FR12
Quantum optomechanics from cooling to continuous measurement
Light can reveal a mirror’s motion and push it at the same time. Follow that two-way interaction from a familiar spring to cooling, quantum state estimation and a regime where one photon changes the way the next photon enters.
A route from an ordinary spring to quantum measurement
Start with high-school waves and mechanics. Each chapter supplies the next mathematical tool, a worked calculation and explained exercises. The last chapter is a source-reading guide with pinned conventions and a numerical reference.
For wider quantum-optics preparation, consult FR10 when it becomes useful. The shared textbook plan supplies the broader sequence; it is not a requirement to finish another entire route first. FR10 · Shared textbook plan
L1 · Find the motion
A moving mirror shifts a resonance. A calibrated spectrum gives a displacement variance.
Chapter 1
How a moving mirror changes light
Use the standing wave between two mirrors to derive the cavity frequency shift, radiation pressure and zero-point displacement.
Chapter 2
Read a mechanical temperature from a noisy trace
Treat a noise spectrum as variance distributed over frequency, fix one- and two-sided conventions, and calibrate its area.
L2 · Let light act back
A bright drive enhances coupling. Sidebands carry energy, and photon fluctuations disturb what they measure.
Chapter 3
How strong light enhances a weak interaction
Expand around a driven steady state, separate enhanced coupling from single-photon nonlinearity, and follow sideband energy flow.
Chapter 4
Why cooling needs a cavity that stores light
Derive cooling from phonon transition rates, separate optical backaction from thermal leakage, and identify instability.
Chapter 5
Why brighter light also shakes the mirror harder
Derive the competition between shot-noise imprecision and radiation-pressure backaction, including finite detection efficiency.
L3 · Use the record
Condition a state on observations, then turn the estimate into a causal feedback force.
Chapter 6
Why the same mirror has two different widths
Derive Gaussian conditioning from one noisy observation and connect record-conditioned states to the unconditional ensemble.
Chapter 7
Turn an observed motion into a cooling force
Estimate momentum from past observations, apply causal damping feedback, and separate conditional uncertainty from ensemble cooling.
L4 · Choose a quantum target
Follow single-quadrature readout toward squeezed mechanics, and retain the full nonlinearity for photon blockade.
Chapter 8
Where backaction goes when you watch one quadrature
Derive two-tone single-quadrature measurement and distinguish mechanical squeezing, positive Wigner functions and non-Gaussian states.
Chapter 9
Read photon blockade through Rabl’s level diagram
Read the setup, displaced levels, sideband spectrum and photon correlations in Rabl 2011 v2, then recalculate a bounded weak-drive curve.
Three questions to try with light
- Which sideband carries mechanical energy away?
- How much can a noisy record tell a feedback controller?
- Can the first photon keep a second photon out?
Cooling uses a linear-response model. Continuous measurement uses a Gaussian quantum model with saved synthetic records. Photon blockade uses weak-drive Fock sectors. Each answers a different question with its own validity range.
Useful neighbouring experiments
- Coherent exchange with a cavity
- Calibrate Brownian motion with an optical trap
- Read optical quadratures with homodyne detection
These existing experiments supply neighbouring ideas and retain their own models. The foundational optomechanical apparatus task Q040 remains tracked in its original quantum-optics issue. Q040
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.
- Lemonde, Didier & Clerk, Antibunching and unconventional photon blockade with Gaussian squeezed states, arXiv:1410.6510v1 (2014)
Section II, Eqs. 2–4: displaced Gaussian photon moments and amplitude squeezing. The course uses the general moment identity; its finite-coupling optomechanical covariance is a separate calculation.
- 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.
- Callen & Welton, Irreversibility and Generalized Noise, Physical Review 83, 34 (1951)
Foundational relation between linear dissipation and generalized-force fluctuations. The publisher abstract was checked; its full text is subscription-restricted. The course derives its oscillator normalization using the openly available spectral treatment in Aspelmeyer II.2.3.
- Marquardt et al., Quantum theory of cavity-assisted sideband cooling of mechanical motion, arXiv:cond-mat/0701416v1 (2007)
Weak-coupling force-noise transition rates and the resolved-sideband cooling limit. The browser uses the single-mode, weak-coupling rate model; strong-coupling normal-mode splitting needs the full coupled response.
- Jacobs & Steck, A straightforward introduction to continuous quantum measurement, arXiv:quant-ph/0611067v1 (2006)
Continuous position measurement, stochastic master equations and Gaussian moment equations. Our strength multiplies the position-measurement dissipator; a double-commutator convention can differ by a factor of two. The course defines its record and diffusion explicitly.
- Rossi et al., Observing and Verifying the Quantum Trajectory of a Mechanical Resonator, arXiv:1812.00928v1 (2018)
Eqs. 1–5 and Figs. 1–3: rotating-frame Gaussian conditioning and verification using a backward estimate. The course derives a laboratory-frame two-quadrature example. Its synthetic records are separate from the experiment and do not reproduce its device parameters.
- Rossi et al., Measurement-based quantum control of mechanical motion, arXiv:1805.05087v2 (2018)
Published in Nature 563, 53–58 (2018). Experimental feedback cooling from an optical position record. Our causal held-force controller is an instructional Gaussian model, distinct from the experimental filter and electronics.
- 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.
- Rabl, The photon blockade effect in optomechanical systems, arXiv:1102.0278v2 (2011)
Pinned v2, April 15, 2011: Eqs. 1, 3–6 and 11–13; Figs. 1–4. The theory predicts photon antibunching. The local weak-drive two-photon Fock calculation takes zero temperature and the high-Q limit; its selected comparison is labelled separately from finite-Q paper curves. Rabl uses the amplitude linewidth; our energy linewidth is twice it.