Abstract

In recent continuous-variable (CV) multipartite entanglement researches, the number of fully inseparable light modes has been increased dramatically by the introduction of a multiplexing scheme in either the time domain or the frequency domain. In this paper, we propose a scheme that a large-scale (≥ 20) CV dual-rail cluster entangled state is established based on a spatial mode comb in a self-imaging optical parametric oscillator, which is pumped by two spatial Laguerre-Gaussian modes with different polarization and identical frequency. A sufficient condition of full inseparability for a CV dual-rail cluster entangled state is used to evaluate the degree of quantum entanglement. It is shown that entanglement exists over a wide range of analyzing frequency and pump parameter. We have found a new scheme that uses the optical parametric cavity to generate a large-scale entanglement based on optical spatial mode comb. The presented system will be hopefully as a practical entangled source for quantum information.

© 2017 Optical Society of America

1. Introduction

Quantum entanglement is the most important optical resource for quantum computation [1], quantum dense coding [2] and quantum teleportation [3], which represent the foundations for construction of a quantum information network. Therefore, research on the multipartite quantum entanglement has been developed in these years. Some of generation schemes of multipartite entanglement are based on the coincident nonlinearities [4,5]. As a special type of multipartite entanglement, the cluster states has been introduced by Raussendorf via an Ising Hamiltonian [6], which are less likely to be destroyed by local operations [7]. Additionally, the cluster states represent an effective way to realize large-scale entanglement. These states could not only speed up computation using quantum algorithms, but could also be used as a medium for quantum information transfer in quantum communication protocols. Therefore, research into the generation of cluster states has become one of the most important research fields today.

To realize usable quantum computation and quantum information processes, large-scale entanglement, i.e., multipartite entanglement between numerous subsystems, has attracted considerable focus for operation in either the frequency domain or the time domain [8, 9]. Large-scale entanglement is an interesting topic that lies at the forefront of the current research and research into such systems has begun in various laboratories. There are two common approaches that can be used to generate a large-scale cluster entangled state: one is use of a quantum optical frequency comb and the other is use of an optical spatial mode comb. In 2011, 15 quadripartite entangled cluster states were generated simultaneously over 60 consecutive Q modes in an optical frequency comb [10]. In another experiment, a cluster state in a quantum optical frequency comb with more than 60 modes that were entangled and simultaneously available was achieved [11]. Experimental demonstrations of continuous-variable (CV) cluster states have included 10,000 time-multiplexed sequentially entangled modes [12]. In 2016, one-million-mode continuous-variable cluster states by unlimited time-domain multiplexing have been generated [13]. In addition to this experimental research, we proposed a theoretical scheme to produce a multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier when operating below threshold [14] and obtained a low-frequency signal beyond the quantum limit in a nondegenerate optical parametric amplifier via frequency-shift detection using frequency combs [15]. In 2014, a CV dual-rail cluster state over an optical spatial mode comb was generated in four-wave-mixing process [16]. In 2016, a scheme for generating CV spatial cluster entangled states based on optical mode combs via a large-Fresnel-number DOPO was proposed by us [17]. Other than frequency comb, spatial mode comb is a different new method for generating cluster entanglement states. Spatial freedom of light is an effective approach to scale the number of entangled states [18], which can bring new extensions and improvements. First, small interval of frequency comb depends on FSR of optical cavity, therefore it is difficult to separate them spatially in experiments. However, spatial separation of spatial modes is relative simple and efficient, by using spatial modulators. On the other hand, spatial modes are easier to detect by using Multi-quadrant detectors, CCD, etc. While the detection of the frequency comb optical field needs preparation of the local light fields with accurate frequency lines and more measurement times. In addition, such kind of spatial multipartite entanglement will be useful for future spatial multichannel quantum information application and quantum image transfer.

In this paper, we propose a new scheme to generate a large-scale CV dual-rail cluster entangled state in a specially designed self-imaging optical parametric oscillator (OPO), which can multiply the number of entangled modes. The rest of this paper is arranged as follows. In section II, our theoretical model for generating a large-scale spatial CV dual-rail cluster entangled state of Laguerre-Gaussian modes is introduced briefly and the evolution equations for the spatial modes and quadrature fluctuation are deduced. Then, the boundary conditions of the optical cavity are used to calculate the amplitude and phase quadratures of these spatial modes. In section III, the entanglement criterion that was proposed by van Loock and Furusawa for full inseparability of the optical fields [19] is used to estimate whether there is any entanglement among the CV dual-rail cluster states. Finally, a brief summary of the work is presented in section IV.

2. Theoretical model and derivation of equations

A large-scale CV dual-rail cluster state of the Laguerre-Gaussian modes is generated using a self-imaging OPO operating below threshold (σ < 1), which is pumped using two spatial Laguerre-Gaussian modes with the same frequency 2ω0 and different polarizations from two-ports respectively. As is shown in Fig. 1, two periodically polarized type-zero phase-matching nonlinear crystals represented by χ(2) are placed within a four-mirror ring cavity. The quasi-phase-matching crystal between flat mirror 2 and 3 (1 and 4) corresponds to zzz (yyy) parametric down-conversion (PDC), where the first letter denotes polarization of pump mode and the other two letters denote polarizations of down-converted modes. The pump fields are denoted by lg1pz, lg1py, which can generate spatial Laguerre-Gaussian modes lgsz, lgiz and lgsy, lgiythrough the PDC process respectively, where s, i = ±1, ±2, ±3 are the azimuthal mode indices, z and y represent the polarizations of the down-converted modes. Two pump fields with energy ħ ωp can be converted into two fields, signal and idler, with energies of ħ ωs and ħ ωi are degenerate in frequency, i.e., ωs = ωi = ω0. To generate a significant effect, the nonlinear interaction must satisfy the energy (ħ ωp=ħ ωs + ħ ωi, ωp = 2ω0), momentum (kp=ks+ki) and orbit angular momentum [20](lp ħ = ls ħ + li ħ) conservation conditions. Actually the efficiency of zzz PDC is bigger than yyy PDC for PPKTP crystals, so this four-mirror cavity of dual-port input and single-port output can be used to control waists and powers of two pump modes separately, and then the phase-matched efficiency of two crystals can be balanced. Such a cavity is a fully transverse degenerate one, which implies that all pump and down-converted modes can resonate simultaneously. From a geometrical point of view, an optical cavity is self-imaging when an arbitrary ray retraces its own path after a single round trip [21,22]. The self-imaging ring cavity requires three lenses of focal length fi, i = 1, 2, 3 and distances cij of image plane of lens i and object plane of lens j are given by c1,2=f1f2f3,c1,3=f1f3f2,c2,3=f2f3f1.

