Coherent cancellation of backaction noise in optomechanical force measurements

Maximilian H. Wimmer, Daniel Steinmeyer, Klemens Hammerer, and Michèle Heurs

Phys. Rev. A 89, 053836 – Published 28 May 2014

Abstract

Optomechanical detectors have reached the standard quantum limit in position and force sensing where measurement backaction noise starts to be the limiting factor for the sensitivity. A strategy to circumvent measurement backaction and surpass the standard quantum limit has been suggested by M. Tsang and C. Caves [Phys. Rev. Lett. 105, 123601 (2010)]. We provide a detailed analysis of this method and assess its benefits, requirements, and limitations. We conclude that a proof-of-principle demonstration based on a micro-optomechanical system is demanding but possible. However, for parameters relevant to gravitational-wave detectors, the requirements for backaction evasion appear to be prohibitive.

Received 24 March 2014

DOI:https://doi.org/10.1103/PhysRevA.89.053836

Authors & Affiliations

Maximilian H. Wimmer¹, Daniel Steinmeyer¹, Klemens Hammerer^1,2, and Michèle Heurs^1,2

¹Institute for Gravitational Physics (Albert Einstein Institute), Leibniz University Hannover, Callinstraße 38, 30167 Hannover, Germany
²Institute for Theoretical Physics, Leibniz University Hannover, Callinstraße 38, 30167 Hannover, Germany

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Vol. 89, Iss. 5 — May 2014

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Images

Figure 1
(a) Optomechanical cavity without noise cancellation. (b) Cavity with coherent quantum noise cancellation. An auxiliary cavity coupled to the main cavity via a beam splitter and an OPA process provides an antinoise path for backaction cancellation.
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Figure 2
Noise power spectral densities $S_{F} (ω)$ of force measurement vs measurement strength $G$ ( $\propto$ power) at a fixed frequency $ω$ . The spectral density without coherent noise cancellation [Eq. (15), dotted blue line] exhibits its minimum at the standard quantum limit $S_{SQL} = 1 / γ_{m} | χ_{m} |$ , Eq. (16), at an optimal $G = 1 / χ_{m}$ . The spectral density with coherent noise cancellation, Eq. (24), is shown for $ω = ω_{m}$ (yellow dot-dashed line) and for off-resonant frequencies (dashed purple line). On resonance no improvement below the SQL is possible; off resonance the SQL can be surpassed by up to a factor of $1 / 2 Q_{m}$ . In the case of imperfect CQNC (solid green line) due to an ancilla cavity linewidth larger than the mechanical linewidth, $κ_{a} > γ_{m}$ , the spectral density in Eq. (28) will be limited by backaction noise, but the minimum still falls below SQL if $κ_{a} < 2 ω_{m}$ . Here $ω_{m} / κ_{a} = 0.1$ .
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Figure 3
Noise power spectral densities achieved for optimal power vs frequency. The standard quantum limit $S_{SQL} (ω)$ [dotted blue line, Eq. (16)] can be surpassed with coherent quantum noise cancellation $S_{CQNC} (ω)$ [dashed purple line, Eq. (25)] by a factor of $1 / 2 Q_{m}$ off resonance. Here $Q_{m} = 1000$ .
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Figure 4
Spectral density of ideal [dashed purple line, Eq. (24)] and nonideal [solid green line, (28)] CQNC normalized to the standard quantum limit, Eq. (16). On resonance no improvement can be achieved, whereas off resonance an improvement of $κ_{a} / 2 ω_{m}$ is still possible.
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Figure 5
Noise spectral density for relative mismatch of $g_{BS}$ and $g_{DC}$ with respect to $g$ (dashed orange line) and the relative mismatch between $g_{BS}$ and $g_{DC}$ (dot-dashed black line). The mismatch $ε_{i}$ is set to 10% from the ideal value. For guidance, SQL and ideal CQNC (both thin gray line) and CQNC with nonideal $κ_{a}$ (solid green line) are plotted as well. The relative mismatch between $g_{BS}$ and $g_{DC}$ shows a frequency dependency which limits the sensitivity at higher frequencies. The mismatch to the optomechanical coupling gives a constant decrease of noise cancellation at all frequencies.
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Figure 6
Quantum noise (dotted blue line) and CQNC reduced noise for the system with parameters given in Table 1. Below 100 MHz CQNC with nonideal $κ_{a}$ (solid green line) gives a radiation pressure backaction noise reduction of about 10%. For ideal CQNC ( $κ_{a}$ matched, dashed purple line) we achieve a noise reduction of 30%. As we cannot increase the power to the optimal value, the reduction is less than $κ_{a} / 2 ω_{m}$ and $1 / 2 Q_{m}$ , respectively.
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