Mirror-field entanglement in a microscopic model for quantum optomechanics

We use a microscopic model, the mirror-oscillator-field (MOF) model proposed by C. R. Galley, R. O. Behunin, and B. L. Hu [Phys. Rev. A 87, 043832 (2013)], to describe the quantum entanglement between a mirror's center-of-mass (c.m.) motion and a field. In contrast with the conventional approach where the mirror-field entanglement is understood as arising from the radiation pressure of an optical field inducing the motion of the mirror's c.m., the MOF model incorporates the dynamics of the internal degrees of freedom of the mirror that couple to the optical field directly. The major advantage in this approach is that it provides a self-consistent treatment of the three pertinent subsystems (the mirror's c.m. motion, its internal degrees of freedom, and the field) including their back-actions on each other, thereby giving a more accurate account of the quantum correlations between the individual subsystems. The optical and the mechanical properties of a mirror arising from its dynamical interaction with a quantum field are obtained without imposing any boundary conditions on the field additionally, as is done in the conventional way. As one of the new physical features that arise from this self-consistent treatment of the coupled optics and mechanics behavior we observe a coherent transfer of quantum correlations from the field to the mirror via its internal degrees of freedom. We find the quantum entanglement between the optical field and the mirror's center-of-mass motion upon coarse-graining over the internal degree of freedom. Further, we show that in certain parameter regimes the mirror-field entanglement is enhanced when the field interacts resonantly with the mirror's internal degree of freedom, a result which highlights the importance of including the internal structure of the mirror in quantum optomechanical considerations.

Figure 1

Schematic representation of the interaction of a mirror with a field via its internal degree of freedom.

Figure 2

Reflection properties from the two different forms of coupling ($q\mathrm{\Phi }$ and $\dot{q}\mathrm{\Phi }$). (a) Reflectance as a function the incident field wavelength (in meters) for different idf to plasma frequency ratios $(r_p=\mathrm{\Omega }/{\mathrm{\Omega }}_P)$; the plasma frequency is fixed at ${\mathrm{\Omega }}_P=1.37\times {10}^{16}$ Hz (for silver) from $q\mathrm{\Phi }$ coupling, choosing $r_p\approx 0.3$ mimics the cutoff behavior for silver. (b) Reflectance and transmittance spectrum from $\dot{q}\mathrm{\Phi }$ coupling to simulate the optical response for a photonic crystal as from the experimental results in [36]. Each resonance corresponds to a separate effective idf with resonance frequencies $\mathrm{\Omega }=\{3.01\times {10}^{15}\mathrm{Hz},2.51\times {10}^{15}\mathrm{Hz},2.43\times {10}^{15}\mathrm{Hz}\}$ and corresponding plasma frequencies ${\mathrm{\Omega }}_P=\{0.5\times {10}^{14}\mathrm{Hz},0.2\times {10}^{14}\mathrm{Hz},0.1\times {10}^{14}\mathrm{Hz}\}$.

Figure 3

(a) Reflectance as a function of the dimensionless parameter ${\mathrm{\Omega }}_P/\mathrm{\Omega }$ (ratio of the plasma frequency to the idf's natural frequency) and the idf-field detuning $\mathrm{\Delta }/\mathrm{\Omega }$. It can be seen that for weaker coupling corresponding to ${\mathrm{\Omega }}_P/\mathrm{\Omega }\ll 1$, the reflection spectrum has a sharper resonance. The effective bilinear coupling strengths for both (b) idf-field $\left({\alpha }_{OF}\right)$ and (c) idf-mdf $\left({\alpha }_{OM}\right)$ increase with increasing plasma frequency as $\sim \sqrt{{\mathrm{\Omega }}_P}$. (d) The effective mdf-field coupling coefficient $\left({\alpha }_{MF}\right)$ in the weak-coupling limit is largely determined by the reflection coefficient, while for strong coupling the fluctuation-mediated part becomes relevant, as can be seen from (33).

Figure 4

Evolution of mirror-field entanglement as measured by the logarithmic negativity $E_{MF}$ (see Appendix pp1 for definition) as obtained from the boundary condition approach (Appendix pp2) and the coupled MOF dynamics. We find that for an isolated idf the time scale for entanglement is largely determined by the effective idf-field coupling (${\alpha }_{OF}$). The two approaches concur in the weak-coupling limit for a strongly damped idf. The parameter values, in natural units where $c=1, \hslash =1$, and $e=\sqrt{4\pi \alpha }$, used here are $m=0.001, \mathrm{\Omega }=100, M=10, \mho =0.1, {\mathrm{\Omega }}_P=5, A_0={10}^{-4}$, and $T=1000$. The effective idf-field coupling strength, $|{\alpha }_{OF}|/\mho \approx 16$, determines the time scale for entanglement dynamics for the case of an undamped idf.

Figure 5

Mirror-field entanglement given by the logarithmic negativity for an undamped idf as a function of the dimensionless idf-field detuning ($\mathrm{\Delta }/\mho$) and dimensionless time ($\mathrm{\Omega }t$). We observe that the entanglement peaks for a resonant idf-field interaction at ($\mathrm{\Delta }/\mho =0$) and for $\mathrm{\Delta }/\mho =-1$ ($\mathrm{\Omega }=\omega +\mho$); the entanglement is sustained for longer times. The oscillation time scales are determined by the effective idf-field coupling ($|{\alpha }_{OF}|$). The parameters values, in units where $c=1, \hslash =1$, and $e=\sqrt{4\pi \alpha }$, used here are $m=0.001, \mathrm{\Omega }=100, M=10, \mho =0.1, {\mathrm{\Omega }}_P=0.05, A_0={10}^{-4}$, and $T=1000$.