Conditional Dynamics of Optomechanical Two-Tone Backaction-Evading Measurements

Matteo Brunelli, Daniel Malz, and Andreas Nunnenkamp

Phys. Rev. Lett. 123, 093602 – Published 29 August 2019

Abstract

Backaction-evading measurements of mechanical motion can achieve precision below the zero-point uncertainty and quantum squeezing, which makes them a resource for quantum metrology and quantum information processing. We provide an exact expression for the conditional state of an optomechanical system in a two-tone backaction-evading measurement beyond the standard adiabatic approximation and perform extensive numerical simulations to go beyond the usual rotating-wave approximation. We predict the simultaneous presence of conditional mechanical squeezing, intracavity squeezing, and optomechanical entanglement. We further apply an analogous analysis to the multimode optomechanical system of two mechanical and one cavity mode and find conditional mechanical Einstein-Podolski-Rosen entanglement and genuinely tripartite optomechanical entanglement. Our analysis is of direct relevance for ultrasensitive measurements and measurement-based control in high-cooperativity optomechanical sensors operating beyond the adiabatic limit.

Received 15 March 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.093602

Physics Subject Headings (PhySH)

Optomechanics Quantum control Quantum entanglement Quantum information processing with continuous variables Quantum metrology Quantum sensing

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

Matteo Brunelli¹, Daniel Malz², and Andreas Nunnenkamp¹

¹Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom
²Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany

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Vol. 123, Iss. 9 — 30 August 2019

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Images

Figure 1
(a) Backaction-evading (BAE) measurement of a single mechanical quadrature. An optomechanical cavity ( $\hat{a}$ ) is driven on the lower and upper mechanical ( $\hat{b}$ ) sideband and is continuously monitored via the output homodyne current. (b) If two mechanical modes ${\hat{b}}_{1}$ and ${\hat{b}}_{2}$ are considered instead (dashed boxes), a two-mode BAE measurement is realized. (c) Monitoring of the output field both introduces backaction and allows us to extract information. Arrows originate from the source terms in the equations of motion derived from Eq. (2). Within RWA, backaction is confined to ${\hat{P}}_{m}$ and reduction of uncertainty to ${\hat{X}}_{m}$ . (d) Counter-rotating terms open new channels, which corrupt the BAE regime and reduce squeezing in ${\hat{X}}_{m}$ but, at the same time, enable joint reduction of uncertainty of other variables, e.g., ${\hat{X}}_{c}$ and ${\hat{P}}_{m}$ ; thus, correlations are enhanced and robust entanglement can be generated subject to measurement.
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Figure 2
Mechanical squeezing (in dB) for $g = 0.01 ω_{m}$ (red), $g = 0.05 ω_{m}$ (yellow), and $g = 0.3 ω_{m}$ (cyan) as predicted by Eq. (5). Other parameters are $γ = 10^{- 4} ω_{m}$ , $\bar{n} = 10$ , $η = 1$ . Solid black lines represent the adiabatic solution $σ_{X_{m}, ad}^{2}$ while dashed lines represent that of a slow cavity $σ_{X_{m}, slow}^{2}$ . For each curve, the part to the left of the black dot ( $g = κ$ ) is in the strong-coupling regime.
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Figure 3
(a) Mechanical squeezing (in dB) assuming the RWA [Eq. (5)] (dashed dark curves) and beyond the RWA (lighter shaded areas). The curves are for $g = 0.01 ω_{m}$ (red), $g = 0.05 ω_{m}$ (yellow), and $g = 0.3 ω_{m}$ (cyan); the dotted curve shows the mean squeezing (averaged over one mechanical period) and the shaded area extends between the minimum and maximum value of squeezing. Solid black lines represent the adiabatic solution $σ_{X_{m}, ad}^{2}$ . (b) Conditional entanglement (measured by the logarithmic negativity) for the same couplings as (a); the vertical dashed line corresponds to $κ = 0.05 ω_{m}$ , and in the inset, we show the temporal evolution of entanglement along this cut for the case $g = 0.05 ω_{m}$ . (c) Conditional cavity squeezing for the same couplings as (a). (d) Zoom-in of panel (b). In all panels, other parameters are: $γ = 10^{- 4} ω_{m}$ , $\bar{n} = 10$ , $η = 1$ .
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Figure 4
(a) Mechanical two-mode squeezing (in dB) assuming the RWA (dashed curves), beyond the RWA (lighter shaded areas), and in the adiabatic limit (dotted darker curves). The curves are for $g = 0.01 ω$ (red, which is zero), $g = 0.05 ω$ (yellow), and $g = 0.3 ω$ (cyan). As in Fig. 3, the dotted gray curve shows the average two-mode squeezing (taken over $2 π / ω$ ). In the inset the conditional cavity squeezing is shown. Other parameters are $Ω = 0.1 ω$ , $γ = 10^{- 4} ω$ , $\bar{n} = 10$ , $η = 1$ . (b) Inseparability structure of the conditional three-mode optomechanical system. The shaded region marks the presence of mechanical two-mode squeezing. Other parameters as in (a). (c) Same as (b) except for $\bar{n} = 100$ .
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