 figure: Fig. 1

Fig. 1 Schematic of experimental setup, the green lines represent pump modes lg1pz and lg1py, and the red lines represent down-converted spatial modes with different polarizations z and y, which are generated by the pump fields lg1pz and lg1py. Two output mode combs are shown in the dashed box, only one part of the down-converted modes are given, and intensity profile of the down-converted modes are given below.

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A structural diagram of a large-scale CV dual-rail cluster state is shown in Fig. 2. All the EPR pairs concatenate into the spatial Laguerre-Gaussian mode sequence (...lg−4, lg3, lg−2, lg1, lg0, lg−1, lg2, lg−3, lg4...) shown in Fig. 2(a) that extends to the optical spatial mode comb. These spatial modes are connected by the curved arrows shown in Fig. 2(a), and comprise a large-scale CV dual-rail cluster state after passing through the single beam splitter, as shown in Fig. 2(b).

 figure: Fig. 2

Fig. 2 (a) Structural diagram: the EPR pairs that were generated in the DOPO, the z and y modes are denoted by the red solid lines and the dashed lines, respectively. The yellow curved arrows (top) connected the zzz EPR pairs that were generated by the pump lg1pz and the yellow curved arrows (bottom) connected the yyy EPR pairs that were generated by the pump lg1py; the vertical arrows denote the pump modes. (b) A large-scale CV dual-rail cluster entangled state: the initial EPR pairs generated by the OPO (top) turn, after passing through a single beam splitter (gray ellipses), into a CV dual-rail cluster state (bottom). Whose ±1/2 weight edges are color coded (contrary to the qubit case, weighted cluster CV states are still stabilizer states).

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Here, the interaction Hamiltonian is:

H^=i[a1iχmb^1pza^sa^i+s1iχmb^1pya^sa^i]+H.C.
where χm, (m = 1 − 5) represents the effective nonlinear coupling parameter for different order parametric process, b^1pz and b^1py denote the annihilation operators for the intra-cavity pump modes, a^s and a^i are the creation operators of the signal and idler modes, respectively.

Here, we only consider two pump modes and twelve down-converted modes for simplicity. In the ideal case with perfect phase matching and without any detuning, the Langevin equations can be expressed as follows:

τb^˙1(z)(t)=γp(z)b^1(z)(t)+ε1χ1a^0(z)(t)a^1(z)(t)χ2a^1(z)(t)a^2(z)(t)χ3a^2(z)(t)a^3(z)(t)+2γpb(z)b^1in(z)(t)+2γpc(z)c^b1(z)(t),τb^˙1(y)(t)=γp(y)b^1(y)(t)+ε1χ1a^0(y)(t)a^1(y)(t)χ2a^1(y)(t)a^2(y)(t)χ3a^2(y)(t)a^3(y)(t)+2γpb(y)b^1in(y)(t)+2γpc(y)c^b1(y)(t),τa^˙0(z)(t)=γ0(z)a^0(z)(t)+χ1b^1(z)(t)a^1(z)(t)2γb0(z)a^0in(z)(t)+2γc0(z)c^0(z)(t),τa^˙0(y)(t)=γ0(y)a^0(y)(t)+χ1b^1(y)(t)a^1(y)(t)2γb0(y)a^0in(y)(t)+2γc0(y)c^0(y)(t),τa^˙1(z)(t)=γ1(z)a^1(z)(t)+χ1b^1(z)(t)a^0(z)(t)2γb1(z)a^1in(z)(t)+2γc1(z)c^1(z)(t),τa^˙1(y)(t)=γ1(y)a^1(y)(t)+χ2b1(y)(t)a^2(y)(t)2γb1(z)a^1in(y)(t)+2γc1(y)c^1(y)(t),τa^˙1(z)(t)=γ1(z)a^1(z)(t)+χ2b1(z)(t)a^2(z)(t)2γb1(z)a^1in(z)(t)+2γc1(z)c^1(z)(t),τa^˙1(z)(t)=γ1(y)a^1(y)(t)+χ1b1(y)(t)a^0(y)(t)2γb1(y)a^1in(y)(t)+2γc1(y)c^1(y)(t),τa^˙1(y)(t)=γ1(y)a^1(y)(t)+χ1b1(y)(t)a^0(y)(t)2γb1(y)a^1in(y)(t)+2γc1(y)c^1(y)(t),τa^˙2(z)(t)=γ2(z)a^2(z)(t)+χ2b1(z)(t)a^1(z)(t)2γb2(z)a^2in(z)(t)+2γc2(z)c^2(z)(t),τa^˙2(y)(t)=γ2(y)a^2(y)(t)+χ3b1(y)(t)a^3(z)(t)2γb2(y)a^2in(z)(t)+2γc2(y)c^2(y)(t),τa^˙2(z)(t)=γ2(z)a^2(z)(t)+χ3b1(z)(t)a^3(z)(t)2γb2(z)a^2in(z)(t)+2γc2(z)c^2(z)(t),τa^˙2(y)(t)=γ2(y)a^2(y)(t)+χ2b1(y)(t)a^1(z)(t)2γb2(y)a^2in(z)(t)+2γc2(y)c^2(y)(t),τa^˙3(z)(t)=γ3(z)a^3(z)(t)+χ3b1(z)(t)a^2(z)(t)2γb3(z)a^3in(z)(t)+2γc3(z)c^3(z)(t),τa^˙3(y)(t)=γ3(y)a^3(y)(t)+χ3b1(y)(t)a^2(y)(t)+2γb3(y)a^3in(y)(t)+2γc3(y)c^3(z)(t).
where τ is the round-trip time of the optical field inside the DOPO, χ1, χ2 and χ3 are the effective nonlinear coupling parameters, b^1(z)(b^1(y)) and a^i(z)(a^i(y)) are the amplitude operators of the pump modes and the down-converted modes inside the cavity, respectively. ε1 and ε−1 represent the pump fields that enter the cavity, and will be described classically. b^iin(z) (b^iin(y)) and a^iin(z)(a^iin(y)) denote the input amplitude operators of the pump modes and the down-converted modes, respectively. c^bi(z)(c^bi(y)) and c^i(z)(c^i(y)) are the excess vacuum noise operators of the pump modes and the down-converted modes, respectively.

