Engineering Superposition States and Tailored Probes for Nanoresonators via Open-Loop Control

We show that a nanoresonator can be prepared in mesoscopic superposition states merely by monitoring a qubit coupled to the square of the resonator’s position. This works for thermal initial states, and does not require a third-order nonlinearity. The required coupling can be generated using a simple open-loop control protocol, obtained with optimal control theory. We simulate the complete preparation process, including environmental noise. Our results indicate the power of open-loop control for state engineering and measurement in quantum nanosystems.

Figure 1

Snapshots of the Wigner function for a harmonic oscillator with effective frequency $f^{\prime }=1/\tau =1\mathrm{MHz}$, subjected to a continuous measurement of the square of position. Luminosity denotes the absolute value of the Wigner function (online: blue is positive and green negative). The $x$ axis is the dimensionless position, $x=a+a^{†}$, and the $y$ axis is $p=-i(a-a^{†})$. Both axes cover the interval [$-7.1$, 7.1]. Top row, temperature $T=0$: (a) $t=0$ (vacuum state); (b) $t=6.25\tau$; (c) $t=13.75\tau$. Bottom row, $T=22.7\mathrm{mK}$: (d) $t=0$ (thermal state); (e) $t=17.625\tau$; (f) $t=18.5\tau$.

Figure 2

A control protocol for generating the effective interaction $\mu {\sigma }_z{x}^2$ between a nanoresonator and a qubit, where the physical interaction is $200\mu {\sigma }_{z}x$, and the effective frequency of the resonator is ${\omega }^{\prime }=\mu$. The control Hamiltonian is $H(t)=\hslash [c_x(t){\sigma }_x+c_y(t){\sigma }_y]$, with $c_x$ shown in (a), and $c_y$ shown in (b). The insets give an expanded view of the first tenth of the protocol.

Figure 3

The Wigner function for a harmonic oscillator measured via a CPB. Plots (a)–(c): Mesoscopic superposition states generated by the measurement for various CPB decoherence rates, $\gamma$, and $T=0$. The phase-space scale is the same as Fig. 1. (a) $\gamma =0$; (b) $\gamma =0.1f^{\prime }$; $\gamma =0.5f^{\prime }$. Plots (d)–(f) are the respective side views of plots (a)–(c). Plots (g)–(i) are a sequence where the resonator is initially prepared in the ground state, with $\gamma =0.1f^{\prime }$ and the bath at $T=30\mathrm{mK}$.