Selective Linear or Quadratic Optomechanical Coupling via Measurement

The ability to engineer both linear and nonlinear coupling with a mechanical resonator is an important goal for the preparation and investigation of macroscopic mechanical quantum behavior. In this work, a measurement based scheme is presented where linear or square mechanical-displacement coupling can be achieved using the optomechanical interaction that is linearly proportional to the mechanical position. The resulting square-displacement measurement strength is compared to that attainable in the dispersive case that has a direct interaction with the mechanical-displacement squared. An experimental protocol and parameter set are discussed for the generation and observation of non-Gaussian states of motion of the mechanical element.

Popular Summary

Experimental advances in quantum optics in recent decades have beautifully elucidated the fundamental and often counterintuitive features of quantum mechanics. Cavity optomechanics aims to extend the realm of quantum optical control to a mechanical one: using it to prepare and study quantum states of motion of micro- or nano-fabricated mechanical oscillators. Simply put, the field of optomechanics uses light to measure and to manipulate the motion of a mechanical device that is sufficiently compliant to move under even the reflection of light. Such optomechanical systems are envisioned to be used in quantum information processing, for high-precision sensors, or to study quantum decoherence. One of the important goals in cavity optomechanics is to experimentally prepare and observe quantum states of mechanical motion. This is, however, technically challenging. In this paper, an avenue is opened that significantly relaxes the experimental requirements needed to achieve this goal.

Currently there are two approaches to cavity optomechanics: a reflective approach and a dispersive approach. In the former, the interaction depends linearly upon the position of the mechanical oscillator; and in the latter, one may choose between position or position-squared coupling by careful placement of the mechanical oscillator within an optical field. This selectivity provides considerable versatility for coherent quantum experiments. Here, a scheme is introduced that illustrates how to perform measurements of the mechanical displacement squared for the reflective approach. Remarkably, it turns out that the resulting position-squared measurements of the mechanical oscillator can be much stronger than that achievable in the dispersive case.

This paper introduces a powerful new optomechanical procedure for the preparation and exploration of quantum behavior of mechanical devices without the need for strong single-photon coupling.

Figure 1

Cavity optomechanics is currently realized using (a) reflective and (b) dispersive approaches. The interaction in the former is proportional to mechanical displacement $X_M$, however, in the latter the interaction can be tuned to being proportional to $X_M$ or $X_M^2$. Pulses of light may be used to probe and manipulate the motional state of the mechanical resonator. The optical setup considered here (c) is a pulse incident upon an optomechanical system with an interaction proportional to $X_M$ and then an optical quadrature measurement performed via homodyne detection. Following the interaction, the Wigner function of the optical field (d) is scimitar shaped due to mechanical-position-induced optical rotations. The distribution of the initial coherent state is indicated by the dashed circle. (The parameters for this plot were chosen to exaggerate the curvature.) For small rotations, the phase quadrature $Q_P$ is proportional to $X_M$ and the amplitude quadrature carries $X_M^2$ information. The amplitude-quadrature measurement outcome indicated by the red dashed line may have originated from two distinct mechanical positions, which provides a means for superposition preparation. By choosing the phase in (e) homodyne detection one can use the amplitude quadrature for $X_M^2$ measurements and quantum-state preparation or use the phase quadrature for $X_M$ measurements to perform quantum-state tomography.

Figure 2

Mechanical Wigner functions of initial states (left), conditional states (center), and unconditional states (right). ($X_M$ is the horizontal axis, $P_M$ is the vertical axis. The plot range is $\pm 5$ for all axes. Color scale: black is for zero magnitude, blue for positive values, and red for negative values.) The initial states are the ground state, $\overline{n}=0$ (a), a thermal state with $\overline{n}=2$ (d), and a momentum-squeezed vacuum state with squeezing parameter $r=0.5$ (g). Conditional states prepared with ${{\rm Y}}_X$ acting on the corresponding initial states with ${\chi }_X=1$, $\Delta Q_X=1.5$, $w=0.8$ are shown in (b) and (e) and $\Delta Q_X=6.4$ has been used in (h). The probabilities of obtaining an outcome in the windows used above are: (b) 14.9%, (e) 14.5%, and (h) 1.1%. Note the disappearance of negativity—a quantum-to-classical transition—for initial thermal occupation (e) and if the measurement outcomes are ignored (c), (f), (h).