To simplify the calculations, suppose that the two pump fields ε1 and ε−1 are identical, and then the losses of the pump modes are defined as γp = γpb + γpc, where γpb, γpc correspond to the output losses and the intra-cavity losses for the pump modes. The output coupling losses and the intra-cavity losses for the down-converted modes are the same, and are represented by γb0(z)=γb0(y)=γb±1(z)=γb±1(y)=γb±2(z)=γb±2(y)=γb3(z)=γb3(z)=γb and γc0(z)=γc0(y)=γc±1(z)=γc±1(y)=γc±2(z)=γc±2(y)=γc3(z)=γc3(z)=γc, where the total loss is γ = γb + γc, and thus γ0(z)=γ0(y)=γ±1(z)=γ±1(y)=γ±2(z)=γ±3(y)=γ3(z)=γ3(y)=γ.

The nonlinear coupling parameter χm is proportional to the overlap integral between the down-converted modes and the pump modes in the transverse plane, i.e., χm = Γp,i,s χ(2). The overlap integral is then defined as Γp,s,i=up(r)us(r)ui(r)dr [23], Here, u(r) represent the expression of the Laguerre-Gaussian modes, u (r) can be simplified in the condition of p=0, the orbit angular momentum conservation, perfect phase matching and the extremely small gouy phase. We use us(r), ui(r) as the notation for the fundamental signal and idler mode basis whose first mode has a waist of ω0, and up(r) for the second harmonic pump mode basis whose first mode has a waist of ω0/2. The overlap coefficients are given by Table 1, the nonlinear coupling parameters are then found to be: χ1 = χ(2), χ2 = 0.707χ(2), and χ3 = 0.433χ(2).

Tables Icon

Table 1. The overlap integrals and normalizations of the down-converted modes and pump modes

By linearization of the operators, b^1(z)=β1(z)+δb^1(z), b^1(y)=β1(y)+δb^1(y), a^i(z)=α^i(z)+δa^i(z), a^i(y)=αi(y)+δa^i(y), a^iin(z)=δa^iin(z) and a^iin(y)=δa^iin(y), (i = 0, ±1, ±2, ±3...), we can obtain the steady-state equations and quantum fluctuation equations for Eq. (2). By solving the steady-state equations, the oscillation threshold εth and the pump parameter σ are expressed as εth = γγp1 and σ = ε/εth, respectively. The steady-state solution is given by β±1(z/y)=ε/γ and α0(z)=α0(y)=α±1(z)=α±1(y)=α±2(z)=α±2(y)=α3(z)=α3(y)=0. The quantum fluctuation equations perform a Fourier transformation, we can then obtain the fluctuation dynamics equations. By applying the definitions of the amplitude and phase quadratures, i.e., X̂ = â+â and Ŷ = (ââ)/i, the amplitude quadratures of the down-converted spatial modes can then be expressed as follows:

Mx(δQ^0(z)(ω)δQ^1(z)(ω)δQ^1(z)(ω)δQ^2(z)(ω)δQ^2(z)(ω)δQ^3(z)(ω))=2γb(δQ^0in(z)(ω)δQ^1in(z)(ω)δQ^1in(z)(ω)δQ^2in(z)(ω)δQ^2in(z)(ω)δQ^3in(z)(ω))+2γc(δQ^c0(z)(ω)δQ^c1(z)(ω)δQ^c1(z)(ω)δQ^c2(z)(ω)δQ^c2(y)(ω)δQ^c3(y)(ω)),My(δQ^0(y)(ω)δQ^1(y)(ω)δQ^1(y)(ω)δQ^2(y)(ω)δQ^2(y)(ω)δQ^3(y)(ω))=2γb(δQ^0in(y)(ω)δQ^1in(y)(ω)δQ^1in(y)(ω)δQ^2in(y)(ω)δQ^2in(y)(ω)δQ^3in(y)(ω))+2γc(δQ^c0(y)(ω)δQ^c1(y)(ω)δQ^c1(y)(ω)δQ^c2(y)(ω)δQ^c2(y)(ω)δQ^c3(y)(ω)).
Here,
Mx=(iωτ+γχ1β1(z)0000χ1β1(z)iωτ+γ000000iωτ+γχ2β2(z)000χ2β2(z)iωτ+γ000000iωτ+γχ3β3(z)0000χ3β3(z)iωτ+γ),My=(iωτ+γ0χ1β1(y)0000iωτ+γ00χ2β2(y)0χ1β1(y)0iωτ+γ000000iωτ+γ0χ3β3(y)0χ2β2(y)00iωτ+γ0000χ3β3(y)0iωτ+γ).

The matrix forms of the phase quadratures can be obtained in a similar manner. Using the boundary conditions [24] of δQ^iout=2γbiδQ^iδQ^iin and δP^iout=2γbiδP^iδP^iin, (i = 0, ±1, ±2, ±3...), we can then calculate the amplitude and phase quadratures for the different polarization modes. The two different polarization modes z and y turn into a CV dual-rail cluster entangled state after passing through a single 50/50 beam splitter with a relative phase of π/2 [25]. The resulting output modes (after the beam splitter) are then labeled z′ and y′, respectively. The beam splitter transformation is given by a^(z)=12(a^(z)+ia^(y)) and a^(y)=12(a^(z)+ia^(y)), and thus the amplitude and phase quadratures of the output modes are as follows: Q^i(z)=12(Q^i(z)P^i(y)), P^i(z)=12(P^i(z)+Q^i(y)), Q^i(y)=12(Q^i(z)P^i(y)), P^i(y)=12(P^i(z)Q^i(y)). Based on Eq. (3), the amplitude and phase quadratures of the modes lg0(z),lg0(y),lg±1(z),lg±1(y)... can now be obtained.

