PhD Defense by Peyman Malekzadeh

Multi-mode Entanglement in Membrane-in-the-Middle Cavity

This thesis develops an optomechanical platform whose central objective is to realize and verify quantum correlations—in particular, light–mechanics entanglement—in a membrane-in-the-middle (MIM) Fabry–Perot cavity system. The work is struc-tured in a bottom-up way so that each part provides what is needed for the final entanglement measurement. It begins by introducing a high-finesse, overcoupled Fabry–Perot cavity using quantum input–output theory, because the ultimate witness of entanglement will be extracted from the outgoing optical field. A high-Q Si3N4 membrane is then placed inside the cavity and treated as a macroscopic mechanical oscillator with well-defined conjugate quadratures, driven by both the thermal bath and the cavity field. The mechanical oscillator is probed at cryogenic temperature in order to operate in a regime where the coherent interaction with light dominates over the thermal decoherence rate.

For this optical–mechanical system, the standard radiation-pressure interaction is derived from the full optomechanical Hamiltonian and linearized around a strong drive. This leads to a concrete model of optomechanical (ponderomotive) squeezing for the MIM system, including the experimentally relevant case where more than one membrane mode, located in two different band gaps, couples to the same cavity mode. Running the cavity in an overcoupled, red-detuned regime allows amplitude fluctuations of light to be transduced by the mechanics and partially fed back into the optical phase, producing sub-shot-noise fluctuations in the output quadrature, which are observed by direct detection of the cavity output light.

 After establishing the stationary picture, the thesis moves to the time domain treatment of pulsed optomechanics. A concrete setup is introduced in which the cavity is kept locked, while the membrane is probed with blue-detuned optical pulses of controlled duration. Because this is a non-stationary regime, the usual frequency-domain spectra are replaced by time-synchronized acquisition and analysis of the output field. This setup enables operating the system in the blue-detuned regime in a way that is suitable for implementing and characterizing a two-mode-squeezing interaction between light and mechanics. However, due to technical limitations, The demonstration shows that the same MIM cavity can be driven in a way that is naturally suited forgenerating EPR-type correlations.  

The thesis then returns to the stationary regime, but now with the explicit aim of observing light–mechanics entanglement. A theoretical model is developed to show that, in the unresolved-sideband but quantum-cooperative regime, the continuously driven cavity already contains the right mixture of beam-splitter and two-mode-squeezing terms to produce steady quantum correlations. The key experimental requirement is to measure both amplitude and phase quadratures (with a vacuum noise penalty) of the outgoing light simultaneously. To this end, the thesis proposes and implements a dual-quadrature (double-homodyne) detection scheme, based on polarization separation of the cavity output beam and a local oscillator, together with a data-analysis pipeline that builds covariance matrices from the recorded time traces and evaluates an EPR-type variance. In this way, all the earlier elements—cavity modeling, MIM coupling, ponderomotive squeezing, pulsed time-domain techniques, and dual-quadrature readout—are combined into a practical route to certifying light–mechanics entanglement in a realistic optomechanical membrane system