Parametric amplification in Josephson junction embedded transmission lines

An electronic transmission line that contains an array of nonlinear elements (Josephson junctions) is studied theoretically. A continuous nonlinear wave equation describing the dynamics of the node flux along the transmission line is derived. It is shown that due to the nonlinearity of the system, a mixing process between four waves with different frequencies is possible. The mixing process can be utilized for amplification of weak signals due to the interaction with a strong pump wave. An analytical solution for the spatial evolution of the wave amplitudes is derived, and found to be in excellent agreement with the results of numerical computations. Simulations of realistic parameters show that the power gain can exceed 20 dB over a bandwidth of more than 2 GHz.

Figure 1

Transmission line scheme. The Josephson junctions and the capacitors are represented by $J.J.$ and $C$, respectively. The voltages and the currents are denoted by $V$ and $I$, respectively. The length of all the cells is constant and is equal to $a$.

Figure 2

Evolution of (a) the normalized power of the pump wave $B_p^2$, signal wave $B_s^2$, and idler wave $B_i^2$, and (b) the phase mismatch $\Theta$ versus normalized position along the transmission line. The numerical solution of Eqs. (A24, A25, A26), and (A31) and the analytical solution Eqs. (35) and (46) are plotted by solid lines and dashed lines, respectively. Note that the corresponding numerical and analytical curves are overlapping. The vertical dashed line is located in the point $x=1/g$ after which the wave amplitudes grow exponentially with exponent $g$. The asymptotic value of the phase ${\Theta }_{\infty }$ expressed by Eq. (62) is plotted by a horizontal dashed line in Fig. 2b.

Figure 3

Signal power gain in the system for the case of Fig. 2 ($B_{p,0}=0.4$) as well as for lower initial pump amplitudes $B_{p,0}=0.2$ and $0.3$. The numerical solution calculated from the solution of Eqs. (A24, A25, A26) and (A31) and the analytical solution Eq. (51) are plotted by solid lines and dashed lines, respectively. Note that the corresponding numerical and analytical curves are overlapping.

Figure 4

Signal power gain as a function of signal frequency and position given by the analytical expression Eq. (51). All the parameters are the same as in Fig. 2 except the varying signal frequency. The black line is located where the gain level is 3 dB below the local peak.

Figure 5

Exponential gain factor $g=\sqrt{-{\Delta }_p}$ versus signal frequency [given by Eq. (52)]. The maximal gain is obtained at $f_s=f_p$. The approximate value $g_{\max }$ given by Eq. (57) is plotted by a horizontal dashed line. The simplified expression for the amplification bandwidth for which $g>0$, given by Eq. (56) is plotted by vertical dashed lines.

Figure 6

Spatial evolution of the system for the same case of Fig. 2 in long distance. In long distance, in which the amplified signal becomes large, the underlying assumption of the analytical solution derivation fails, and the analytical solution becomes invalid.