Word Measures on Wreath Products II

Every word $w$ in $F_r$, the free group of rank $r$, induces a probability measure (the $w$-measure) on every finite group $G$, by substitution of random $G$-elements in the letters. This measure is determined by its Fourier coefficients: the $w$-expectations $E_w[\chi]$ of the irreducible characters of $G$. For every finite group $G$, every stable character $\chi$ of $G\wr S_n$ (trace of a finitely generated $FI_G$-module), and every word $w\in F_r$, we approximate $E_w[\chi]$ up to an error term of $O(n^{-\pi(w)})$, where $\pi(w)$ is the primitivity rank of $w$. This generalizes previous works by Puder, Hanany, Magee and the author. As an application we show that random Schreier graphs of representation-stable actions of $G\wr S_n$ are close-to-optimal expanders. The paper reveals a surprising relation between stable representation theory of wreath products and not-necessarily connected Stallings core graphs.Comment: 40 pages, 13 figure

Stable Invariants and Their Role in Word Measures on Groups

Every word in a free group induces a word measure -- a probability measure defined via the word map -- on every compact group. This paper presents a conjectural picture about the role of a plethora of stable invariants of words in word measures on groups. These invariants generalize the stable commutator length and include, among others, two invariants recently defined by Wilton: the stable primitivity rank and a non-oriented analog of stable commutator length we call stable square length. The conjectures say, roughly, that these stable invariants control the asymptotics of the expected values of stable characters, under word measures. We reinforce these conjectures by proving a version for word measures on wreath products, and by introducing a related formula for stable irreducible characters of the symmetric group.Comment: 53 pages, 3 figures, with an appendix by Danielle Ernst-West, Doron Puder and Matan Seide

Optical Backaction-Evading Measurement of a Mechanical Oscillator

Quantum mechanics imposes a limit on the precision of a continuous position measurement of a harmonic oscillator, as a result of quantum backaction arising from quantum fluctuations in the measurement field. A variety of techniques to surpass this standard quantum limit have been proposed, such as variational measurements, stroboscopic quantum non-demolition and two tone backaction-evading (BAE) measurements. The latter proceed by monitoring only one of the two non-commuting quadratures of the motion. This technique, originally proposed in the context of gravitational wave detection, has not been implemented using optical interferometers to date. Here we demonstrate continuous two-tone backaction-evading measurement in the optical domain of a localized GHz frequency mechanical mode of a photonic crystal nanobeam cryogenically and optomechanically cooled in a $^3$He buffer gas cryostat close to the ground state. Employing quantum-limited optical heterodyne detection, we explicitly show the transition from conventional to backaction-evading measurement. We observe up to 0.67 dB (14%) reduction of total measurement noise, thereby demonstrating the viability of BAE measurements for optical ultrasensitive measurements of motion and force in nanomechanical resonators

Optical backaction-evading measurement of a mechanical oscillator.

Quantum mechanics imposes a limit on the precision of a continuous position measurement of a harmonic oscillator, due to backaction arising from quantum fluctuations in the measurement field. This standard quantum limit can be surpassed by monitoring only one of the two non-commuting quadratures of the motion, known as backaction-evading measurement. This technique has not been implemented using optical interferometers to date. Here we demonstrate, in a cavity optomechanical system operating in the optical domain, a continuous two-tone backaction-evading measurement of a localized gigahertz-frequency mechanical mode of a photonic-crystal nanobeam cryogenically and optomechanically cooled close to the ground state. Employing quantum-limited optical heterodyne detection, we explicitly show the transition from conventional to backaction-evading measurement. We observe up to 0.67 dB (14%) reduction of total measurement noise, thereby demonstrating the viability of backaction-evading measurements in nanomechanical resonators for optical ultrasensitive measurements of motion and force

Dissipative Quantum Feedback in Measurements Using a Parametrically Coupled Microcavity

Micro- and nanoscale optical or microwave cavities are used in a wide range of classical applications and quantum science experiments, ranging from precision measurements, laser technologies to quantum control of mechanical motion. The dissipative photon loss via absorption, present to some extent in any optical cavity, is known to introduce thermo-optical effects and thereby impose fundamental limits on precision measurements. Here, we theoretically and experimentally reveal that such dissipative photon absorption can result in quantum feedback via in-loop field detection of the absorbed optical field, leading to the intracavity field fluctuations to be squashed or antisquashed. Strikingly, this modifies the optical cavity susceptibility in coherent response measurements and causes excess noise and correlations in incoherent interferometric optomechanical measurements using a cavity. We experimentally observe such unanticipated dissipative dynamics in optomechanical spectroscopy of sideband-cooled optomechanical crystal cavitiess at both cryogenic temperature (approximately 8 K) and ambient conditions. The dissipative feedback introduces effective modifications to the optical cavity linewidth and the optomechanical scattering rate and gives rise to excess imprecision noise in the interferometric quantum measurement of mechanical motion. Such dissipative feedback differs fundamentally from a quantum nondemolition feedback, e.g., optical Kerr squeezing. The dissipative feedback itself always results in an antisqueezed out-of-loop optical field, while it can enhance the coexisting Kerr squeezing under certain conditions. Our result has wide-ranging implications for future dissipation engineering, such as dissipation enhanced sideband cooling and Kerr squeezing, quantum frequency conversion, and nonreciprocity in photonic systems