an effective dielectric function $\epsilon(ic\kappa)$, linear impedance, or a microscopic potential $V$ encoding the local response.
• Expressing the total fluctuational energy by integrating over the fluctuating currents (or fields) coupled to these elements. This is mathematically formalized through the introduction of a potential operator $V$ and the T-operator

$T = V(I + G_0 V)^{-1} \ ,$

where $G_0$ is the free electromagnetic Green’s function.

• For two circuit elements $A$ and $B$, the Casimir energy is given by

$$\mathcal{E} = \frac{\hbar c}{2\pi} \int_0^\infty d\kappa \, \log \det \left[ I - F_A U^{(AB)} F_B U^{(BA)} \right] \ ,$$

where $F_{A,B}$ encode the on-shell scattering amplitudes (dependent on the local electromagnetic response of the elements), and $U^{(AB)}$ represent translation matrices mapping basis functions between the geometries or locations of the two elements (Emig, 2010).

In circuit layouts, the translation matrices $U$ encapsulate not only physical distances between components but also orientation, wiring, and the possible presence of additional circuit branches or shunt elements. Any quantum or thermal fluctuation in, e.g., a superconducting resonator, produces a scattered field that then interacts with neighboring conductors, yielding an energy shift whose derivative gives the force.

2. Circuit Modeling: From Dielectric Functions to Response Functions

Each circuit component is characterized via a linear response function, mapping the traditional macroscopic dielectric description onto frequency-dependent circuit parameters:

• For lumped elements, the response is determined by capacitance $C$, inductance $L$, resistance $R$, or their frequency representations (e.g., complex impedance $Z(\omega)$).
• The dielectric response in a circuit can be described with generalized models, such as:
  • Plasma model: $$\epsilon_p(ic\kappa) = 1 + \left(\frac{2\pi}{\lambda_p \kappa}\right)^2$$
  • Drude model: $$\epsilon_D(ic\kappa) = 1 + \frac{(2\pi)^2}{(\lambda_p\kappa)^2 + \pi c\kappa/\sigma}$$
  • where $\sigma$ is the conductivity and $\lambda_p$ the plasma wavelength.

Changing doping, temperature, or composition (as in gapped metals or off-stoichiometric patches) modifies these parameters and hence alters the induced Casimir-Lifshitz force (Boström et al., 2023, Boström et al., 14 Feb 2024, Pal et al., 12 Jun 2024).

Circuit geometry is encoded at the level of multipole expansions or modal decompositions: for quasi-1D systems (waveguides, TLs), translation matrices $U$ reduce to phase shifts and scattering matrices; for more complex or mixed topologies, $U$ reflects the spatial arrangement and effective circuit distances.

3. Quantum Fluctuations, Dissipation, and Circuit Fluctuation Force

At zero or finite temperature, the circuit-induced Casimir-Lifshitz force arises from the spectrum of quantum and thermal fluctuations intrinsic to all dissipative circuit elements:

• Any resistor is accompanied by a quantum noise current source, whose spectral density (at $T=0$) is given by

$S_{I_N}(\omega) = \frac{2\hbar\omega}{R(\omega)}.$

• In a general RLC circuit containing a capacitor with capacitance $C$, embedded in series/parallel with other components, the circuit’s impedance $Z(\omega)$ determines the voltage noise across and energy stored in that capacitor (Shahmoon et al., 2016). The zero-point energy is then:

$$U = \hbar \int_{0}^{\infty} \frac{d\omega}{2\pi} \frac{R(\omega)C\omega}{[1+X(\omega)C\omega]^2 + [R(\omega)C\omega]^2}.$$

• The generalized force exerted by the fluctuational potential on a mechanical coordinate $\xi$ (such as the plate gap of a variable capacitor) is

$$f = -\frac{\partial U}{\partial \xi} = -\frac{\partial U}{\partial C}\frac{\partial C}{\partial \xi}.$$

• This framework allows for experimental tunability: by changing $R$, $L$, $C_0$ or by modulating $C$ via an external actuator, it is possible to engineer both the sign (attractive vs. repulsive) and scaling (e.g., $1/y^2$, $1/\sqrt{y}$) of the circuit-induced force (Shahmoon et al., 2016).

For transmission-line circuits, both quantum electrodynamics (canonical quantization of voltage/current fields) and the fluctuation-dissipation theorem (classical circuit analysis plus noise sources) yield the same Casimir-Lifshitz force expressions. In one spatial dimension, the force between two impedances $Z_1(\omega)$ and $Z_2(\omega)$ separated by distance $\ell$ in a TL with characteristic impedance $Z_0$ is

$$f = \frac{\hbar c}{\pi \ell^2} \int_0^\infty du \, u \frac{r_1(iu) r_2(iu) e^{-2u}}{1 - r_1(iu) r_2(iu) e^{-2u}}$$

where $$r_j(\omega) = \frac{Z_j(\omega) - Z_0}{Z_j(\omega) + Z_0}$$ (TE/TM analog in 1D) (Shahmoon, 2017).

