Mechanical resonators for storage and transfer of electrical and optical quantum states

We study an optomechanical system in which a microwave field and an optical field are coupled to a common mechanical resonator. We explore methods that use these mechanical resonators to store quantum-mechanical states and to transduce states between the electromagnetic resonators from the perspective of the effect of mechanical decoherence. Besides being of fundamental interest, this coherent quantum state transfer could have important practical implications in the field of quantum information science, as it potentially allows one to overcome intrinsic limitations of both microwave and optical platforms. We discuss several state transfer protocols and study their transfer fidelity using a fully quantum-mechanical model that utilizes quantum state diffusion techniques. This work demonstrates that mechanical decoherence should not be an insurmountable obstacle in realizing high-fidelity storage and transduction.

Figure 1

A schematic diagram for an optical cavity coupled to a thin dielectric membrane mechanical resonator, which in turn is coupled to a resonant electrical $LC$ circuit. The system can be pumped or read out by both microwave and optical drives.

Figure 2

An equivalent system of two coupled adjustable beam splitters which can be a formal analog to the two-mode optomechanical system in the linearized approximation. In this open quantum system, each oscillator is also coupled to its respective reservoir (not shown).

Figure 3

A schematic diagram of the coupling $\pi$-pulse sequence for quantum state memory tests. The sign of the coupling constant for retrieval must be the opposite of the sign for storage in order to cancel the phase accumulation during the pulses.

Figure 4

Memory fidelity as a function of mechanical resonator quality. (a) For $\overline{n}=0$. Input states $|\alpha \rangle$ (red circles) with $\alpha =1$, $|{\psi }_{SF}\rangle$ (black diamonds), $|{\psi }_{cat}\rangle$ (blue squares). The solid line is the analytic formula for the coherent fidelity from Eq. (17). (b) For $\overline{n}=3$. (c) For squeezed states $|\alpha ,\xi \rangle$ with $\alpha =1$ and $\overline{n}=0$ for various squeezing parameters $\xi$. In all cases, $\Delta T=64{\omega }_m^{-1}$ and the peak coupling is ${\Omega }_{\mu }=0.1{\omega }_m$. The horizontal dotted black line indicates $95\%$ fidelity for reference.

Figure 5

(a) Memory fidelity of $|{\psi }_{SF}\rangle$ as a function of mechanical resonator quality for various wait times for $\overline{n}=3$. The fidelity decreases exponentially with decreasing $Q$ and with increased wait times. At low $Q$, the fidelity saturates to the thermalized value. (b) The memory fidelity versus scaled wait time for $\overline{n}=0$. (c) The memory fidelity versus scaled wait time for different temperatures, showing how all the curves collapse onto one universal curve. In all cases, the peak coupling is ${\Omega }_{\mu }=0.1{\omega }_m$.

Figure 6

Distribution of fidelities for $|{\psi }_{SF}\rangle$ for $Q_m=10000$ (a), $Q_m=1000$ (b), and $Q_m=500$ (c) for 1000 independent QSD trajectories. In all cases, $\Delta T=64{\omega }_m^{-1}$, $\overline{n}=3$, and the peak coupling is ${\Omega }_{\mu }=0.1{\omega }_m$. Note the different horizontal axis in each plot.

Figure 7

(a) Transfer fidelity vs mechanical resonator quality for $|{\psi }_t\rangle$ for the separated pulse scheme (red dots), the simultaneous pulse scheme (blue squares), intuitive pulse scheme (green diamonds), and counterintuitive pulse scheme (black stars) with $\overline{n}=3$. (b) Number of photons or phonons in each resonator for intuitive, separated coupling pulses for zero temperature where $\Delta T=64{\omega }_m^{-1}$. (c) The pulse profile for separated pulses. (d) Number of photons or phonons in each resonator for simultaneous coupling pulses for zero temperature. (e) The pulse profile for simultaneous pulses. In all cases, the peak couplings are ${\Omega }_o={\Omega }_{\mu }=0.1{\omega }_m$.

Figure 8

The fidelity versus pulse area and pulse separation for $Q_m=100000$, $\overline{n}=0$, and ${\Omega }_o={\Omega }_{\mu }=0.1{\omega }_m$. The negative horizontal axis values represent the peak separation for counterintuitively ordered pulses, while the positive horizontal axis values represent the peak separation for intuitively ordered pulses.

Figure 9

Schematic of a three-state system as conventionally used for implementing a STIRAP transfer process.

Figure 10

The number of photons or phonons in each resonator for the STIRAP-like coupling pulses. The pulse area is $10\pi$ and the peak separation is –$14\%$.