Engineering of nonclassical motional states in optomechanical systems

We propose to synthesize arbitrary nonclassical motional states in optomechanical systems by using sideband excitations and photon blockade. We first demonstrate that the Hamiltonian of the optomechanical systems can be reduced, in the strong single-photon optomechanical coupling regime when the photon blockade occurs, to one describing the interaction between a driven two-level trapped ion and the vibrating modes, and then show a method to generate target states by using a series of classical pulses with desired frequencies, phases, and durations. We further analyze the effect of the photon leakage, due to small anharmonicity, on the fidelity of the expected motional state, and study environment induced decoherence. Moreover, we also discuss the experimental feasibility and provide operational parameters using the possible experimental data.

Figure 1

(a) Schematic diagram for optomechanical systems with the radiation-pressure type interaction: the cavity can be in either the optical, or the microwave, or the radiowave regime; and the mechanical resonator can be doubly clamped beams, singly clamped cantilevers, radial breathing modes of microtoroids, and membranes. (b) Three different transition processes are presented by the black (carrier), red ($k$-phonon red-sideband excitation), and blue ($k$-phonon blue-sideband excitation) arrow lines.

Figure 2

Schematic diagram for the information leakage to the third level due to the small arharmonicity. Here, the two lowest horizontal lines in each column linked by the vertical black line with two arrows denote two-level approximation with the carrier process. The gray vertical line with the arrow pointed to the top line in each column simply denotes the information leakage in the carrier process. However, each red (dark red) slanted line with two arrows pointed to two black lines in different columns denotes the red sideband excitation (information leakage in the red sideband excitation). The first and second letter in the states, e.g, $|g,0\rangle$, denote that the cavity field and the mechanical resonator are the ground state $|g\rangle$ and the vacuum state $|0\rangle$, respectively.

Figure 3

Fidelities are plotted as a function of $\left|\delta \right|/\Omega$ for preparing states $|2\rangle$ in (a) and $\left(\right|0\rangle -|2\rangle )/\sqrt{2}$ in (b) using both numerical (black solid curve) and approximately analytical (red dash curve) results for $\eta =0.1$.

Figure 4

Fidelities are plotted for preparing states $|2\rangle$ in (a) and $\left(\right|0\rangle -|2\rangle )/\sqrt{2}$ in (b) as a function of the cavity decay rate ${\gamma }_c/\Omega$ with different temperatures (${\overline{n}}_m=0,1,5$) for ${\gamma }_m={\gamma }_c/10$, $\left|\delta \right|/\Omega =10$, and $\eta =0.1$.