3. Sufficient conditions for overall inseparability

The quantum entanglements of the CV dual-rail cluster states are characterized via the correlations of their amplitude and phase quadratures. Based on the entanglement criterion proposed by van Loock and Furusawa for the inseparability [13, 19], we consider the criterion for all possible separable bipartitions in our set of output modes and obtain sufficient conditions for overall inseparability:

Qs.i=((Q^sz+Q^sy)(Q^iz+Q^iy))2<1/2,
Ps.i=((P^sz+P^sy)(P^iz+P^iy))2<1/2,
Qs.i=((Q^izQ^iy)+(Q^szQ^sy))2<1/2,
Ps.i=((P^iz+P^iy)(P^sz+P^sy))2<1/2.
s + i = 1 corresponds to Eqs. (5) and (6), whereas s + i = −1 corresponds to Eqs. (7) and (8), and 1/2 is the shot-noise limit. The quantum entanglements of the spatial CV dual-rail cluster states are vividly illustrated in Fig. 3. The quantum entanglement of Eqs. (5) to (8) can be measured via two-tone balanced homodyne detection using spatially tailored local oscillator (LO) modes at ω0, where the quadrature combinations can be fed into a spectrum analyzer that is used to display the noise power in experiment. Calculations show that the results of Eqs. (5) and (6) [and those of Eqs. (7) and (8)] are identical.

 figure: Fig. 3

Fig. 3 Visualization of quantum entanglements of Eqs. (5) to (8) in the CV dual-rail cluster states shown in Fig. 2(b).

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Figure 4 shows the quantum entanglement of Qi,s (Pi,s) (i, s = 0, 1, i, s = 0, −1, i, s = −1, 2, i, s = 1, −2, i, s = −2, 3, i, s = 2, −3, i, s = −3, 4, i, s = 3, −4, i, s = −4, 5, i, s = 4, −5) versus the normalized analyzing frequency Ω = ωτ/γ. The entanglement decreases gradually with the normalized analyzing frequency. It satisfies the sufficient conditions for overall inseparability over a wide range of normalized analyzing frequency when we select the proper pump parameter. As is shown, the trends of all curves are similar.

 figure: Fig. 4

Fig. 4 Quantum entanglement versus normalized frequency Ω = ωτ/γ with γp = 0.025, γ = 0.02, γb = 0.018, γc = 0.002, χ1 = χ(2), χ2 = 0.707χ(2), χ3 = 0.433χ(2), χ4 = 0.250χ(2), χ5 = 0.140χ(2) and σ = 0.8. (I): Q1,0 = Q0,−1 = P1,0 = P0,−1, blue line; (II): Q2,−1 = Q1,−2 = P2,−1 = P1,−2, red line; (III): Q3,−2 = Q2,−3 = P3,−2 = P2,−3, purple line; (IV): Q4,−3 = Q3,−4 = P4,−3 = P3,−4, green line; (V): Q5,−4 = Q4,−5 = P5,−4 = P4,−5, black line.

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Figure 5 shows the quantum entanglement of Qi,s (Pi,s) (i, s = 0, 1, i, s = 0, −1, i, s = −1, 2, i, s = 1, −2, i, s = −2, 3, i, s = 2, −3, i, s = −3, 4, i, s = 3, −4, i, s = −4, 5, i, s = 4, −5) versus the pump parameter σ = ε/εth (which has been normalized with respect to the pump threshold). The entanglements grow with increasing pump parameter and the largest entanglement can be obtained near the threshold. This satisfies the sufficient conditions for overall inseparability when the DOPO is operating below threshold (σ < 1).

 figure: Fig. 5

Fig. 5 Quantum entanglement versus pump parameter σ = ε/εth with γp = 0.025, γ = 0.02, γb = 0.018, γc = 0.002, χ1 = χ(2), χ2 = 0.707χ(2), χ3 = 0.433χ(2), χ4 = 0.250χ(2), χ5 = 0.140χ(2) and Ω = 0.2. (I): Q1,0 = Q0,−1 = P1,0 = P0,−1, blue line; (II): Q2,−1 = Q1,−2 = P2,−1 = P1,−2, red line; (III): Q3,−2 = Q2,−3 = P3,−2 = P2,−3, purple line; (IV): Q4,−3 = Q3,−4 = P4,−3 = P3,−4, green line; (V): Q5,−4 = Q4,−5 = P5,−4 = P4,−5, black line.

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As clearly shown in Figs. 4 and 5, the first-order quantum entanglement satisfy the condition Q0,1 = Q0,−1 because the spatial mode EPR pairs lg0(z)lg1(z) and lg0(y)lg1(y) correspond to the same nonlinear coupling parameter χ1, and similarly, Q−1,2 = Q1,−2, Q−2,3 = Q2,−3, Q−3,4 = Q3,−4, Q−4,5 = Q4,−5. Additionally, for the relation of nonlinear coupling parameters: χ1 > χ2 > χ3 > χ4 > χ5, with the nonlinear coupling parameter decreases gradually, the quantum entanglement decreases. Therefore the quantum entanglements of Q0,1 and Q0,−1 are the largest, and the quantum entanglements of Q−4,5 and Q4,−5 are the smallest. We confirm that at least 20 entangled modes can be generated by considering periodicity of entanglement. And the squeezing degree of the noise correlation between the different modes is expected to be detected in experiments, and the squeezing at zero frequency can reach 10.1dB, 9.1dB, 6.2dB, 3.7dB, 2.2dB, respectively.

