Building mechanical Greenberger-Horne-Zeilinger and cluster states by harnessing optomechanical quantum steerable correlations

Greenberger-Horne-Zeilinger (GHZ) and cluster states are two typical kinds of multipartite entangled states and can respectively be used for realizing quantum networks and one-way computation. We propose a feasible scheme for generating Gaussian GHZ and cluster states of multiple mechanical oscillators by pulsed cavity optomechanics. In our scheme, each optomechanical cavity is driven by a blue-detuned pulse to establish quantum steerable correlations between the cavity output field and the mechanical oscillator, and the cavity outputs are combined at a beam-splitter array with given transmissivity and reflectivity for each beam splitter. We show that by harnessing the light-mechanical steerable correlations, the mechanical GHZ and cluster states can be realized via homodyne detection on the amplitude and phase quadratures of the output fields from the beam-splitter array. These achieved mechanical entangled states can be viewed as the output states of an effective mechanical beam-splitter array with the mechanical inputs prepared in squeezed states with the light-mechanical steering. The effects of detection efficiency and thermal noise on the achieved mechanical states are investigated. The present scheme does not require externally injected squeezing and it can also be applicable to other systems such as light-atomic-ensemble interface, apart from optomechanical systems.

Figure 1

A generic dispersive optomechanical system driven by blue-detuned laser pulses.

Figure 2

(a) The schematic plot of the generation of the mechanical GHZ state, in which the output fields from $N$ blue-detuned-pulse driven optomechanical cavities are combined at a BS array, with given transmissivity and reflectivity of each BS as shown. The cavity output field ${\widehat{c}}_1$ from the BS array is under homodyne detection on the phase quadrature and the other output fields ${\widehat{c}}_{j^{\prime }}(j^{\prime }=2,3,...,N)$ are homodyne detected on the amplitude quadratures, and the detection results $I_{x_{j^{\prime }}}$ and $I_{p_1}$ are fed back to displace the mechanical modes with appropriate gains. (b) The combination of the cavity outputs and the detection strategy in (a) can lead to an effective BS scheme for generating the mechanical GHZ state of the mechanical modes ${\widehat{b}}_j$, with the squeezed input states of the modes ${\widehat{d}}_j$ which are achieved via the measurements with the optomechanical steering in (a).

Figure 3

The dependence of the measure $V_{N,\min }^G$ verifying the GHZ correlations for $N=4$ on the initial thermal phonon number ${\overline{n}}_0$, with the detection efficiency $\eta =1.0$ (solid line), $\eta =0.9$ (dashed line), and $\eta =0.7$ (dotted line). The thin and bold lines represent the squeezing parameters $r=1.0$ and $r=1.5$, respectively.

Figure 4

(a) Quadripartite linear cluster state. (b) The schematic plot of the generation of the mechanical linear cluster state, in which the output fields ${\widehat{c}}_1$ and ${\widehat{c}}_2$ are under homodyne detection on the phase quadratures and the other two output fields ${\widehat{c}}_3$ and ${\widehat{c}}_4$ are homodyne detected on the amplitude quadratures, and the detection results are fed back to displace the mechanical modes via classical channels. (c) The combination of the cavity output fields and the detection leads to an effective mechanical BS transformation to generate the mechanical linear cluster state achieved in (b). Here, $\pm i$ denotes $\pm \frac{\pi }{2}$ rotation in phase space of the corresponding cavity output modes (similarly hereafter).

Figure 5

The dependence of the measures $V_{12,min}^{cl}$ and $V_{23,min}^{cl}$ verifying the linear cluster state correlations on the initial thermal phonon number ${\overline{n}}_0$ for the squeezing parameter $r=1.5$ and different detection efficiencies $\eta$.

Figure 6

(a) Quadripartite T-shape cluster state. (b) The schematic plot of the generation of the mechanical T-shape cluster state. The cavity output field ${\widehat{c}}_2$ from the BS array is under homodyne detection on the amplitude quadrature and the other three output fields are detected on the phase quadratures. (c) The effective mechanical BS transformation for generating the mechanical T-shape cluster state achieved in (b).

Figure 7

(a) Quadripartite square cluster state. (b) The schematic plot of the generation of the mechanical square cluster state. The BS array output fields ${\widehat{c}}_1$ and ${\widehat{c}}_2$ are under homodyne detection on the phase quadrature and the other two output fields ${\widehat{c}}_3$ and ${\widehat{c}}_3$ are detected on the amplitude quadratures. (c) The effective BS transformation for generating the mechanical square cluster state achieved in (b). Here the sign $-1$ denote $\pi$ rotation in phase space of the corresponding cavity output mode.