Selectable linear or quadratic coupling in an optomechanical system

There has been much interest recently in the analysis of optomechanical systems incorporating dielectric nano- or microspheres inside a cavity field. We analyze here the situation when one of the mirrors of the cavity itself is also allowed to move. We reveal that the interplay between the two oscillators yields a cross-coupling that results in, e.g., appreciable cooling and squeezing of the motion of the sphere, despite its nominal quadratic coupling. We also discuss a simple modification that would allow this cross-coupling to be removed at will, thereby yielding a purely quadratic coupling for the sphere.

Figure 1

Schematic model of our system. A dielectric nano- or microparticle is coupled quadratically to a cavity field that is bounded by a moving mirror. The particle is held in place by an externally generated harmonic potential, shown dotted, whose origin is left open in this work. The cavity field is pumped and decays only through the immobile mirror. Different pumping configurations are explored in Appendix 5.

Figure 2

Occupation number for the motion of the sphere, presented on a logarithmic scale. Note the “resonant” condition, indicated by the dashed line, corresponding to the sphere frequency that minimizes this occupation number at each particular ${\omega }_1$. Cooling of the motion of the sphere by almost three orders of magnitude is seen, despite the nominal quadratic coupling. The plot here has been optimized over $\tilde{\Delta }$ and $P_{\mathrm{in}}$. The mirror and sphere are in contact with separate baths at temperatures of 50 mK and 1 K, respectively. The rest of the parameters used for generating this plot are in Sec. 2. Inset: Cut of the main figure along the line ${\omega }_1=10{\kappa }_{\mathrm{c}}$ (blue, lower curve). The red, upper curve shows the occupation number in the absence of any cooling. The dashed line indicates the value of ${\omega }_2$ for which the lowest occupation number is reached.

Figure 3

(a) Normal-mode frequencies, extracted from the eigenvalues of $\mathbf{A}$. For low powers, the red (top) curve represents the cavity mode, the green (middle, at low powers) curve the mirror, and the blue (bottom) curve the sphere. (b) Occupation numbers for the sphere (blue; upper curve at low powers) and mirror (green; lower). Note the narrow range of powers for which the systems hybridize. At this point, the two frequencies and occupation numbers are equal. We have chosen ${\omega }_1=10{\kappa }_{\mathrm{c}}$ and the optimal values for ${\omega }_2$ and $\tilde{\Delta }$. The mirror and sphere are both in contact with baths at a temperature of 1 K. The dashed line denotes the value of $P_{\mathrm{in}}$ for which the occupation number is minimized. (${\omega }_1=10{\kappa }_{\mathrm{c}}$, ${\omega }_2=3.4{\kappa }_{\mathrm{c}}$, $\tilde{\Delta }=-27.2{\kappa }_{\mathrm{c}}$, $g_1=1.0\times {10}^{-3}{\kappa }_{\mathrm{c}}$, $g_2=-2.4\times {10}^{-10}{\kappa }_{\mathrm{c}}$, $\chi =3.7\times {10}^{-3}$; other parameters as in the body of the paper.)

Figure 4

(a) Quadrature variances, $\langle {\widehat{x}}_j^2\rangle$ (solid curves) and $\langle {\widehat{p}}_j^2\rangle$ (dashed curves); the green, upper curves represent the mirror ($j=1$), the blue, lower curves the sphere ($j=2$). The solid black horizontal line represents the $\frac{1}{2}$-quantum ground-state vacuum variance. Note that only the $p$ quadrature of the sphere is squeezed below the vacuum level. The baths for both oscillators were held at zero temperature, and $g_2$ was a factor of 100 larger than the parameters in the body of the paper. (b) Corresponding squeezing for the sphere. (${\omega }_1=20{\kappa }_{\mathrm{c}}$, ${\omega }_2=10{\kappa }_{\mathrm{c}}$, $\tilde{\Delta }=-10{\kappa }_{\mathrm{c}}$, $g_1=7.2\times {10}^{-4}{\kappa }_{\mathrm{c}}$, $g_2=-8.0\times {10}^{-9}{\kappa }_{\mathrm{c}}$, $\chi =4.5\times {10}^{-3}$; other parameters as in the body of the paper.)

Figure 5

Different pumping geometries in one dimension. (a) For symmetric pumping, a node or antinode always lies at the center of the cavity. (b) For pumping from the immobile mirror, the nodal structure shifts with the moving mirror. (c) For pumping from the moving mirror, the nodal structure is fixed with respect to the immobile mirror.