Topological Casimir effect in a quantum LC circuit: Real-time dynamics

We study novel contributions to the partition function of the Maxwell system defined on a small compact manifold $\mathbb{M}$ with nontrivial mappings ${\pi }_1[U(1)]\cong \mathbb{Z}$. These contributions cannot be described in terms of conventional physical propagating photons with two transverse polarizations and instead emerge as a result of tunneling transitions between topologically different but physically identical vacuum winding states. We argue that if the same system is considered in the background of a small external time-dependent electromagnetic field, then real physical photons will be emitted from the vacuum, similar to the dynamical Casimir effect where photons are radiated from the vacuum due to time-dependent boundary conditions. The fundamental technical difficulty for such an analysis is that the radiation of physical photons on mass shell is inherently a real-time Minkowskian phenomenon while the vacuum fluctuations interpolating between topological $|k⟩$ sectors rest upon a Euclidean instanton formulation. We overcome this obstacle by introducing auxiliary topological fields, which allows for a simple analytical continuation between Minkowski and Euclidean descriptions, and develop a quantum mechanical technique to compute these effects. We also propose an experimental realization of such small effects using a microwave cavity with appropriate boundary conditions. Finally, we comment on the possible cosmological implications of this effect.

Figure 1

An illustration of the mechanism of photon emission in the conventional DCE (top) and in the TCE or Maxwell system on a compact manifold (bottom). In the conventional DCE, the accelerating Casimir plates turn some of the virtual photons into real on-shell propagating photons which leave the system. In the case of the TCE, the tunneling transitions between infinitely degenerate vacuum winding states $|m⟩$ are represented by instanton solutions $A_{\mathrm{top}}^{\mu }$. These instanton configurations cannot be expressed in terms of physical transverse propagating E&M fields. Precisely, these topological configurations are eventually responsible for the emission of real photons in a time-dependent background; see Secs. 4 and 5 for details.