It should be noted that the weights here are 1/2 and −1/2, because all modes passing through a single 50/50 beam splitter. Under this condition, the entanglement criterion is satisfied indeed, even though it is not optimum.

4. Conclusions

In this work, we proposed a new scheme to generate a large-scale CV dual-rail cluster state of Laguerre-Gaussian modes based on a spatial mode comb in a self-imaging OPO when operating below threshold; the scheme can be distinguished from other schemes of the frequency domain and the time domain. Their constant entanglement versus the normalized analyzing frequency and the pump parameter can be observed under sufficient conditions for full inseparability. The dual-rail cluster states of the 20 fully inseparable light modes can be generated using strong pump power, a high χ(2) and a higher Γp,i,s. If we can achieve the following conditions, e.g., perfect mode-matching, alignment of the interactional modes and special transverse pump structure, a larger parametric interaction and a larger-scale entanglement (i.e., more than 20 spatial modes) will be realized. This scheme can pave a new way to generate large-scale CV cluster entangled states using spatial mode combs, which can then be applied extensively in the fields of the quantum computing, and the relevance of continuous variables for cluster states is also important to universal quantum computing because of potential for scalability and a fault tolerance [11,26].

Funding

National Natural Science Foundation of China (NSFC) (11504218, 61108003, 91536222, 61405108); Natural Science Foundation of Shanxi Province, China (2013021005-2); National Key Research and Development Plan (2016YFA0301404).

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11. M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014). [CrossRef]   [PubMed]  

12. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013). [CrossRef]  

13. J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

14. R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013). [CrossRef]  

15. R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015). [CrossRef]   [PubMed]  

16. R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014). [CrossRef]  

17. R. G. Yang, J. J. Wang, J. Zhang, K. Liu, and J. R. Gao, “Generation of continuous-variable spatial cluster entangled states in optical mode comb,” J. Opt. Soc. Am. B 33(12), 2424–2429 (2016). [CrossRef]  

18. K. Liu, J. Guo, C. X. Cai, J. X. Zhang, and J. R. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178 (2016). [CrossRef]   [PubMed]  

19. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003). [CrossRef]  

20. M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009). [CrossRef]   [PubMed]  

21. L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009). [CrossRef]  

22. J. A. Arnaud, “Degenerate Optical Cavities,” Appl. Opt. 8, 189–196 (1969). [CrossRef]   [PubMed]  

23. M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006). [CrossRef]  

24. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984). [CrossRef]  

25. G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009). [CrossRef]  

26. N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014). [CrossRef]   [PubMed]  

References

  • View by:

  1. M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
    [Crossref]
  2. J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
    [Crossref] [PubMed]
  3. T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
    [Crossref]
  4. O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
    [Crossref]
  5. R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
    [Crossref]
  6. H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
    [Crossref] [PubMed]
  7. P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
    [Crossref]
  8. R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).
  9. R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
    [Crossref]
  10. M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
    [Crossref] [PubMed]
  11. M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
    [Crossref] [PubMed]
  12. S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
    [Crossref]
  13. J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).
  14. R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
    [Crossref]
  15. R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015).
    [Crossref] [PubMed]
  16. R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
    [Crossref]
  17. R. G. Yang, J. J. Wang, J. Zhang, K. Liu, and J. R. Gao, “Generation of continuous-variable spatial cluster entangled states in optical mode comb,” J. Opt. Soc. Am. B 33(12), 2424–2429 (2016).
    [Crossref]
  18. K. Liu, J. Guo, C. X. Cai, J. X. Zhang, and J. R. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178 (2016).
    [Crossref] [PubMed]
  19. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
    [Crossref]
  20. M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
    [Crossref] [PubMed]
  21. L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
    [Crossref]
  22. J. A. Arnaud, “Degenerate Optical Cavities,” Appl. Opt. 8, 189–196 (1969).
    [Crossref] [PubMed]
  23. M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
    [Crossref]
  24. M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
    [Crossref]
  25. G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
    [Crossref]
  26. N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
    [Crossref] [PubMed]

2016 (3)

2015 (1)

2014 (3)

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref] [PubMed]

2013 (2)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

2012 (1)

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

2011 (1)

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

2009 (4)

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

2006 (2)

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

2005 (1)

R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
[Crossref]

2004 (1)

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

2003 (3)

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

2002 (1)

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

2001 (1)

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

1984 (1)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

1969 (1)

Andersen, U. L.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

Armstrong, S. C.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Arnaud, J. A.

Bachor, H-A.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Bartlett, S. D.

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

Bloomer, R.

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Briegel, H. J.

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Browne, D. E.

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

Cai, C. X.

Cao, Z. L.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Chalopin, B.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Chen, M.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

Collett, M. J.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Delaubert, V.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Dong, P.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Dong, R. F.

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

Fabre, C.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Feng, S.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Furusawa, A.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

Gao, J. R.

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Gu, M.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Guo, J.

Harb, C. C.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

ichi Yoshikawa, J.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Jennings, G.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Jing, J.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Jing, J. T.

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

Johnson, T. J.

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

Kaji, T.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Lam, P. K.

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Lassen, M.

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Leuchs, G.

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

Liu, K.

Lopez, L.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Maître, A.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Makino, K.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

Menicucci, N. C.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref] [PubMed]

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Miwa, Y.

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Peng, K.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Pfister, O.

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
[Crossref]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Pooser, R.

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

R. Pooser and O. Pfister, “Observation of triply coincident nonlinearities in periodically poled KTiOPO4,” Opt. Lett. 19, 2635–2637 (2005).
[Crossref]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Pysher, M.

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Ralph, T. C.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Raussendorf, R.

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Riviére de la Souchére, A.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

Sanders, B. C.

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

Shahrokhshahi, R.

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

Shiozawa, Y.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

Sornphiphatphong, C.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Suzuki, S.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Sych, D.

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

Treps, N.

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

Ukai, R.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

van Loock, P.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

Wang, J. J.

Weedbrook, C.

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

Xie, C.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Xie, D.

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

Xue, Z. Y.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Yan, Y.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Yang, M.

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

Yang, R. G.

