Left-handed superlattice metamaterials for circuit QED

Quantum simulation is a promising field where a controllable system is used to mimic another system of interest, whose properties one wants to investigate. One of the key issues for such simulation is the ability to control the environment the system couples to, be it to isolate the system or to engineer a tailored environment of interest. One strategy recently put forward for environment engineering is the use of metamaterials with negative indices of refraction. Here we build on this concept and propose a circuit-QED simulation of many-body Hamiltonians using superlattice metamaterials. We give a detailed description of a superlattice transmission line coupled to an embedded qubit, and show how this system can be used to simulate the spin-boson model in regimes where analytical and numerical methods usually fail, e.g., the strong coupling regime.

Figure 1

Composite left-handed superlattice and right-handed transmission line coupled to a qubit. The superlattice unit cell is formed by two cells, indicated by red (dark gray) and green (light gray) backgrounds. The equivalent discrete circuit model for the RHTL is illustrated.

Figure 2

Incoming and outgoing voltage and current of a lattice cell with impedances $Z_1$ and $Z_2$. The matrix describing the relations between outgoing and incoming values in terms of $Z_1$ and $Z_2$ is the ABCD matrix.

Figure 3

Mode number vs frequency of a left-handed superlattice transmission line with $\epsilon =2$. The periodic structure gives rise to a band gap with the high-frequency band revealing the left-handedness of the system.

Figure 4

Width of the lower frequency band as a function of the superlattice parameter $\epsilon$.

Figure 5

Numerical (red dots) and theoretical (black line) approximations to the mode density $D\left(\omega \right)$ of the coupled system for a superlattice with 200 cells and $\epsilon =2$.

Figure 6

Voltage profile for the modes with index 50 to 53 for a transmission line with $n_{\mathrm{sl}}=200$ and $\epsilon =2$. The superlattice is positioned at the left side and the RHTL on the right side in the plot with their coupling at $0.01\mathrm{m}$. Due to the discreteness of the superlattice, the lines do not represent the real voltage profile for each position along the circuit but only at circuit nodes which are represented by dots.

Figure 7

Typical solution to Eq. (18) on inverted axes for 20 superlattice cells with $\epsilon =2$ and an initial energy splitting of the qubit of ${\mathrm{\Delta }}_0=1.2{\omega }_{\mathrm{sl}}$. Blue (solid) line shows the coupling strength $g$ (horizontal axis) for a given renormalized tunnelling element (vertical axis). Orange (dashed) line indicates the value to which ${\mathrm{\Delta }}_{\mathrm{eff}}$ converges.

Figure 8

Effective tunneling rate of the qubit as a function of the coupling constant (both in units of the superlattice resonance frequency ${\omega }_{\mathrm{sl}}$) for $\epsilon =2$ and 200 superlattice cells. Each color represents a different initial bare tunneling element, which coincides with the effective tunneling element for zero coupling. Shaded regions represent the two energy bands. Localized, weakly localized, and delocalized phases are clearly visible.

Figure 9

Phase diagram of the spin-boson model of a hybrid transmission line environment with $\epsilon =1.1$ and 200 superlattice cells. Four distinct phases are shown. The colored background shows the energy region (separated by the band edges ${\omega }_{\text{1-}}, {\omega }_{1+}$, and ${\omega }_2$) ${\mathrm{\Delta }}_{\mathrm{eff}}$ is lying in as a contour plot, whereas black dots are actual jumps found from numerics. For smaller coupling strengths jumps are less likely to be found numerically due to the decreased jump strength and the discrete nature of the superlattice.