All-optical coherent quantum-noise cancellation in cascaded optomechanical systems

Coherent quantum-noise cancellation (CQNC) can be used in optomechanical sensors to surpass the standard quantum limit (SQL). In this paper, we investigate an optomechanical force sensor that uses the CQNC strategy by cascading the optomechanical system with an all-optical effective negative-mass oscillator. Specifically, we analyze matching conditions and losses and compare the two possible arrangements in which either the optomechanical or negative-mass system couples first to light. While both of these orderings yield a sub-SQL performance, we find that placing the effective negative-mass oscillator before the optomechanical sensor will always be advantageous for realistic parameters. The modular design of the cascaded scheme allows for better control of the subsystems by avoiding undesirable coupling between system components while maintaining a performance similar to the integrated configuration proposed earlier. We conclude our work with a case study of a micro-optomechanical implementation.

Figure 1

(a) Sketch of the cascaded setup. The optomechanical sensor (green) consists of a mechanical oscillator inside an optical cavity, depicted in a membrane-in-the-middle setup. The effective negative-mass oscillator (blue) comprises two optical modes, coupled via a beam-splitter and down-conversion process. A nonlinear periodically poled potassium titanyl phosphate (PPKTP) crystal exemplifies the coupling for the down-conversion and a wave plate for the beam-splitter coupling. (b) Simplified depiction of the possible arrangements of the cascaded scheme.

Figure 2

Force noise for a mismatch between the ancilla cavity linewidth ${\kappa }_{\mathrm{a}}$ and the damping rate of the mechanical oscillator ${\gamma }_{\mathrm{m}}$. For ${\kappa }_{\mathrm{a}}<{\omega }_{\mathrm{m}}$ an improvement of ${\kappa }_{\mathrm{a}}/2{\omega }_{\mathrm{m}}$ can be achieved off resonance (solid green line). For ${\kappa }_{\mathrm{a}}\gg {\omega }_{\mathrm{m}}$, the effect of CQNC is completely canceled for low frequencies, and the sensitivity is worse than the SQL for high frequencies (red dashed line). The shaded areas mark the bounds for sub-SQL sensitivity, from below the fundamental limit given by Eq. (26) and from above the SQL given by Eq. (15). Parameters are given in Table 1.

Figure 3

Force noise for imperfect matching of measurement strength. For mismatched coupling strength compensated by linewidth mismatch, perfect noise cancellation can be recovered at low frequencies (solid green line for $\epsilon =\delta =0.9$). For matched linewidth but mismatched coupling strength, noise cancellation is limited, but sub-SQL performance is possible (dashed red line for $\delta =0.9$). In the case of matched coupling strength but mismatched linewidth, we find a frequency (33) where perfect noise cancellation is possible (dash-dotted purple line, with $\epsilon =0.9$). The shaded areas mark the bounds for sub-SQL sensitivity, from below the fundamental limit given by Eq. (26) and from above the SQL given by Eq. (15). Parameters are given in Table 1.

Figure 4

Flow chart between the mode of the ancilla and the meter cavity. The solid line shows the original back-action flow, and the dashed line shows the noise introduced by the relative mismatch $g_{\mathrm{r}}$ of the beam-splitter and down-conversion coupling.

Figure 5

Force noise for relative mismatch of beam-splitter and down-conversion coupling (a) for $g_{\mathrm{r}}=0.2$ and (b) for $g_{\mathrm{r}}=-0.2$. The relative mismatch introduces additional noise by modifying the effective back-action term of the NMO. Perfect noise cancellation is not possible, but sub-SQL levels are achievable for low frequencies. For high frequency, no noise reduction is possible. Traversing through the OMS first seems advantageous for noise cancellation. The shaded areas mark the bounds for sub-SQL sensitivity, from below the fundamental limit given by Eq. (26) and from above the SQL given by Eq. (15). Parameters are given in Table 1.

Figure 6

Force noise normalized to $Q_{\mathrm{m}}$ for the parameters given by Table 2 and temperature $T=4\text{K}$. For low frequencies, sub-SQL performance is possible for the integrated setup (solid green line) and the case $\mathrm{NMO}\mapsto \mathrm{OMS}$ (dashed blue line). No sub-SQL levels are possible for the case $\mathrm{OMS}\mapsto \mathrm{NMO}$ (dash-dotted orange line). The shaded area shows levels above the SQL.