Where backaction goes when you watch one quadrature

Select a quantity in the rotating frame

Write free displacement as $q(\tau)=X\cos\tau+P\sin\tau$ . Rotating quadratures $X$ and $P$ satisfy $[X,P]=i$ . Multiply displacement by a synchronous cosine modulation. The result contains a stationary $X$ term and terms at twice the mechanical frequency.

For resolved sidebands and coupling and damping slow relative to mechanical rotation, average the fast terms away. This rotating-wave approximation leaves an interaction containing only the selected quadrature. It commutes with that quadrature and sends ideal measurement backaction into its conjugate.

\cos\tau\,q(\tau)=\tfrac12X+\tfrac12X\cos2\tau+\tfrac12P\sin2\tau,\qquad H_{\rm BAE}=-\hbar h X X_c

X_c is the coupled optical quadrature. The coefficient h includes modulation amplitude and the factor one half.

Source locations: Clerk

Specify which disturbance is evaded

The quadrature uncertainty product still applies. Thermal noise enters the observed quadrature, and pump imbalance, detuning, finite sideband resolution or residual fast terms can leak backaction into it. A backaction-evading claim therefore needs an observable, a time scale and a bandwidth.

Ordinary one-tone position readout continuously follows laboratory-frame position. Mechanical rotation brings earlier momentum kicks back into that coordinate. Backaction evasion requires changing the drive protocol and the measured quadrature.

Source locations: Aspelmeyer · Clerk

Narrow one direction and widen the other

Ground-state variance is $1/2$ in both quadratures. A pure squeezed state multiplies one by $e^{-2r}$ and the other by $e^{2r}$ , where $r$ is squeezing strength. A minimum-area circle becomes a minimum-area ellipse.

Worked example: choose $r=\ln2$ . Then $V_X=1/8$ , $V_P=2$ and their product is $1/4$ . The narrow variance is a quarter of vacuum, about six decibels of squeezing. Occupation is nevertheless $(1/8+2-1)/2=0.5625$ . A state can be narrower in one direction and contain more energy than vacuum.

Unequal red and blue pumps, with stronger red pumping, can instead engineer an environment into which motion relaxes as a squeezed state. This squeezed reservoir differs from balanced two-tone measurement. Wollman and colleagues used reservoir engineering experimentally; the ellipse here is an analytic illustration.

V_X=\frac{e^{-2r}}2,\quad V_P=\frac{e^{2r}}2,\quad V_XV_P=\frac14,\quad \bar n=\sinh^2r

These formulas describe zero-mean pure squeezed vacuum. Thermal noise and detection losses generally increase phase-space area.

X

P

Analytic equal-area contours for vacuum and pure squeezed vacuum with squeezing parameter log two. One variance is a quarter of vacuum; the conjugate variance is four times vacuum.

Source locations: Wollman et al.

A positive phase-space picture can still be nonclassical

A squeezed Gaussian state has a positive Wigner function everywhere. It is still nonclassical: no classical probability mixture of coherent states has a quadrature variance below vacuum. Wigner negativity is a stronger non-Gaussian feature.

The one-phonon Fock state $|1\rangle$ has a negative Wigner function at the origin. A linear Hamiltonian, Gaussian initial state and Gaussian continuous observation preserve Gaussian form. Preparing such negative regions therefore needs photon-counting conditioning, sufficient nonlinearity or another non-Gaussian operation.

The final chapter tests photon antibunching. Nonclassical outgoing photons do not establish mechanical Wigner negativity. Define and validate the optical and mechanical observables separately.

W_1(X,P)=\frac{2(X^2+P^2)-1}{\pi}e^{-(X^2+P^2)}

This analytic ideal one-phonon Wigner function illustrates a non-Gaussian target. The current record reconstruction remains Gaussian.

Predict, change, compare, explain

Separate position localization from quadrature squeezing

Compare covariance principal axes with vacuum variance. Predict how lower efficiency changes area and the narrowest direction.
Save the record and covariance and check uncertainty. Explain how rotation returns momentum backaction to position in this one-tone model.
Specify the modulation and approximation needed for a two-tone backaction-evading protocol as a new design.

Exercises with explanations

A zero-correlation Gaussian state has variances one quarter and one. Is it physical and squeezed?
Read the explanation
Their product is one quarter, saturating uncertainty. One quarter lies below vacuum variance one half, so the state is squeezed. Widening the conjugate quadrature preserves the quantum bound.
Does occupation below one guarantee squeezing or Wigner negativity?
Read the explanation
No. A low-temperature thermal state can have occupation below one with both variances above vacuum and a positive Wigner function. Measure the relevant quadrature or non-Gaussian statistic.

Read the sources

Derivations, worked examples and explanatory figures are authored for the stated models. Synthetic records and model references are labelled separately from experimental results. The source locations below identify the reviewed theory and the limited numerical comparison.

Aspelmeyer, Kippenberg & Marquardt, Cavity Optomechanics, arXiv:1303.0733v1 (2013); RMP 86, 1391 (2014)
Pinned arXiv v1: II.1–II.2 cavity linewidth and mechanical spectra; III.1–III.3 radiation pressure and linearization; V.2 and VII.1 dynamical backaction and cooling; VI.1–VI.2 measurement and single-quadrature readout; X.6 nonlinear quantum optomechanics. Equations and examples here are derived in the course convention. This is a review, not a new experimental dataset.
Clerk, Marquardt & Jacobs, Back-action evasion and squeezing of a mechanical resonator using a cavity detector, arXiv:0802.1842v1 (2008)
Two-tone measurement of one rotating quadrature and the effects of counter-rotating terms. The course derives the ideal rotating-wave interaction as a reading node; the single-tone measurement Lab does not implement this protocol.
Wollman et al., Quantum squeezing of motion in a mechanical resonator, arXiv:1507.01662v1 (2015)
Fig. 1 and Eqs. 1–3: unequal red and blue pumps engineer a squeezed reservoir. This is an experimental paper. Course ellipses are original analytic Gaussian illustrations, with no imported experimental points.