Tunable refraction in a two-dimensional quantum-state metamaterial

In this paper we consider a two-dimensional quantum-state metamaterial comprising an array of qubits (two-level quantum objects). Here we propose that it should be possible to manipulate the propagation of quantum information. We show that a quantum metamaterial such as the one considered here exhibits several different modes of operation, which we have termed Aharonov-Bohm, intermediate, and quantum-Zeno. We also see interesting behavior which could be thought of as either quantum birefringence (where the material acts like a beam splitter) as well as the emergence of quantum correlations in the circuit's measurement statistics. Quantum-state metamaterials as proposed here may be fabricated from a variety of technologies from superconducting qubits to quantum dots and would be readily testable in existing state-of-the-art laboratories.

Figure 1

Set up, for (a) $2\times 2$ and (b) $3\times 3$ quantum metamaterial where each node represents a qubit and each edge the qubit-qubit coupling coupling. The directed couplings (arrowed, blue) correspond to ${\Xi }_{ij}\left(\varphi \right)={\sigma }_i^{+}{\sigma }_j^{-}exp\left[i\varphi \right]+{\sigma }_i^{-}{\sigma }_j^{+}exp[-i\varphi ]$ and the undirected couplings to ${\Xi }_{ij}\left(0\right)$. Node 1 is considered to be an input node (magenta), and the output nodes are for (a) 2–4 and (b) 5–9 (green). The Lindblad operators $L_j$ describe the interaction with the source ($L_1$) and detectors ($L_{2-9}$).

Figure 2

Plots of detector counts as a function of control phase $\varphi$ and decay constant for the $2\times 2$ array, top row (a, b, c), and the $3\times 3$ array, bottom row (d, e, f). In the left-hand panes ${\gamma }_{\mathrm{in}}={\gamma }_{\mathrm{out}}=0.1$ (a, d), while ${\gamma }_{\mathrm{in}}={\gamma }_{\mathrm{out}}=1$ in the middle panes (b, e), and then ${\gamma }_{\mathrm{in}}={\gamma }_{\mathrm{out}}=5$ in the right-hand panes (c, f). In each case the time over which the statistics for each trajectory were collected was adjusted by a factor of $1/{\gamma }_{\mathrm{in}}$ to allow for comparison.