Yokoyama, S.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Yonezawa, H.

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

Yoshikawa, J.

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

Zhai, S. Q.

R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015).
[Crossref] [PubMed]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

Zhai, Z. H.

Zhang, J.

R. G. Yang, J. J. Wang, J. Zhang, K. Liu, and J. R. Gao, “Generation of continuous-variable spatial cluster entangled states in optical mode comb,” J. Opt. Soc. Am. B 33(12), 2424–2429 (2016).
[Crossref]

R. G. Yang, J. Zhang, Z. H. Zhai, S. Q. Zhai, K. Liu, and J. R. Gao, “Scheme for efficient extraction of low-frequency signal beyond the quantum limit by frequency-shift detection,” Opt. Express 23, 21323–21333 (2015).
[Crossref] [PubMed]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Zhang, J. X.

K. Liu, J. Guo, C. X. Cai, J. X. Zhang, and J. R. Gao, “Direct generation of spatial quadripartite continuous variable entanglement in an optical parametric oscillator,” Opt. Lett. 41, 5178 (2016).
[Crossref] [PubMed]

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

Zhao, F.

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Photonics (1)

J. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,” Appl. Photonics 1(6), 777 (2016).

J. Eur. Opt. Soc.-Rapid (1)

M. Lassen, V. Delaubert, C. C. Harb, P. K. Lam, N. Treps, and H-A. Bachor, “Generation of Squeezing in Higher Order Hermite-Gaussian Modes with an Optical Parametric Amplifier,” J. Eur. Opt. Soc.-Rapid 1, 06003 (2006).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Opt. Soc.Am. B (1)

R. G. Yang, J. Zhang, S. Q. Zhai, K. Liu, J. X. Zhang, and J. R. Gao, “Generating multiplexed entanglement frequency comb in a nondegenerate optical parametric amplifier,” J. Opt. Soc.Am. B 30(2), 314–318 (2013).
[Crossref]

Nat. Photonics (1)

S. Yokoyama, R. Ukai, S. C. Armstrong, C. Sornphiphatphong, T. Kaji, S. Suzuki, J. ichi Yoshikawa, H. Yonezawa, N. C. Menicucci, and A. Furusawa, “Ultra-large-scale continuous-variable cluster states multiplexed in the time domain,” Nat. Photonics 7, 982–986 (2013).
[Crossref]

New J. Phys. (1)

G. Leuchs, R. F. Dong, and D. Sych, “Triplet-like correlation symmetry of continuous variable entangled states,” New J. Phys. 11, 113040 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (9)

P. Dong, Z. Y. Xue, M. Yang, and Z. L. Cao, “Generation of cluster states,” Phys. Rev. A 73, 033818 (2006).
[Crossref]

M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, “Quantum computing with continuous-variable clusters,” Phys. Rev. A 79, 062318 (2009).
[Crossref]

T. J. Johnson, S. D. Bartlett, and B. C. Sanders, “Continuous-variable quantum teleportation of entanglement,” Phys. Rev. A 66, 042326 (2002).
[Crossref]

O. Pfister, S. Feng, G. Jennings, R. Pooser, and D. Xie, “Multipartite continuous-variable entanglement from concurrent nonlinearities,” Phys. Rev. A 70, 020302(R) (2004).
[Crossref]

P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67, 052315 (2003).
[Crossref]

R. Pooser and J. T. Jing, “Continuous-variable cluster-state generation over the optical spatial mode comb,” Phys. Rev. A 90, 043841 (2014).
[Crossref]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

L. Lopez, B. Chalopin, A. Riviére de la Souchére, C. Fabre, A. Maître, and N. Treps, “Multimode quantum properties of a self-imaging optical parametric oscillator: Squeezed vacuum and Einstein-Podolsky-Rosen-beams generation,” Phys. Rev. A 80, 043816 (2009).
[Crossref]

R. Raussendorf, D. E. Browne, and H. J. Briegel, “Measurement-based quantum computation on cluster states,” Phys. Rev. A 68, 022312 (2003).
[Crossref]

Phys. Rev. Lett. (6)

M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
[Crossref] [PubMed]

M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
[Crossref] [PubMed]

N. C. Menicucci, “Fault-tolerant measurement-based quantum computing with continuous-variable cluster states,” Phys. Rev. Lett. 112, 120504 (2014).
[Crossref] [PubMed]

M. Lassen, G. Leuchs, and U. L. Andersen, “Continuous Variable Entanglement and Squeezing of Orbital Angular Momentum States,” Phys. Rev. Lett. 102, 163602 (2009).
[Crossref] [PubMed]

J. Jing, J. Zhang, Y. Yan, F. Zhao, C. Xie, and K. Peng, “Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables,” Phys. Rev. Lett. 90, 167903 (2003).
[Crossref] [PubMed]

H. J. Briegel and R. Raussendorf, “Persistent Entanglement in Arrays of Interacting Particles,” Phys. Rev. Lett. 86, 910–913 (2001).
[Crossref] [PubMed]

Quantum Inf. Comput. (1)

R. Shahrokhshahi and O. Pfister, “Large-scale multipartite entanglement in the quantum optical frequency comb of a depleted-pump optical parametric oscillator,” Quantum Inf. Comput. 12, 953–969 (2012).

