Effects of lasing in a one-dimensional quantum metamaterial

Electromagnetic pulse propagation in a quantum metamaterial, an artificial, globally quantum coherent optical medium, is numerically simulated. We show that a one-dimensional quantum metamaterial based on superconducting quantum bits, initialized in an easily reachable factorized excited state, demonstrates lasing in the microwave range, accompanied by the chaotization of qubit states and generation of higher harmonics. These effects may provide a tool for characterization and optimization of quantum metamaterial prototypes.

Figure 1

(a) One-dimensional (1D) quantum metamaterial based on charge qubits placed in a waveguide formed by two bulk superconductors. The charge qubit is formed by a superconducting island, separated from the bulk superconductors by identical Josephson junctions, and plays the role of artificial atoms. The green wave represents distribution of the electromagnetic wave in the metamaterial. The control circuitry of qubits (gate electrodes) is not shown. (b) The cross-sectional view of the 1D quantum metamaterial. The superconducting waveguides are separated by the distance $D$. The charge qubits align at regular intervals $L$. ${\varphi }_n$ is the superconducting phase of the $n\mathrm{th}$ qubit island. $A_{z,n},B_{y,n}$, and $E_{z,n}$ are the vector potential, the magnetic field, and the electric field of the $n\mathrm{th}$ unit cell, respectively.

Figure 2

(a) The initial distribution of vector potential $a_z$ in a unit cell (Gaussian pulse). (b) Time evolution of energy in the system. Bottom to top: field-qubit interaction energy $E_{\mathrm{int}}$ (mauve), electromagnetic field energy $E_{\mathrm{field}}$ (blue), qubit energy $E_{\mathrm{qubit}}$ (green), and total energy $E_{\mathrm{total}}$(red).

Figure 3

(left) Electric field amplitude and (right) the probability amplitude of the excited state in the metamaterial at $\tau =300,320,340,360,380,400,420$, and 600 (top to bottom). The origin of $x$ is shifted to keep the wave packet at the center of the panel.

Figure 4

Qubit energy $E_{\mathrm{qubit}}$ as a function of time for various field amplitudes. Right to left: $P=0.05$ (green), 0.2 (blue), 0.5 (pink), 1 (turquoise), 1.5 (gray). At $P=0.02$ (brown) lasing did not start up to $\tau =40000$.

Figure 5

Field frequency spectrum (top) for $\tau =120\text{–}240$ and (bottom) for $\tau =600\text{–}720$. Note the appearance of both higher-harmonic (at $\sim 4\pi =2{\omega }_g$) and subharmonic peaks after the lasing occurred.

Figure 6

Field frequency spectrum for $\tau =0\text{–}60$ (red) and for $\tau =60\text{–}120$ (blue) in the case of $\epsilon =2{\omega }_g=4\pi$. The subharmonic component $\frac{\epsilon }{2}={\omega }_g$ increases due to parametric lasing.

Figure 7

Field energy $E_{\mathrm{field}}$ as a function of time in the presence of dissipation. Left to right: $\sigma =1\times {10}^{-4}$ (red), $5\times {10}^{-4}$ (green), $2.5\times {10}^{-3}$ (blue). The cross points indicate the field energy changing with time according to the typical quality factor ${10}^4$.