Optomechanics assisted by a qubit: From dissipative state preparation to many-partite systems

We propose and analyze nonlinear optomechanical protocols that can be implemented by adding a single atom to an optomechanical cavity. In particular, we show how to engineer the environment in order to dissipatively prepare the mechanical oscillator in a superposition of Fock states with fidelity close to 1. Furthermore, we demonstrate that a single atom in a cavity with several mechanical oscillators can be exploited to realize nonlinear many-partite systems by stroboscopically driving the mechanical oscillators. This can be used to prepare nonlinear many-partite states by applying either coherent protocols or engineering dissipation. The analysis of the protocols is carried out using a perturbation theory for degenerate Liouvillians and numerical tools. Our results apply to other systems where a qubit is coupled to a mechanical oscillator via a bosonic mode, e.g., in cavity quantum electromechanics.

Figure 1

Fidelity for the preparation of the non-Gaussian state ${\rho }_{\mathrm{A}}$ as a function of ${\mathcal{C}}_{\mathrm{m}}$ for different qubit cooperativities ${\mathcal{C}}_{\mathrm{q}}$. Upper solid (blue) line: ${\mathcal{C}}_{\mathrm{q}}=\infty$. Upper dashed (red) line: ${\mathcal{C}}_{\mathrm{q}}=100$. Dash-dotted (green) line: ${\mathcal{C}}_{\mathrm{q}}=20$ [dotted (black) line: comparison to the analytic result for ${\mathcal{C}}_{\mathrm{q}}=20$]. Lower solid (orange) line: ${\mathcal{C}}_{\mathrm{q}}=10$. Lower dashed (purple) line: ${\mathcal{C}}_{\mathrm{q}}=5$.

Figure 2

(a) Fidelity to prepare ${\tilde{\rho }}_{\mathrm{A}}$ (${\rho }_{\mathrm{A},\mathrm{m}}$) as a function of $\zeta$ for ${\gamma }_{\mathrm{aux}}/{\gamma }_{\mathrm{eff}}=1$ and different cooperativities. The size of the Hilbert space for the mechanical oscillator is chosen as $N=10$. Upper solid (blue) line: ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=\infty$. Upper dashed (red) line: ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=100$. Dash-dotted (green) line: ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=20$. Lower solid (orange) line: ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=10$. Inset: Comparison of the simulation for ${\mathcal{C}}_{\mathrm{m}}={\mathcal{C}}_{\mathrm{q}}=100$ and different sizes of the Hilbert space. Solid (blue) line, $N=30$; dashed (red) line, $N=10$. (b) Fidelity to prepare ${\rho }_{\mathrm{A},\mathrm{m}}$ after the measurement. Parameters and color scheme as in (a). (c) Fidelity to prepare ${\rho }_{\mathrm{A},\mathrm{m}}$ as a function of ${\gamma }_{\mathrm{aux}}/{\gamma }_{\mathrm{eff}}$ for $\zeta =0.2$, ${\mathcal{C}}_{\mathrm{m}}={\mathcal{C}}_{\mathrm{q}}=1000$, and different jump operators ${\widehat{J}}_1$. Upper solid (blue) line: ${\widehat{J}}_1={\widehat{\sigma }}^{+}-\zeta {\widehat{b}}^{†}$. Dashed (red) line: ${\widehat{J}}_1={\widehat{\sigma }}^{+}$. Dash-dotted (green) line: ${\widehat{J}}_1={\widehat{\sigma }}^{+}+\zeta {\widehat{b}}^{†}$. Lower solid (orange) line: ${\widehat{J}}_1={\widehat{b}}^{†}$.

Figure 3

(a) Fidelity for the preparation of the non-Gaussian entangled state ${\tilde{\rho }}_{\mathrm{A}}$ as a function of $\zeta$ in the perturbative regime ${\gamma }_{\mathrm{aux}}/{\gamma }_{\mathrm{eff}}=0.1$. Different colors show various cooperativities and compare the numerical result (solid line) to the prediction within perturbation theory (dashed line). From top to bottom: blue, ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=\infty$; red, ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=100$; green, ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=20$; orange, ${\mathcal{C}}_{\mathrm{q}}={\mathcal{C}}_{\mathrm{m}}=10$. (b) Fidelity for the preparation of ${\rho }_{\mathrm{A},\mathrm{m}}$ after taking a measurement of the qubit as described in the main text. Parameters and color scheme are as in (a).

Figure 4

Dynamics of two stroboscopically driven mechanical oscillators coupled to an initially excited qubit. Top: Comparison between the population of the mechanical mode under full evolution [olid (blue) line] and the adiabatically eliminated one [dashed (red) line]; all parameters are given in units of $g_{\mathrm{q}}{g}_{\mathrm{m}}/\Delta$ and no dissipation is included. Left: Conditions (i)–(v) are fulfilled. Right: Condition (i) is not fulfilled. Bottom: Evolution under the influence of dissipation over time $[\Delta /g_{\mathrm{m}}{g}_{\mathrm{q}}]$ for different cooperativities: upper solid (blue) line, ${\mathcal{C}}_{\mathrm{m}}={\mathcal{C}}_{\mathrm{q}}=\infty$; dashed (red) line, ${\mathcal{C}}_{\mathrm{m}}={\mathcal{C}}_{\mathrm{q}}=1000$; dotted (green) line, ${\mathcal{C}}_{\mathrm{m}}={\mathcal{C}}_{\mathrm{q}}=100$, lower solid (orange) line, ${\mathcal{C}}_{\mathrm{m}}={\mathcal{C}}_{\mathrm{q}}=10$.