Generation of cluster states in optomechanical quantum systems

We consider an optomechanical quantum system composed of a single cavity mode interacting with $N$ mechanical resonators. We propose a scheme for generating continuous-variable graph states of arbitrary size and shape, including the so-called cluster states for universal quantum computation. The main feature of this scheme is that, differently from previous approaches, the graph states are hosted in the mechanical degrees of freedom rather than in the radiative ones. Specifically, via a $2N$-tone drive, we engineer a linear Hamiltonian which is instrumental to dissipatively drive the system to the desired target state. The robustness of this scheme is assessed against finite interaction times and mechanical noise, confirming it as a valuable approach towards quantum state engineering for continuous-variable computation in a solid-state platform.

Figure 1

(a) Optomechanical system consisting of one optical cavity mode $a$ coupled to $N$ noninteracting mechanical resonators $b_1,\cdots ,b_N$. The cavity dissipates with a damping rate $\kappa$, and it is driven by $M$ classical laser fields. (b) The state of the mechanical resonators can be prepared in different graph state geometries, e.g., from left to right, a linear, a dual-rail, and a generic graph state.

Figure 2

Time evolution of the fidelity for a four-modes linear graph state with fixed switching time $t_{\mathrm{s}}=20{\kappa }^{-1}$. We have set for the $j\text{th}$ oscillator the frequency ${\mathrm{\Omega }}_j/2\pi =j\mathrm{MHz}$.

Figure 3

Fidelity of the steady state of the four-mode linear graph as a function of evolution time per switching step. The fidelity shown in this plot is that of the steady state of the mechanical oscillators after applying all the switching steps. We have set for the $j\text{th}$ oscillator the frequency ${\mathrm{\Omega }}_j/2\pi =j\mathrm{MHz}$.

Figure 4

Contour plots of the fidelity between the target state and the state generated using the protocol described in the text. The fidelity is shown as a function of the temperature of the mechanical bath (horizontal axis) and the mechanical damping rate (vertical axis) for different target graph states and levels of squeezing. Each mechanical oscillator has a frequency ${\mathrm{\Omega }}_j/2\pi =j\mathrm{MHz}$ $(j=1,...,N$, with $N=2,4,8)$. The solid white lines correspond to a fidelity of $0.99$, the dashed lines to $0.90$, and the dotted lines to $0.80$. Each data point is taken for an optimal evolution time (see text).

Figure 5

Optomechanical system consisting of one optical cavity mode and $N$ mechanical resonators modes. The cavity mode couples only to the first mechanical mode, and all the mechanical modes couple to each other via nearest-neighbor interaction.

Figure 6

Fidelity as a function of the target squeezing between a linear graph state of $N$ nodes and the state generated using the protocol described in the text. Each data point is taken for an optimal evolution time (see text). Each mechanical oscillator has a frequency ${\mathrm{\Omega }}_j/2\pi =11j\mathrm{MHz}$ $(j=1,...,N$, with $N=1,\cdots ,6)$ and mechanical damping $\gamma /2\pi =32\mathrm{Hz}$. The cavity mode decays with rate $\kappa /2\pi =0.2\mathrm{MHz}$. The bath temperature for panel (a) is $T=15\mathrm{mK}$ and for panel (b) is $T=1\mathrm{mK}$.