N-qubit W state of spatially separated single molecule magnets

Xin-You Lü; Pei-Jun Song; Ji-Bing Liu; Xiaoxue Yang

doi:10.1364/OE.17.014298

1. Introduction

Nowadays quantum entanglement, as a striking feature of quantum mechanics, has played a central role in the field of quantum information due to its potential applications in quantum teleportation, quantum cryptography, quantum computer, etc[1, 2, 3, 4, 5, 6, 7, 8, 9]. Along with the progress of multiparty quantum communication theory [10, 11, 12, 13], the multipartite entanglement attracts increasing attentions of physicists in recent years. Compared with bipartite entanglement, there usually exist many inequivalent classes for multipartite entangled states (e.g., GHZ state[14], W state[15, 16] and cluster state[17] etc.), which cannot be transformed into each other under local operations and classical communication (LOCC) protocols. Recently, a large number of theoretical and experimental schemes have been proposed for generating the multipartite GHZ state [18, 19, 20, 21, 22], W state [23, 24, 25, 26], Dicke states [27, 28, 29] and cluster state [30, 31, 32, 33, 34, 35] and so on [36, 37, 38, 39, 40]. For example, Zou et al. [20] proposed a scheme to prepare a four-particle GHZ state of distant atoms that are trapped separately into leaky cavities. Yamamoto et. al. [23] proposed a theoretical scheme for realizing the three-photon polarization entangled W state based on the spontaneous parametric down-conversion process. Xiao et al. [28] proposed a scheme for generating the N-partite Dicke state of largely detuned atoms, through detecting the leaky polarized photons from an optical cavity. They also show that their scheme has a very high success probability since the atomic spontaneous emission is strongly suppressed. More recently, Li et al. [35] realized a continuous-variable entangled cluster states of four distinct atomic ensembles in a high-finesse ring cavity based on the dispersive interaction between the atomic ensembles and the cavity mode. However, most of the present schemes for realizing multipartite entanglement employ the atoms or trapped ions as the entangled qubits. The atoms and trapped ions are gaseous medium, which exists some defects in the flexibility of device fabrication.

In recent years, the SMM, as a solid-state media with nanoscale size, has attracted much interests due to its quantum magnetic properties at macroscopic scales and potential applications in information storage, quantum computing etc. [41, 42, 43, 44, 45, 46]. For example, Misiorny et al. [43] proposed a scheme for realizing the magnetic switching process based on the spin reversal of SMM. Michael et al. realized the Grover’s algorithm in the SMM system and show that this system can be used to build dense and efficient memory devices based on the Grover algorithm. Far more than these, it also has been shown that the SMM will reveal many quantum characteristics analogous to atoms including separate level structure, quantum coherence effects when it is subjected to a dc magnetic field perpendicular to its easy anisotropy axis [47, 48, 49, 50, 51]. Based on this, the electromagnetically induced transparency [47], four-wave mixing [48], and propagation of microwave soliton [49] have been realized in SMM system. All above characters of SMM have present a feasible platform to realize many quantum information processes using solid-state qubits.

On the other hand, it should be pointed out that entanglement between spatially separated subsystems is very useful for distributed quantum computation [52]. The cavity-fiber-cavity system [53, 54, 55, 56, 57], which consists of distant optical cavities connected by fiber, is an effective system for realizing entanglement between spatially separated qubits. Based on this system, the two-dimensional [54] and three-dimensional [55] bipartite entangled states of spatially separated atoms have been theoretically proposed in recent years. However, until now, no related theoretical or experimental work has been carried out to realize the multipartite entangled state of distant SMMs in cavity-fiber-cavity system. This correlative research is significant for the progress of multi-node quantum entanglement network. Inspired by this point, in this paper, we proposed a schemes for generating N-qubit W state of spatially separated SMMs in a cavity-fiber-cavity system. In the present schemes, an assistant SMM is trapped into a central cavity and N entangled SMMs are individually trapped into the other N cavities, which connect with the cental cavity together via N fibers. The major advantages of our scheme are as follows. (1) The framework consisting of entangled qubits can be expediently designed according to our needs. For example, the four-qubit W state with the configurations of square and regular tetrahedron can be realized easily based on our scheme. (2) The effects of SMM’s spontaneous decay and photons leakage out of fiber can be efficiently suppressed via employing the dispersive atom-field interaction and strong coupling intensity of cavity fiber. (3) It is robust with respect to the deviations of some experimental parameters.