4. Spectral Range, Ohmic Approximation, and Experimental Regimes

The spectral distribution of fluctuations determines practical observability and modeling simplifications:

• If both resonance frequency $\Omega$ and damping $\gamma$ of the oscillator (or RLC circuit) depend on the geometric parameter $d$, the Casimir-like force includes high-frequency contributions up to a cutoff set by the frequency dependence of $\gamma$.
• In most electrical circuits, only the resonance (e.g., via $C(d)$) shows significant $d$-dependence, while $\gamma$ is weakly dependent. In this regime, low-frequency ("Ohmic") contributions dominate, and the circuit-induced force reduces to the form (Barash, 27 Aug 2025):

$$f(d) \approx -\frac{\partial F}{\partial d} \approx -\left( \frac{T}{\Omega(d)} + \ldots \right) \frac{\partial \Omega(d)}{\partial d} .$$

For a planar electromechanical capacitor,

$f_T = -\frac{T}{2d} ,$

at high temperatures.

• At low $T$, and for weak dissipation, the zero-point contribution dominates:

$$f_0 \propto -\frac{\hbar}{4\sqrt{\epsilon_0 L S d}} ,$$

where $L$ is the circuit inductance and $S$ the plate area.

This reduction is crucial for extending the theory to lumped-element circuit models and ensures that circuit-induced Casimir-Lifshitz effects can be meaningfully isolated in real devices.

5. Geometry, Non-Additivity, and Material/Topological Control

The circuit-induced Casimir-Lifshitz force is highly non-additive and depends sensitively on both circuit geometry and topological properties:

• Multipole expansions and translation matrices $U$ encode complex geometrical arrangements. Parasitic branches, shielding, and the three-body effect (presence of a ground plane or multiple circuit paths) can produce strong non-additive screening or enhancement (Emig, 2010).
• The topology of the electromagnetic mode structure (including possible nonreciprocity from motion, artificial materials, or breaking of time-reversal symmetry) can yield repulsive forces. For example, relative motion (layer sliding) introduces nonreciprocal phase shifts and magnetoelectric coupling, generating repulsion even in otherwise identical dielectrics (Maslovski, 2011).
• The electromagnetic response in topologically nontrivial materials (Chern insulators, axionic TIs) or in engineered circuits with analogous Hall conductance can suppress or reverse the sign of Casimir-Lifshitz interactions (Lu, 2021).

Manipulation of the material parameters—such as phase (stoichiometry) in gapped metals, chemical potential in graphene, or by using nonlocal dielectric functions—enables active tuning and engineering of the Casimir-Lifshitz force between integrated circuit elements (Boström et al., 2023, Boström et al., 14 Feb 2024, Pal et al., 12 Jun 2024).

6. Practical Applications in Modern Circuits and NEMS/MEMS

The realization and control of circuit-induced Casimir-Lifshitz forces have immediate consequences for nanotechnology, quantum electronics, and microsystems engineering:

• In NEMS/MEMS, Casimir-Lifshitz and circuit-induced forces can be comparable and must be controlled to avoid stiction or collapse of nanoscale gaps. Repulsive forces, tunable by circuit design or environmental control, offer routes to anti-stiction solutions (Schmidt et al., 2022, Shahmoon et al., 2016).
• In quantum circuits, fluctuation-induced forces act as tunable potentials or energy level shifts for superconducting qubits or as engineered couplings in hybrid systems.
• By changing circuit parameters (e.g., varying phase in a gapped metal surface, changing bias or doping), actuation, switching, and noncontact manipulation at the nanometer scale can be achieved, as in quantum levitation, nanoparticle trapping, or reversible switching between attraction and repulsion at sub-10 nm separations (Pal et al., 12 Jun 2024).
• The circuit-induced Casimir-Lifshitz force can be isolated experimentally by modulating only circuit parameters (e.g., $R$, $L$, or environment), leaving the geometric separation fixed, and measuring response signatures in high-$Q$ systems (Shahmoon et al., 2016, Barash, 27 Aug 2025).
• Theoretical generalizations permit calculation in arbitrary layered or nested circuit geometries using recursion relations for Fresnel coefficients (in the language of impedance matching) (Tomas, 2011).

7. Methodological and Theoretical Generalizations

Recent advances offer unifying and universal computational schemes for Casimir-Lifshitz forces in both bulk and circuit settings:

• Operator-based formulations (bulk T-operator and surface operator $M$) enable a reduction in complexity, allowing volume or surface integral formulations that extend to arbitrary circuit topologies and interpenetrating geometries (Bimonte et al., 2021), with the Casimir free energy given by

$$F = k_B T \sum_n' \log \det \left( \frac{M(i\xi_n)}{M_\infty(i\xi_n)} \right) .$$

• Gelfand-Yaglom theorem and functional determinant techniques simplify calculations in systems where the elements are modeled as localized potentials (analogous to points or short segments in circuits) (Ttira et al., 2011).
• Comparison of QED (canonical quantization and scattering) and fluctuation-dissipation (noise-plus-Kirchhoff circuit analysis) approaches has confirmed equivalence but shown the latter’s superiority in handling general circuit geometries and dissipative elements (Shahmoon, 2017).

The implication is that circuit-induced Casimir-Lifshitz forces are subject to quantum engineering at the interface of condensed matter, photonics, and circuit electrodynamics, with a suite of robust mathematical tools available for prediction, design, and experimental extraction.