Multimode phononic correlations in a nondegenerate parametric amplifier

Multimode phononic correlations in a nondegenerate parametric amplifier
S. Chakram, Y. S. Patil, M. Vengalattore
New Journal of Physics, 17, 063018 (2015)
arXiv:1412.8536 (2014)

Abstract

We describe the realization of multimode phononic correlations that arise from nonlinear interactions in a mechanical nondegenerate parametric amplifier. The nature of these correlations differs qualitatively depending on the strength of the driving field in relation to the threshold for parametric instability. Below this threshold, the correlations are manifest in a combined quadrature of the coupled mechanical modes. In this regime, the system is amenable to back-action evading measurement schemes for the detection of weak forces. Above threshold, the correlations are manifest in the amplitude difference between the two mechanical modes, akin to intensity difference squeezing observed in optical parametric oscillators. We discuss the crossover of correlations between these two regimes and applications of this quantum-compatible mechanical system to nonlinear metrology and out-of-equilibrium dynamics.

Bibtex

@article{chakramMultimode2015,
author={S Chakram and Y S Patil and M Vengalattore},
title={Multimode phononic correlations in a nondegenerate parametric amplifier},
journal={New Journal of Physics},
volume={17},
number={6},
pages={063018},
url={http://stacks.iop.org/1367-2630/17/i=6/a=063018},
year={2015}
}

Figure 1

(a) Distinct eigenmodes of the membrane resonator are coupled through parametric excitation of the supporting substrate.  (b) The strength of the two-mode interaction is enhancedwhen the parametric interaction betweenmembrane eigenmodes at ωi,j is mediated by a substrate excitation at ωS = ωi + ωj .

Figure 2

Pump (blue) and signal/idler (red) amplitudes as a function of the normalized parametric drive, μ =|Fs|/|Fs,cr|.

Figure 3

Normalized variances of amplified and squeezed collective quadratures as a function of the normalized parametric drive. Black lines indicate the variances formatched frequencies (δω = 0 ) and loss rates (δγ = 0). Solid lines indicate the amplified (blue) and squeezed (red) variances for δγ = 0.5, δω = −0.5. The dashed lines indicate the amplified and squeezed variances for δγ = δω ≠ 0, the case ofmatched asymmetrieswhere the peak noise squeezing again approaches a factor of 1/2 as μ → 1. The dashed horizontal line represents the thermomechanical variance given by kBT/mω^2.

Figure 4

(a) Peak noise squeezing as a function of loss asymmetry δγ for δω = 0 with the squeezing going linearly from 1/2 → 1 as δγ = 0 → 1. (b) Peak squeezing as a function of the frequency asymmetry δω, for δγ = 0. (c) Peak squeezing as a function of δγ and δω.

Figure 5

(a) Normalized variance of the fluctuations in the y+ (amplified, blue) and y− (squeezed, red) quadratures for the case of zero detuning (solid lines) and a detuning of Δ = γ (dashed lines). The amplified quadrature diverges at the instability threshold. For Δ = 0, this occurs at μ = 1 (solid black vertical line). For Δ = γ, this occurs at μ = sqrt(1 +(Δ/γ)^2) = sqrt(2) (dashed black vertical line). The black horizontal line is at 1/2. (b) Peak noise squeezing as a function of normalized detuning Δ/γ . Both these graphs are shown for the case of no loss or frequency asymmetry, i.e. δγ = δω = 0.

Figure 6

Variance of the fluctuations in the difference y− (red) and sum y+ (blue) quadratures vs the parametric drive μ. The dashed lines are computed for δγ = δω = 0. The solid lines are computed for δγ = 0.31, δω = 0.09. The amplitude difference quadrature is squeezed for all values above threshold to a value of 1/2 . As can be seen, amplitude difference squeezing is extremely robust to experimental imperfections such as a frequency mismatch or loss asymmetry. Inset: schematic of the mean value of the membrane mode amplitude Ai,j and the fluctuations δαi,j and δβi,j . δβ represent amplitude fluctuations while δα are related to fluctuations of the phase.

Figure 7

Two-mode correlations in the vicinity of the threshold for parametric instability. Variance of the fluctuations of the normalized difference (y−) and sum (y+) quadratures above and below threshold vs the normalized parametric drive μ. The dashed lines represent the variances for δγ = δω = 0. The solid lines represent the variances for δγ = 0.31, δω = 0.09. The dashed horizontal line represents the thermomechanical variance given by kBT/mω^2.

Figure 8

Corrections to squeezing spectra due to finite measurement duration. Normalized variance of amplified (y+) and squeezed (y−) quadratures for a measurement time of 300 s (~100 ring down periods) (black solid lines). Also shown for comparison are the corresponding amplified (blue) and squeezed (red) quadratures in steady state. These are computed for δγ = 0.31, δω = 0.09 (γ ~ 2π × 100mHz) and correspond to the experimental parameters in [28]. The divergence at the instability threshold is attenuated for finite measurement times due to the divergent response times.