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Observation and Feedback-Suppression of Measurement Back-Action

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Quantum Limits on Measurement and Control of a Mechanical Oscillator

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Abstract

Realizing a quantum-noise-limited measurement of the position of a harmonic oscillator has been a 30 year old white whale of experimental physics, stretching back to the conception of interferometric gravitational wave antennae. Here we perform an interferometric position measurement with an imprecision 40 dB below that at the standard quantum limit, and observe back-action commensurate with it, due to radiation-pressure quantum fluctuations. We demonstrate how feedback can be used in this regime to suppress back-action.

Moby Dick seeks thee not. It is thou, thou, that madly

seekest him!

Herman Melville

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Notes

  1. 1.

    This is derived as follows. Assume the body is in thermal equilibrium at temperature T, so that it is described by the canonical thermal state \(\hat{\rho }=e^{-\beta \hat{H}}/Z\), with \(Z :=\,\mathrm {Tr}\, e^{-\beta \hat{H}}\), and \(\beta :=\,(k_B T)^{-1}\). Then, the average energy is given by, \(\langle {\hat{H}}\rangle :=\,\mathrm {Tr}\, \hat{H}\hat{\rho } = -\frac{1}{Z}\partial _\beta Z\), while its second moment is, \(\langle {\hat{H}\rangle ^2} :=\,\mathrm {Tr}\, \hat{H}^2 \hat{\rho } = \frac{1}{Z}\partial _\beta ^2 Z = -\partial _\beta \langle {\hat{H}}\rangle +\langle {\hat{H}}\rangle ^2\). Subtracting these two expressions give the variance in the energy:

    $$\begin{aligned} \mathrm {Var}\left[ {\hat{H}}\right] :=\,\langle \hat{H}^2\rangle -\langle {\hat{H}}\rangle ^2 = -\partial _\beta \langle {\hat{H}\rangle } = k_B T^2\, \partial _T \langle {\hat{H}}\rangle = k_B T^2 C_V. \end{aligned}$$

    Here, \(C_V :=\,\partial _T \langle {\hat{H}}\rangle \) is the specific heat at constant volume as defined conventionally. To refer the above variance in energy to an apparent variance in temperature, we again use the definition of the specific heat, \(\delta T = \delta E/C_V\), to arrive at, \(\mathrm {Var}\left[ {T}\right] = \mathrm {Var}\left[ {E}\right] /C_V^2 = k_B T^2/C_V\).

  2. 2.

    Longitudinal elastic modes have a similar effect, but their frequency being larger, the resulting phase noise isn’t relevant in our experiment.

  3. 3.

    A function f(t) is (asymptotically) stable if \(\left| {f(t\rightarrow \infty )}\right| < \infty \); it is causal if \(f(t<0) = 0\). These properties can be expressed in terms of the Fourier transform,

    $$\begin{aligned} f[\Omega ] = \int _{-\infty }^{\infty } f(t) e^{i\Omega t}\, \mathrm {d}t, \end{aligned}$$

    extended to the complex plane (i.e. \(\Omega \mapsto \Omega +i\Gamma \)). Firstly the bound (an instance of the triangle inequality),

    $$\begin{aligned} \left| {f[\Omega +i\Gamma ]}\right| = \left| {\int f(t) e^{(i\Omega -\Gamma )t}\, \mathrm {d}t}\right| \le \int \left| {f(t)}\right| e^{-\Gamma t}\, \mathrm {d}t, \end{aligned}$$

    together with stability implies that, \(\left| {f[\Omega +i\Gamma ]}\right| < \infty \,\, \mathrm {for}\,\, \Gamma > 0\); thus, all singularities of \(f[\Omega +i\Gamma ]\) are in the lower-half plane (real axis included). Secondly, causality in time domain can be expressed as the identity \(f(t)=\Theta (t)f(t)\), where \(\Theta \) is the Heaviside step function; taking its Fourier transform gives,

    $$\begin{aligned} f[\Omega ] = \frac{i}{\pi }\int _{-\infty }^{\infty } \frac{f[\Omega ']}{\Omega '-\Omega }\,\mathrm {d}\Omega ', \end{aligned}$$

    a constraint imposed by casuality for real frequencies (separating out the real and imaginary parts of this equation gives the Kramers-Kronig relation). We now consider the integral of \(f[\Omega ']/(\Omega '-\Omega )\) along a contour C in the complex plane that consists of the real line and an arc at infinity enclosing the upper-half plane; by stability, the latter integral is zero; by causality, the former satisfies the above constraint; thus, we can show that,

    $$\begin{aligned} \oint _C \frac{f[\Omega ']}{\Omega '-\Omega } \mathrm {d}\Omega ' = 0. \end{aligned}$$

    Therefore (by Cauchy’s theorem) \(f[\Omega ]\) is analytic in the upper-half plane [63,64,65,66]. (A stronger version of this line of inference goes by the name of Titchmarsh’s theorem, which identifies causal and stable functions as boundary values of analytic functions [67, Section 17.8].)

  4. 4.

    This is generally true of symmetrized spectra of the observables of linear Markovian systems driven by white-noise. Firstly, the spectra of the observables of such systems is a rational function of the frequency, i.e. a ratio of polynomials (Markovianity is essential for this to be true since arbitrarily long time delays preserve linearity but do not have a rational frequency response). Secondly, considering \(\Omega \) to be extended into the complex plane, it can be shown that the spectrum of an observable, say y, satisfies the relation, \(\bar{S}_{yy}[\Omega ]=\bar{S}_{yy}^*[-\Omega ^*]\) (for real frequencies, this reduces to, \(\bar{S}_{yy}[\Omega ]=\bar{S}_{yy}[-\Omega ]\), given in Eq.  2.1.16). This symmetry implies a characteristic symmetry for the roots of the numerator and denominator polynomials: purely real roots occur in positive/negative pairs, while purely imaginary ones in conjugate pairs, and complex roots occur in conjugate positive/negative quadruplets. Thus, both the numerator and denominator can be factorised such that the factors have zeros symmetric about the real-axis. Collecting those factors with zeros and poles in either half-plane gives the required factorisation.

  5. 5.

    A theorem due to Bode [63] asserts that for a causal stable filter, a magnitude response that falls off as some polynomial power, \(\Omega ^n\), has to impart at least a phase change of \(n\pi /2\); thus this filter adds a phase of \(\pi /2\).

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

    Vivishek Sudhir

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Sudhir, V. (2018). Observation and Feedback-Suppression of Measurement Back-Action. 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_6

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