Inductive van der Waals Force between Two Quantum Loops

Abstract

We study the van der Waals-London force, which is typically associated with fluctuating dipoles in atoms, in a mesoscopic circuit consisting of two inductively coupled superconducting loops. We investigate the ``inductive" van der Waals-London interaction using both semiclassical and quantum electrodynamic (QED) approaches. The semiclassical model predicts a repulsive interaction due to anticorrelated current fluctuations. In contrast, the QED framework, which incorporates virtual photon exchange, reveals a predominantly attractive force. A key contribution comes from a state-independent two-photon exchange, which is absent in the semiclassical description and undetectable by spectroscopy. Our study introduces a new experimental platform for measuring the van der Waals force between individual artificial atoms via controlled mesoscopic circuits.

Figures (4)

• Figure 1: Schematic diagrams of the two interacting quantum loops: (a) In the semiclassical model, the loops are represented by an inductive coupling of the two quantum LC circuits. (b) In QED, vacuum photons mediate the interaction between the loops.
• Figure 2: (a) Representation of the interaction between the loop state (solid lines) and the photons (wavy lines). The two types of interaction, $W$ and $X$, involve the single (single wavy line) and double (double wavy line) photons, respectively. (b) Leading-order single-photon exchange through $W$ leads to the magnetic interaction between two current-flowing states.
• Figure 3: These diagrams illustrate the inductive van der Waals-London interaction between two loops in the current-free ground state via the exchange of two photons. The sum of diagrams (i) and (ii) is equivalent to the semiclassical result. The second-order $X$ process (diagram (iii)) is the predominant contribution. Diagram (i-a), related to the retardation effect, is negligible. Symbols $g$ and $e$ represent the ground and the excited states of a loop, respectively.
• Figure 4: (a) The semiclassical interaction energy, $\Delta E_\mathrm{sc}$ (see Eq. \ref{['eq:Esc']}), and (b) the mutual force, $F_3$ (see Eq. \ref{['eq:F3']}), calculated from the QED approach, as a function of the distance $R$ between the two circular loops with $b/a=0.05$. Note that the dominant QED term yields an attractive mutual force, $F_3$, which contrasts sharply with the repulsive interaction energy $\Delta E_\mathrm{sc}$, or $\Delta E_i+\Delta E_{ii}$ (see Eq. \ref{['eq:DE1+DE2']}).