Degenerate squeezing in a dual-pumped integrated microresonator: Parasitic processes and their suppression

Using a general Hamiltonian treatment, we theoretically study the generation of degenerate quadrature squeezing in a dual-pumped integrated microring resonator coupled to a waveguide. Considering a dual-pump four-wave mixing configuration in an integrated ${\text{Si}}_3{\text{N}}_4$ platform, and following the coupled-mode theory approach, we investigate the effects of parasitic quantum nonlinear optical processes on the generation of squeezed light. Considering five resonance modes in this approach allows us to include the most important four-wave mixing processes involved in such a configuration. We theoretically explore the effects of the pump detunings on different nonlinear processes and show that the effects of some of the parasitic processes are effectively neutralized by symmetrically detuning the two pumps. This yields a significant enhancement in the output squeezing quality without physically changing the structure, but suffers from the trade-off of requiring substantially higher pump power for a fixed target level of squeezing.

Figure 1

(a) Schematic of microring resonator, with the nominal radius $R$, side-coupled to a physical channel and an effective phantom channel, which is introduced to account for scattering losses. The $x$ coordinate for the phantom channel increases in the opposite direction to that of the real channel. The inset shows the electric field intensity of the fundamental mode at $1550\text{nm}$ inside the ring. The parameters for the ring and channel are given in Sec. 10. (b) The range ${\mathcal{K}}_J$ of $k$, given by Eq. (8), associated with the $J\mathrm{th}$ ring resonance.

Figure 2

Four-wave mixing processes that occur in a ring resonator when two resonances $p_1$ and $p_2$ are pumped. DP-SFWM leads to squeezing of the $s$ mode while the unwanted processes of SP-SFWM and BS-FWM generate noise and impurify the output. The corresponding conjugate of each process is shown with the dotted arrows with the same color. For the sake of illustration we present these processes in four different levels. The lowest level shows the two BS-FWM processes and the second level shows the HP-SFWM process. In the third level we present the two SP-SFWM processes, and the DP-SFWM process is shown in the fourth level.

Figure 3

Illustration of the hot (shown in dashed red) and cold (shown in solid blue) cavity resonances. The resonance frequencies do not shift equally since the modes $p_1$ and $p_2$ are affected by SPM and XPM, while the modes $m, s$, and $n$ are affected only by XPM. Considering the frequency shift caused by SPM to be $U$, the modes $p_1$ and $p_2$ move $3U$, and the other modes move $4U$ toward lower frequencies. The detuning of the pumps from the hot-cavity resonances are indicated by ${\delta }_{p_1}$ and ${\delta }_{p_2}$.

Figure 4

Squeezing (solid lines) and antisqueezing (dashed lines) spectra for different input powers. The pumps are on resonance with the hot resonator. Black solid line is shot-noise level. The two outer lines correspond to 15 dBm and the two inner lines correspond to 11 dBm.

Figure 5

Squeezing (solid lines) and antisqueezing (dashed lines) spectra for different combinations of nonlinear processes. Black solid line is shot-noise level. The pumps are on resonance with the hot resonator. The highest squeezing is achieved when only DP-SFWM is included and the lowest is obtained when all the processes are involved. As can be seen at 0 GHz, the squeezing levels of the different nonlinear process combinations can be listed from bottom to top as (1) DP-SFWM, (2) DP-SFWM and BS-FWM, (3) DP- and SP-SFWM, (4) no HP-SFWM, and (5) all processes. Similarly for the antisqueezing level at at 0 GHz we have (1) DP-SFWM and BS-FWM, (2) DP-SFWM, (3) no HP-SFWM, (4) all processes, and (5) DP- and SP-SFWM.

Figure 6

Squeezing in dB (top) when only DP-SFWM is included and (bottom) when both SP- and DP-SFWM are included as functions of the detunings of the first and the second pumps from the hot-cavity resonances. Both cases were simulated using the same total input power of $16\text{dBm}$.

Figure 7

Highest squeezing (solid lines) achievable for different pump powers with and without SP-SFWM. The values were obtained by searching for the highest values in the 2D detuning space (similar to Fig. 6) for each total input power in the range; we also plot the value of the antisqueezing (dotted lines) under these conditions. The top and the bottom lines correspond to the SP- and DP-SFWM, and the two middle lines correspond to only DP-SFWM. The inset shows the difference between the highest squeezing when including SP-SFWM and the case when only DP-SFWM is considered.

Figure 8

Squeezing (top) and ratio of the total generated photons in the modes $m$ and $n$ to the number of generated photons in the mode $s$ (bottom) as a function of the detuning of the first pump from the hot-cavity resonance for three different scenarios of fixing the product of the energies at $69.6{\text{pJ}}^2$ (solid lines), total energy in the ring at $16.7\text{pJ}$ (dashed lines), and antisqueezing at $2.64\text{dB}$ (dotted lines). The pumps are detuned symmetrically, i.e., ${\delta }_{p_1}=-{\delta }_{p_2}$, and the calculations included DP-SFWM and SP-SFWM.