Sub-Poissonian phonon lasing in three-mode optomechanics

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Sub-Poissonian phonon lasing in three-mode optomechanics

Niels Lörch and Klemens Hammerer

Phys. Rev. A 91, 061803(R) – Published 9 June 2015

Abstract

We propose to use the resonant enhancement of the parametric instability in an optomechanical system of two optical modes coupled to a mechanical oscillator to prepare mechanical limit cycles with sub-Poissonian phonon statistics. Strong single-photon coupling is not required. The requirements regarding sideband resolution, circulating cavity power, and environmental temperature are in reach with state of the art parameters of optomechanical crystals. Phonon antibunching can be verfied in a Hanburry Brown–Twiss measurement on the output field of the optomechanical cavity.

Received 2 March 2015

DOI:https://doi.org/10.1103/PhysRevA.91.061803

Authors & Affiliations

Niels Lörch and Klemens Hammerer

Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Callinstraße 38, 30167 Hannover, Germany and Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

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Vol. 91, Iss. 6 — June 2015

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Images

Figure 1
Left: Two optical modes $a$ and $b$ are coupled to a mechanical mode $c$ . The $b$ mode is resonantly driven by a laser of strength $E$ and frequency $ω_{L} = ω_{b}$ . The $a$ mode is detuned from $b$ by $Δ = ω_{m}$ , the mechanical resonance frequency, as depicted in the plot on the right. The nonlinear interaction of the three modes $a$ , $b$ , and $c$ gives rise to optomechanical limit cycles with strongly sub-Poissonian phonon number statistics. A third optical mode $d$ can be used to reduce the effective temperature of the mechanical oscillator's bath and to read out the phonon statistics in a Hanburry Brown–Twiss measurement.
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Figure 2
(a) Optically mediated (anti)damping $γ_{opt} (ζ)$ (bold line) as a function of mechanical amplitude $ζ$ according to Eq. (6). The steady state $ζ_{0}$ is reached when $γ_{opt} (ζ_{0}) = - γ$ (dashed line). (b) Intracavity photon number ${| α |}^{2}$ in mode $a$ (blue dashed line), ${| β |}^{2}$ in mode $b$ (red dash-dotted line), and total photon number $n_{cav} = {| α |}^{2} + {| β |}^{2}$ (black solid line) is plotted as a function of mechanical amplitude $ζ$ according to Eq. (5). The optically induced diffusion $D_{opt} = g_{0}^{2} \frac{κ}{2} {(| α |}^{2} + {| β |}^{2}) / h_{ζ}$ of the mechanical oscillator scales exactly like the red dash-dotted line with a scale as given on the right $y$ axis. (c) Schematic phase space trajectory of the mechanical oscillator approaching the limit cycle attractor with amplitude $ζ_{0}$ . In the corotating frame of the oscillator the $X$ quadrature relates to its amplitude and the $Y$ quadrature to its phase.
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Figure 3
(a) Fano factor as a function of effective mechanical bath occupation number $\bar{n}$ and total number of photons in the cavity $n_{ph} [κ γ / 4 g_{0}^{2}] = \sqrt{R}$ according to Eq. (13). (b) Plot of $[g^{(2)} (0) - 1] [{(g_{0} / κ)}^{2}]$ as a function of the same parameters in units of the squared single-photon strong-coupling parameter ${(g_{0} / κ)}^{2}$ according to Eqs. (15) and (16). The condition for both sub-Poissonian statistics ( $F < 1$ ) and antibunching $[g^{(2)} (0) < 1]$ is visualized by the red contour line $\bar{n} = 1 - 2 / \sqrt{R}$ in both plots. (c) Plot of $g_{opt}^{(2)} (0)$ for optimal choice of $R$ as a function of $g_{0} / κ$ and $\bar{n}$ according to Eq. (17). All units in this figure are dimensionless.
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Figure 4
(a) Comparison of analytical results (black solid line) from Eq. (13) to numerical results for Fano factor $F$ with increasing $g_{0} / κ = 0.25, 0.5, 1$ (square, diamond, circle). The parameter $g_{0} E / κ^{2} = 0.04$ is fixed to stay well inside the regime of validity of the adiabatic elimination. In this plot $\bar{n} = 0$ but for finite temperature the agreement of numerics with Eq. (13) is equally good. (b) Negativity (quotient of smallest and largest value) of the mechanical oscillator's Wigner function in steady state calculated with QuTiP's steady state solver. The bath occupation is $\bar{n} = 0.25, 0.5, 1, 2$ for the increasing curves. The driving field $E = 0.07 κ$ is constant in both plots, to stay well in the regime of $n_{ph} ≪ 1$ for numerical simplicity. Each point in (b) is optimized over $R$ by varying $γ$ . All units in this figure are dimensionless.
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