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Experimental Platform: Cryogenic Near-Field Cavity Optomechanics

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Quantum Limits on Measurement and Control of a Mechanical Oscillator

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Radiation-pressure coupling between a mechanical oscillator and an electromagnetic cavity allows, in principle, for an exquisitely sensitive measurement of the oscillator’s motion. In practice, it takes careful engineering to realize this goal. Here we describe the salient properties of our system that enable a quantum-noise-limited performance.

Quantum  phenomena  do  not  occur  in  a Hilbert space, they occur in a laboratory.

Asher Peres

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Notes

  1. 1.

    In this case, intuition is garnered from a different perspective on the measurement process: ideally, the meter and the system gets tightly entangled in the course of the measurement interaction; monogamy of entanglement then requires that the system has to be exclusively entangled with the meter.

  2. 2.

    We here ignore the two-dimensional extension of the beam, justified by the fact that in the cases of interest, the characteristic extension along this dimension is much smaller than the length of the beam. In doing so, we treat the two possible transverse displacements, along the coordinate \(x-\) and \(y-\)directions, using independent Euler-Bernoulli equations; the small coupling between these two mode families, arising from a finite Poisson’s ratio (\(\varsigma _{\text {SiN}}\approx 0.24\)), is therefore ignored. Within this approximation, effects arising from rotation and shear in the transverse direction are also ignored [24, 25].

  3. 3.

    A common-path interferometer (for example using the frequency modulation spectroscopy technique described in Sect. 3.2.4), is technically easier to implement. However, it is not preferred in a cryogenic environment due to the large (mW range) LO powers that might scatter in the cold environment.

  4. 4.

    By design, this cryostat is not meant for cold operation without buffer gas. In order to achieve low temperatures without having buffer gas in the sample volume, we first condense \(^{3}\)He: this is done by filling the \(1\,\text {K}\) pot with liquid \(^{4}\)He and pumping on it to cool it down to \({<}2\, \text {K}\); finally the sorption pump is heated to \(25\, \text {K}\) to eject out all \(^{3}\)He gas and pressurise it. Once condensed, droplets of liquid \(^{3}\)He accumulate at the bottom of the cryostat (the so-called “\(^{3}\)He tail”); pumping on the condensed liquid using the sorption pump evaporatively cools the liquid to as low as \(0.3\, \text {K}\). Once all the condensed \(^{3}\)He is evaporated, the sample volume is in vacuum and the sample temperature slowly rises.

  5. 5.

    In confined spaces, even in the regime of \(\text {Kn}>10\), recoil events may not be independent, leading to an excess thermal force [41] with a characteristic time-scale inversely related to the dimensions of the constriction—this effect is not observed here.

  6. 6.

    It is interesting to note that in the extreme regime, \(\text {Kn}\ll 0.1\), the spectrum of gas damping samples the inter-particle collisions of the gas atoms [42, 43] and necessitates a non-Newtonian fluid model—however this regime is not relevant here.

  7. 7.

    The Reynolds number, (beam velocity)(transverse length)/(kinematic viscosity of He), determines this. For the beam undergoing thermal motion, its root mean square velocity on resonance is \(\sqrt{2 n_\text {m,th}} x_\text {zp}\frac{\Omega _\text {m}}{2\pi } \approx 10^{-4}\, \text {m/s}\). For beam transverse dimension of \(0.5\,\upmu \mathrm{m}\), it follows that, \(\mathrm{Re} < 10^{-4}\).

  8. 8.

    Due to the nature of the problem, it seems reasonable to imagine that for a narrow longitudinal (along the beam) constriction with a transverse opening, squeeze-films get evacuated much more quickly than diffusion would suggest. In fact molecules in a squeeze film execute Lévy walks [45], lending plausibility to their absence after even a short pumping time.

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  1. LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA

    Vivishek Sudhir

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Sudhir, V. (2018). Experimental Platform: Cryogenic Near-Field Cavity Optomechanics. In: Quantum Limits on Measurement and Control of a Mechanical Oscillator. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-69431-3_5

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