Anomalous optical bistability and robust entanglement of mechanical oscillators

Quantum coherence is one of the most intriguing applications of quantum mechanics, and has led to interesting phenomena and uncommon results. Here we show that in a stark contrast to the usual red-detuned condition to observe bistability in single-mode optomechanics, the optical intensities exhibit bistability for all values of cavity-laser detuning due to intermode coupling induced by the two-photon coherence. Interestingly, an unconventional bistability with “ribbonshaped” hysteresis can be observed for blue-detuned laser frequencies. We also demonstrate that the two-photon coherence leads to a strong entanglement between the movable mirrors in the adiabatic regime. Surprisingly, the mirror-mirror entanglement is shown to persist for environment temperatures of the phonon bath up to 12 K using experimental parameters. PACS numbers: 42.50.Wk, 07.10.Cm, 42.50.Ex, 85.85.+j ∗ Corresponding author: [email protected] 1 ar X iv :1 40 9. 83 38 v1 [ qu an tph ] 2 9 Se p 20 14 The entanglement of macroscopic systems provides insight into the fundamental questions regarding the quantum to classical transition. In this respect, mechanical oscillators are of particular interest because of their resemblance to prototypical classical systems. In addition to the theoretical proposals [1–5] that predict entanglement between a mechanical oscillator and a cavity field, the recent experimental realization [6] of entanglement between the motion of a mechanical oscillator and a propagating microwave in an electromechanical circuit makes optomechanical coupling a promising platform for generating macroscopic entanglement. Other interesting theoretical proposals include the entanglement of the mirrors of two different cavities illuminated by entangled light beams [7], and the entanglement of two mirrors of a double-cavity set up coupled to squeezed light [4, 8, 9]. Optomechanical coupling is also shown to exhibit nonlinear effects such as squeezing [10–13], optical bistability [12, 14–19], optomechanically induced transparency [20, 21], and photon blockade [22, 23], among others. A two-mode laser with a gain medium containing an ensemble of three-level atoms in a cascade configuration is shown to exhibit quenching of spontaneous emission [24] and squeezed light [25–27] due to the two-photon coherence between the upper and lower level of the atoms. In such a laser, the two-photon coherence can be generated in two ways: either by injecting the atoms in a coherent superposition of the upper and lower levels of each atom (injected coherence), or coupling the same levels with a strong laser (driven coherence). These coherences are shown to generate entanglement between the cavity modes of the laser [5, 6, 8, 30], and more recently to entangle the movable mirrors of the doubly-resonant cavity [2, 32, 34]. In this work, we consider a two-mode laser with the two movable mirrors of the doublyresonant cavity coupled to the cavity fields via radiation pressure. The laser system consists of a gain medium of three-level atoms in a cascade configuration. We rigorously derive a master equation for the two-mode laser coupled to thermal reservoirs, which generalizes previous results that are only valid for the case of driven coherence [34]. Using this master equation and the mirror-field interaction Hamiltonian we obtain Langevin equations, which are used to study the bistability and entanglement between the two movable mirrors. We show that, in contrast to the conventional bistability in single-mode optomechanics [14, 15, 35] that is shown to exist only for red-detuned frequencies, the mean photon numbers exhibit bistability for all values of detuning due to the intermode coupling induced by the