Quantum state reconstruction of an oscillator network in an optomechanical setting

We introduce a scheme to reconstruct an arbitrary quantum state of a mechanical oscillator network. We assume that a single element of the network is coupled to a cavity field via a linearized optomechanical interaction, the time dependence of which is controlled by a classical driving field. By designing a suitable interaction profile, we show how the statistics of an arbitrary mechanical quadrature can be encoded in the cavity field, which can then be measured. We discuss the important special case of Gaussian state reconstruction and study numerically the effectiveness of our scheme for a finite number of measurements. Finally, we speculate on possible routes to extend our ideas to the regime of single-photon optomechanics.

Figure 1

Sketch of the system under consideration: $a$ is the annihilation operator of the single intracavity mode, whereas $b_1,\cdots ,b_N$ refer to the mechanical oscillators composing the network. Only the mechanical mode $b_1$ is directly coupled to the radiation field. The driving field (red arrow) can be modulated in order to realize the time-varying linearized radiation-pressure coupling needed to reconstruct a selected quadrature of the oscillator network. Only the field $a_{\mathrm{out}}$ leaking from the cavity is eventually measured via homodyne detection. Repeating the protocol for a sufficient set of quadratures, the state of the entire mechanical network can be reconstructed.

Figure 2

The fidelity $\mathcal{F}$ of a single mode thermal state $\left(T=1\right)$ (left) and a single mode squeezed thermal state $(T=1,r=0.2)$ (right) against the number of measurements $\mathcal{N}$ in each collection of measurement results determining $\langle P^n\rangle$. The fidelities are calculated by comparing the reconstructed covariance matrix with the one of ${\rho }_0$ (see text). In order to exemplify the performance of the reconstruction method, we evaluated the fidelity several times for a fixed $\mathcal{N}$. The points represent the average fidelity thus obtained, whereas the error bars give the respective standard deviation.

Figure 3

The fidelity $\mathcal{F}$ of a two mode thermal state $\left(T=1.5\right)$ (left) and a two mode squeezed thermal state $(T=1.5,r=0.2)$ (right) of the normal modes of a network interacting with a springlike coupling against the number of measurements $\mathcal{N}$ in each collection of measurement results determining $\langle P^n\rangle$. The fidelities are calculated and illustrated as in Fig. 2

Figure 4

Interaction profiles for the case of a single mode reconstruction. The various $\theta$ suffice to reconstruct a single mode Gaussian state.

Figure 5

Interaction profiles for the case of a two mode reconstruction. The various $\mathrm{\Theta }$ suffice to reconstruct a two mode Gaussian state. Each $\mathrm{\Theta }$ corresponds to a pair $\left\{({\theta }_1,{\theta }_2)\right\}=\{(-\pi /2,-\pi /2),(0,0),(0,-\pi /2),(-\pi /2,0),(-3\pi /4,-3\pi /4),(-\pi /4,-\pi /4)\}$. The parameters of the interaction profiles depend on the choice of interaction times $\tau$. These were chosen independently for each profile $\mathrm{\Theta }$ in order to promote their distinguishability. These times were $\tau =5/\nu ,15/\nu ,3/\nu ,25/\nu ,25/\nu ,3/\nu$, respectively, where $\nu$ is the smallest eigenfrequency of the two mode network.

Figure 6

The fidelity $\mathcal{F}$ of a single mode thermal state $\left(T=1\right)$ (left) and a single mode squeezed thermal state $(T=1,r=0.2)$ (right) against the number of measurement results determining $\langle P_{\text{out}}^n\rangle$. The fidelities are calculated and illustrated as in Fig. 2. The calculation is repeated for various values of the genuine optical losses $\epsilon =0,0.4,0.8$, demonstrating that reconstruction is still possible but requires a larger sampling of the observable $P_{\text{out}}$ [see Eq. (21)].