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Figures (5)

Fig. 1
Fig. 1 Schematic of experimental setup, the green lines represent pump modes lg 1 p z and lg 1 p y , and the red lines represent down-converted spatial modes with different polarizations z and y, which are generated by the pump fields lg 1 p z and lg 1 p y . Two output mode combs are shown in the dashed box, only one part of the down-converted modes are given, and intensity profile of the down-converted modes are given below.
Fig. 2
Fig. 2 (a) Structural diagram: the EPR pairs that were generated in the DOPO, the z and y modes are denoted by the red solid lines and the dashed lines, respectively. The yellow curved arrows (top) connected the zzz EPR pairs that were generated by the pump lg 1 p z and the yellow curved arrows (bottom) connected the yyy EPR pairs that were generated by the pump lg 1 p y ; the vertical arrows denote the pump modes. (b) A large-scale CV dual-rail cluster entangled state: the initial EPR pairs generated by the OPO (top) turn, after passing through a single beam splitter (gray ellipses), into a CV dual-rail cluster state (bottom). Whose ±1/2 weight edges are color coded (contrary to the qubit case, weighted cluster CV states are still stabilizer states).
Fig. 3
Fig. 3 Visualization of quantum entanglements of Eqs. (5) to (8) in the CV dual-rail cluster states shown in Fig. 2(b).
Fig. 4
Fig. 4 Quantum entanglement versus normalized frequency Ω = ωτ/γ with γp = 0.025, γ = 0.02, γb = 0.018, γc = 0.002, χ1 = χ(2), χ2 = 0.707χ(2), χ3 = 0.433χ(2), χ4 = 0.250χ(2), χ5 = 0.140χ(2) and σ = 0.8. (I): Q1,0 = Q0,−1 = P1,0 = P0,−1, blue line; (II): Q2,−1 = Q1,−2 = P2,−1 = P1,−2, red line; (III): Q3,−2 = Q2,−3 = P3,−2 = P2,−3, purple line; (IV): Q4,−3 = Q3,−4 = P4,−3 = P3,−4, green line; (V): Q5,−4 = Q4,−5 = P5,−4 = P4,−5, black line.
Fig. 5
Fig. 5 Quantum entanglement versus pump parameter σ = ε/εth with γp = 0.025, γ = 0.02, γb = 0.018, γc = 0.002, χ1 = χ(2), χ2 = 0.707χ(2), χ3 = 0.433χ(2), χ4 = 0.250χ(2), χ5 = 0.140χ(2) and Ω = 0.2. (I): Q1,0 = Q0,−1 = P1,0 = P0,−1, blue line; (II): Q2,−1 = Q1,−2 = P2,−1 = P1,−2, red line; (III): Q3,−2 = Q2,−3 = P3,−2 = P2,−3, purple line; (IV): Q4,−3 = Q3,−4 = P4,−3 = P3,−4, green line; (V): Q5,−4 = Q4,−5 = P5,−4 = P4,−5, black line.

Tables (1)

Tables Icon

Table 1 The overlap integrals and normalizations of the down-converted modes and pump modes

Equations (8)

