Long-range waveguide-quantum electrodynamics with left-handed transmission lines

Abstract

While engineering long-range light-matter interactions is the principal aim in waveguide-QED, ironically most of the building blocks rest on local short-range couplings, such as nearest-neighbor-coupled cavity arrays employed in canonical models. Here, we propose a waveguide-QED system with native long-range interactions, comprising a single emitter coupled to a left-handed transmission line (LHTL). Interestingly, the LHTL emulates a synthetic photonic lattice with a slow logarithmic decay of hopping amplitudes over a distance set entirely by the ratio of UV and IR cutoffs of line dispersion. Its intrinsic long-range nature manifests both in the properties of atom-photon bound and scattering states, which exhibit algebraic localization and accelerated photon propagation respectively. Using a method of 'running exponents', we develop a unified picture connecting waveguide dispersion to bound state and light front profiles obtained in the strong long-range hopping regime. These results motivate how transmission lines can enable multi-qubit information processing with tunable-range interactions.

Figures (6)

• Figure 1: Left-handed waveguide-QED.A Circuit representation for a transmon qubit capacitively coupled to a LHTL. B LHTL dispersion for different values of UV cutoff $\omega_c$. C Spectral density seen by the qubit for the same set of cutoff frequencies.
• Figure 2: Hopping amplitudes for transmission lines.A LHTL maps to a tight-binding photonic lattice with logarithmic fall off of hopping amplitudes $\xi(z)$ for $z<1$, and exponential fall off for $z>1$ ($z \equiv n/n_\star$), while RHTL maps to a power-law hopping network with a constant exponent $\alpha = 2$. B Position-dependent local (solid) and global (dashed) hopping exponents for LHTL and RHTL obtained using the running exponent method. C Long-range regimes for 1D power-law hopping network, with $\alpha \to \infty$ corresponding to nearest-neighbor models.
• Figure 3: Algebraic bound-states in LH waveguide-QED.A Spatial profile of bound state for $n_\star \to \infty$, for different emitter frequencies ($g/\omega_0 = 0.1$) shown in different color markers. The dashed black line shows the fit to $1/n^4$ power-law. The top-right inset shows the photon intensity profiles near the qubit location for the three cases. The lower-left inset shows a slightly different asymptote values for two different emitter frequencies (detunings). B Spatial profiles of bound state, for different values of $\omega_{\rm UV} = \omega_{c}$, each showing the exponential decay in the region $z > 1 (n> n_{\star})$ (indicated with the gray gradient on the right). C Position-dependent local bound state exponent, $2\beta(z)$, for left-handed (solid-orange) and right-handed (dashed-blue) waveguide-QED systems, calculated using the running exponent method.
• Figure 4: Qubit-Dynamics for infinite UV cutoff.A Contour in Laplace $\tilde{s}$-space used for calculating qubit dynamics. Solid red line and the black cross show the branch cut due to the continuum and bound-state pole $y_{p}$ outside the continuum, respectively. B Profile of spectral weight function $M(x)$ plotted in rescaled frequency units $x$ ($=2\omega/\omega_0$), calculated for $\Delta/\omega_0 = 1.5,\, g/\omega_0 = 0.2$. Solid line shows the full function, while patterned lines show the approximations used for capturing the contribution of peak near the dressed emitter frequency ($x \leq 2 \Delta/\omega_0$) (dot-dashed) and contributions from the band-edge (dashed line), respectively. C Emitter population dynamics for $\Delta$ across the LHTL band-edge ($\omega_0/2$). D Time-slices at different emitter frequencies taken from (C). The arrow indicates increasing relaxation time with increasing $\Delta$ (decreasing decay rate). E Relaxation time $T_1 \equiv 1/\omega_0\gamma$ analytically calculated at emitter frequencies in (D). F Crossover from exponential relaxation $e^{-\gamma t}$ to long-time power-law tail $t^{-3}$.
• Figure 5: Accelerated light cones in LH waveguide-QED.A Simulated space-time diagrams of photon intensity $\ln |C_n(t)|^2$ for a qubit parked inside the pass band of LHTL ($\Delta/\omega_0 = 1$, $\omega_c/\omega_0 = 30$, $g/\omega_0 = 0.08$). Simulated constant intensity fronts, $t_\epsilon(n)$ for $\epsilon = 10^{-12}$, are plotted as solid white-circles $(N = 1501)$, while the short-time analytical result is shown with the dashed-black line. Dot-dashed white line corresponds to the linear light cone $t = n/v_\text{max}$. B Empty circles represent the local slope for light-cone obtained from simulations, and the solid-orange curve represents the analytical result obtained in Eq. \ref{['eq: early-time local cone exponent']}. C,D,E show the space-time density profiles of scattered photons within the accelerated windows for three different choices of $n_{\star} = 2,\, 40,\, 100$. In each density plot, circle, square, and triangle markers show $\epsilon$-fronts for three different choices of $\log(\epsilon) = -12$, $-14$, and $-16$, respectively ($N = 4001$).