Left-Handed Transmission Lines for the Dynamical Casimir Effect

Abstract

Left-handed (LH) superconducting transmission lines, built with series capacitors and shunt inductors instead of the series inductors and shunt capacitors of a conventional right-handed (RH) line, have a spectrum with a nonzero lower band edge and modes that become sparse at high frequencies. We study the dynamical Casimir effect in a LH line terminated by a flux-modulated SQUID and compare it with an RH line built from the same elements and with a long coplanar waveguide terminated by the same SQUID. We derive the quantum Hamiltonian and show that the coupling between two modes factorizes into their amplitudes at the SQUID. Together, the LH spectrum and the mode dependence of this coupling determine which modes are populated and how fast. With equal elements, the LH coupling grows with frequency and exceeds the RH one above a crossover pump frequency set by the cell inductance and capacitance, and only the LH line has resonances above the RH band edge. Pumping two modes at once lets a weakly coupled mode borrow coupling from a strongly coupled one, while crossings of resonance lines populate three or four modes without increasing the growth rate. Compared with the coplanar waveguide, the LH line has a similar coupling but keeps the created particles in the target mode, whereas the nearly equidistant waveguide spreads them over many pairs and needs more pump energy.

The paper asks how a left-handed (LH) superconducting transmission line—series capacitors and shunt inductors—changes dynamical Casimir photon generation under periodic modulation of a terminating SQUID. It compares this geometry with a right-handed (RH) array built from identical elements and with a long coplanar-waveguide cavity. Circuit quantization gives boundary-mediated couplings $g_{ij}\propto u_i^N u_j^N/\sqrt{\omega_i\omega_j}$, implying $g_{ij}^2=g_{ii}g_{jj}$. Normal-mode calculations, rotating-wave solutions and numerical examples show that LH coupling is suppressed near its lower band edge but increases at higher frequencies, where modes are sparse. For equal cell elements, the approximate single-mode pump-frequency crossover is $\Omega_c\simeq2/\sqrt{LC}$: RH coupling is larger below it and LH coupling above it. Two-mode pumping can increase excitation of a weakly coupled LH target by pairing it with a strongly coupled higher-frequency mode, while producing equal photon numbers in both modes. At resonance crossings, two creation processes evolve independently, whereas creation plus mode exchange can populate three modes with reduced growth rate $\sqrt{g_{kl}^2-g_{km}^2}$. Near the LH band edge, fixed-edge-index asymptotics give coupling $g\sim N^{-3}$ versus spacing $\Delta\omega\sim N^{-2}$. In one waveguide comparison, total created energy relative to target energy is 1 for LH versus 1.8–6 for the waveguide. The analysis assumes a linearized SQUID and resonantly isolated processes where applicable. Accidental multimode coincidences remain possible; high-frequency parasitic RH elements are excluded, and the ideal LH line has no continuum limit.