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H ^ = i [ a 1 i χ m b ^ 1 p z a ^ s a ^ i + s 1 i χ m b ^ 1 p y a ^ s a ^ i ] + H . C .
τ b ^ ˙ 1 ( z ) ( t ) = γ p ( z ) b ^ 1 ( z ) ( t ) + ε 1 χ 1 a ^ 0 ( z ) ( t ) a ^ 1 ( z ) ( t ) χ 2 a ^ 1 ( z ) ( t ) a ^ 2 ( z ) ( t ) χ 3 a ^ 2 ( z ) ( t ) a ^ 3 ( z ) ( t ) + 2 γ p b ( z ) b ^ 1 i n ( z ) ( t ) + 2 γ p c ( z ) c ^ b 1 ( z ) ( t ) , τ b ^ ˙ 1 ( y ) ( t ) = γ p ( y ) b ^ 1 ( y ) ( t ) + ε 1 χ 1 a ^ 0 ( y ) ( t ) a ^ 1 ( y ) ( t ) χ 2 a ^ 1 ( y ) ( t ) a ^ 2 ( y ) ( t ) χ 3 a ^ 2 ( y ) ( t ) a ^ 3 ( y ) ( t ) + 2 γ p b ( y ) b ^ 1 i n ( y ) ( t ) + 2 γ p c ( y ) c ^ b 1 ( y ) ( t ) , τ a ^ ˙ 0 ( z ) ( t ) = γ 0 ( z ) a ^ 0 ( z ) ( t ) + χ 1 b ^ 1 ( z ) ( t ) a ^ 1 ( z ) ( t ) 2 γ b 0 ( z ) a ^ 0 i n ( z ) ( t ) + 2 γ c 0 ( z ) c ^ 0 ( z ) ( t ) , τ a ^ ˙ 0 ( y ) ( t ) = γ 0 ( y ) a ^ 0 ( y ) ( t ) + χ 1 b ^ 1 ( y ) ( t ) a ^ 1 ( y ) ( t ) 2 γ b 0 ( y ) a ^ 0 i n ( y ) ( t ) + 2 γ c 0 ( y ) c ^ 0 ( y ) ( t ) , τ a ^ ˙ 1 ( z ) ( t ) = γ 1 ( z ) a ^ 1 ( z ) ( t ) + χ 1 b ^ 1 ( z ) ( t ) a ^ 0 ( z ) ( t ) 2 γ b 1 ( z ) a ^ 1 i n ( z ) ( t ) + 2 γ c 1 ( z ) c ^ 1 ( z ) ( t ) , τ a ^ ˙ 1 ( y ) ( t ) = γ 1 ( y ) a ^ 1 ( y ) ( t ) + χ 2 b 1 ( y ) ( t ) a ^ 2 ( y ) ( t ) 2 γ b 1 ( z ) a ^ 1 i n ( y ) ( t ) + 2 γ c 1 ( y ) c ^ 1 ( y ) ( t ) , τ a ^ ˙ 1 ( z ) ( t ) = γ 1 ( z ) a ^ 1 ( z ) ( t ) + χ 2 b 1 ( z ) ( t ) a ^ 2 ( z ) ( t ) 2 γ b 1 ( z ) a ^ 1 i n ( z ) ( t ) + 2 γ c 1 ( z ) c ^ 1 ( z ) ( t ) , τ a ^ ˙ 1 ( z ) ( t ) = γ 1 ( y ) a ^ 1 ( y ) ( t ) + χ 1 b 1 ( y ) ( t ) a ^ 0 ( y ) ( t ) 2 γ b 1 ( y ) a ^ 1 i n ( y ) ( t ) + 2 γ c 1 ( y ) c ^ 1 ( y ) ( t ) , τ a ^ ˙ 1 ( y ) ( t ) = γ 1 ( y ) a ^ 1 ( y ) ( t ) + χ 1 b 1 ( y ) ( t ) a ^ 0 ( y ) ( t ) 2 γ b 1 ( y ) a ^ 1 i n ( y ) ( t ) + 2 γ c 1 ( y ) c ^ 1 ( y ) ( t ) , τ a ^ ˙ 2 ( z ) ( t ) = γ 2 ( z ) a ^ 2 ( z ) ( t ) + χ 2 b 1 ( z ) ( t ) a ^ 1 ( z ) ( t ) 2 γ b 2 ( z ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( z ) c ^ 2 ( z ) ( t ) , τ a ^ ˙ 2 ( y ) ( t ) = γ 2 ( y ) a ^ 2 ( y ) ( t ) + χ 3 b 1 ( y ) ( t ) a ^ 3 ( z ) ( t ) 2 γ b 2 ( y ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( y ) c ^ 2 ( y ) ( t ) , τ a ^ ˙ 2 ( z ) ( t ) = γ 2 ( z ) a ^ 2 ( z ) ( t ) + χ 3 b 1 ( z ) ( t ) a ^ 3 ( z ) ( t ) 2 γ b 2 ( z ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( z ) c ^ 2 ( z ) ( t ) , τ a ^ ˙ 2 ( y ) ( t ) = γ 2 ( y ) a ^ 2 ( y ) ( t ) + χ 2 b 1 ( y ) ( t ) a ^ 1 ( z ) ( t ) 2 γ b 2 ( y ) a ^ 2 i n ( z ) ( t ) + 2 γ c 2 ( y ) c ^ 2 ( y ) ( t ) , τ a ^ ˙ 3 ( z ) ( t ) = γ 3 ( z ) a ^ 3 ( z ) ( t ) + χ 3 b 1 ( z ) ( t ) a ^ 2 ( z ) ( t ) 2 γ b 3 ( z ) a ^ 3 i n ( z ) ( t ) + 2 γ c 3 ( z ) c ^ 3 ( z ) ( t ) , τ a ^ ˙ 3 ( y ) ( t ) = γ 3 ( y ) a ^ 3 ( y ) ( t ) + χ 3 b 1 ( y ) ( t ) a ^ 2 ( y ) ( t ) + 2 γ b 3 ( y ) a ^ 3 i n ( y ) ( t ) + 2 γ c 3 ( y ) c ^ 3 ( z ) ( t ) .
M x ( δ Q ^ 0 ( z ) ( ω ) δ Q ^ 1 ( z ) ( ω ) δ Q ^ 1 ( z ) ( ω ) δ Q ^ 2 ( z ) ( ω ) δ Q ^ 2 ( z ) ( ω ) δ Q ^ 3 ( z ) ( ω ) ) = 2 γ b ( δ Q ^ 0 i n ( z ) ( ω ) δ Q ^ 1 i n ( z ) ( ω ) δ Q ^ 1 i n ( z ) ( ω ) δ Q ^ 2 i n ( z ) ( ω ) δ Q ^ 2 i n ( z ) ( ω ) δ Q ^ 3 i n ( z ) ( ω ) ) + 2 γ c ( δ Q ^ c 0 ( z ) ( ω ) δ Q ^ c 1 ( z ) ( ω ) δ Q ^ c 1 ( z ) ( ω ) δ Q ^ c 2 ( z ) ( ω ) δ Q ^ c 2 ( y ) ( ω ) δ Q ^ c 3 ( y ) ( ω ) ) , M y ( δ Q ^ 0 ( y ) ( ω ) δ Q ^ 1 ( y ) ( ω ) δ Q ^ 1 ( y ) ( ω ) δ Q ^ 2 ( y ) ( ω ) δ Q ^ 2 ( y ) ( ω ) δ Q ^ 3 ( y ) ( ω ) ) = 2 γ b ( δ Q ^ 0 i n ( y ) ( ω ) δ Q ^ 1 i n ( y ) ( ω ) δ Q ^ 1 i n ( y ) ( ω ) δ Q ^ 2 i n ( y ) ( ω ) δ Q ^ 2 i n ( y ) ( ω ) δ Q ^ 3 i n ( y ) ( ω ) ) + 2 γ c ( δ Q ^ c 0 ( y ) ( ω ) δ Q ^ c 1 ( y ) ( ω ) δ Q ^ c 1 ( y ) ( ω ) δ Q ^ c 2 ( y ) ( ω ) δ Q ^ c 2 ( y ) ( ω ) δ Q ^ c 3 ( y ) ( ω ) ) .
M x = ( i ω τ + γ χ 1 β 1 ( z ) 0 0 0 0 χ 1 β 1 ( z ) i ω τ + γ 0 0 0 0 0 0 i ω τ + γ χ 2 β 2 ( z ) 0 0 0 χ 2 β 2 ( z ) i ω τ + γ 0 0 0 0 0 0 i ω τ + γ χ 3 β 3 ( z ) 0 0 0 0 χ 3 β 3 ( z ) i ω τ + γ ) , M y = ( i ω τ + γ 0 χ 1 β 1 ( y ) 0 0 0 0 i ω τ + γ 0 0 χ 2 β 2 ( y ) 0 χ 1 β 1 ( y ) 0 i ω τ + γ 0 0 0 0 0 0 i ω τ + γ 0 χ 3 β 3 ( y ) 0 χ 2 β 2 ( y ) 0 0 i ω τ + γ 0 0 0 0 χ 3 β 3 ( y ) 0 i ω τ + γ ) .
Q s . i = ( ( Q ^ s z + Q ^ s y ) ( Q ^ i z + Q ^ i y ) ) 2 < 1 / 2 ,
P s . i = ( ( P ^ s z + P ^ s y ) ( P ^ i z + P ^ i y ) ) 2 < 1 / 2 ,
Q s . i = ( ( Q ^ i z Q ^ i y ) + ( Q ^ s z Q ^ s y ) ) 2 < 1 / 2 ,
P s . i = ( ( P ^ i z + P ^ i y ) ( P ^ s z + P ^ s y ) ) 2 < 1 / 2 .

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