Asymmetric Frequency Conversion in Nonlinear Systems Driven by a Biharmonic Pump

A novel mechanism of asymmetric frequency conversion is investigated in nonlinear dispersive devices driven parametrically with a biharmonic pump. When the relative phase between the first and second harmonics combined in a two-tone pump is appropriately tuned, nonreciprocal frequency conversion, either upward or downward, can occur. Full directionality and efficiency of the conversion process is possible, provided that the distribution of pump power over the harmonics is set correctly. While this asymmetric conversion effect is generic, we describe its practical realization in a model system consisting of a current-biased, resistively shunted Josephson junction. Here, the multiharmonic Josephson oscillations, generated internally from the static current bias, provide the pump drive.

Figure 1

Asymmetric parametric frequency converter. (a) Minimal model based on a time-varying mutual inductance $M(t)$ with frequencies ${\omega }_p$ and $2{\omega }_p$. (b) Frequency landscape showing various frequency mixing processes between the modulation frequency ${\omega }_m$ and the sidebands at ${\omega }_{\pm }$; solid and dashed arrows indicate the amplitudes of different modes and their conjugates, respectively. The modulation frequency ${\omega }_m$ is chosen to coincide with the center resonance ${\omega }_A$ of the low frequency oscillator, and the pump tone ${\omega }_p$ is chosen to coincide with the center resonance ${\omega }_B$ of the high frequency oscillator. The colors of the mixing arrows indicate the colors of the relevant pump frequencies mediating the process, as shown on the separate pump axis. The black arrows indicate reflections.

Figure 2

Asymmetric frequency conversion. (a) Asymmetry between frequency conversion amplitudes $|t_d{|}^2-|t_u{|}^2$ [Eq. (3)] plotted as a function of the strength of the second harmonic (${\xi }_2$) of the pump and phase frustration $\alpha$ between the harmonics. Maximum asymmetry is realized for maximal frustration $\alpha =\pi /2\pm n\pi$, with down-conversion occurring when the second harmonic leads the first harmonic ($\alpha =+\pi /2+2n\pi$) and up-conversion occurring when the second harmonic lags behind the first harmonic ($\alpha =-\pi /2+2n\pi$). (b) Maximal asymmetry plotted as a function of modulation frequency parametrized as dimensionless detuning ${\delta }_{\pm }={\omega }_m/{\mathrm{\Gamma }}_B$. (c) Interference of different scattering amplitudes showing the realization of asymmetry as in (a). Solid lines indicate the usual frequency conversion in the presence of a single monochromatic pump at frequency ${\omega }_p$ while the dashed arrows indicate additional processes mediated by its second harmonic $2{\omega }_p$.

Figure 3

Asymmetry ratio defined as the net difference in up-converted and down-converted power, normalized to the total converted power $(|t_d{|}^2+|s_d{|}^2-|t_u{|}^2-|s_u{|}^2)/(|t_d{|}^2+|s_d{|}^2+|t_u{|}^2+|s_u{|}^2)$, plotted as a function of the strength of the second harmonic ${\xi }_2$, for three different detunings (or modulation frequencies). This ratio is independent of the pump strength ${\xi }_1$. The maximum ratio of unity, corresponding to 100% asymmetry, is achievable for optimal strengths of $2{\omega }_p$ pump and sufficiently small detunings from the carrier (${\omega }_m\ll {\omega }_p$).

Figure 4

Analytical calculation for the RSJ. (a) Static $I-V$ characteristics of the RSJ calculated using the perturbative series method, showing the variation of voltage $V$ across the junction with current $I_J$ flowing through it. The solid black line corresponds to the exact analytical result $V=R\sqrt{I_B^2-I_0^2}$, while the green (dotted), blue (dashed), and violet (dotted-dashed) curves correspond to the series calculation with one $[K=1]$, two $[K=2]$, and three $[K=3]$ Josephson harmonics, respectively. (b) Asymmetry ratio (cf. Fig. 3) for the RSJ, calculated with the pump configuration derived in Eq. (6). This ratio is almost independent of the modulation frequency (parametrized here as a dimensionless variable ${\mathrm{\Omega }}_m={\omega }_m/{\omega }_B$ with ${\omega }_B=I_{B}R/{\phi }_0$) for small ${\mathrm{\Omega }}_m$, and increases as the junction is biased towards the strongly nonlinear regime by increasing $x$ (or decreasing $I_B$) [27].