The remainder of this paper is organized into four parts as follows. In section II, we describe the model under consideration and derive the effective Hamiltonian of system. In section III, We discuss the generation of N-qubit W state of distant SMMs from qualitative and quantitative (corresponding to the case of N=4) aspects. In section IV, we analyze the influences of spontaneous decay of system on the generation of entanglement. Finally, we conclude with a brief summary in section V.

Fig. 1. (Color online) (a) The basal configuration for cavity-fiber-cavity system. An assistant SMM is trapped in a central cavity (cavity M) and N entangled SMMs are individually trapped in N cavities (cavity n), which connect with the central cavity together via N fibers. (b) The level configuration of each SMM.

Download Full Size | PDF

2. Model and Hamiltonian

As shown in Fig. 1(a), we consider a cavity-fiber-cavity system, which consists of N+1 distant cavities and N fibers. The cavity n (n=1,2,3…N) connects with the central cavity (cavity M) via fiber n. An assistant SMM (SMM M) is trapped into the cavity M, and induced by a classical field Ω_M with its magnetic field of the form (e⃗ _yH_M/2)e ^-iω_Mt + c.c. and a quantized electromagnetic field (cavity field) G_M with its magnetic field of the form (e⃗ _x℘M/2) âM^e ^-iv_Mt + c.c.. The SMM n is trapped into cavity n, and induced by a classical field Ω_n with its magnetic field of the form (e⃗ _yH_n/2)e-iwnt + c.c. and a quantized electromagnetic field (cavity field) G_n with its magnetic field of the form (e⃗ _x℘M/2) â_ne ^-iv_nt + c.c.. ωj and v_j (j = M,n) denote the frequencies of classical fields and cavity fields, respectively. $℘ j = c \sqrt{\frac{{ħ v}_{j} ε_{0}}{V_{eff}^{j}}}$ denotes the intensity of the cavity field and e⃗ _k (k = x,y) is the polarization unit vector. Let a uniform dc magnetic field H0 perpendicular to the easy anisotropy axis (z-axis) of the SMM apply to each of them along the x-axis. Then, under dipole approximation, the Hamiltonian of this SMM-cavity system can be given by [47, 51]

\hat{ℋ} = {\hat{ℋ}}_{0 S} + ℋ_{0 F} + \hat{풱},

{\hat{ℋ}}_{0 S} = - D {\hat{S}}_{M z}^{2} + {\hat{ℋ}}_{t r} - g μ_{B} {\hat{S}}_{M x} H_{0} + \sum_{n = 1}^{N} (- D {\hat{S}}_{n z}^{2} + {\hat{ℋ}}_{t r} - g μ_{B} {\hat{S}}_{n x} H_{0}),

{\hat{ℋ}}_{0 F} = ħ v_{M} {\hat{a}}_{M}^{†} {\hat{a}}_{M} + \sum_{n = 1}^{N} ħ v_{n} {\hat{a}}_{n}^{†} {\hat{a}}_{n},

\hat{풱} = [- \frac{g μ_{B}}{2} ({\hat{\vec{S}}}_{M} \cdot {\vec{H}}_{M} e^{- i ω_{M} t} + {\hat{\vec{S}}}_{M} \cdot {\vec{℘}}_{M} {\hat{a}}_{M})

where z is the easy anisotropy axis; $\hat{ℋ}$ _tr is the operator of the transverse anisotropy energy; D, g, and μ_B are the longitudinal anisotropy constant, the Landé factor, and the Bohr magneton, respectively. For simplicity, we choose that the above parameters are the same for each SMM without loss of generality. Ŝ_x, Ŝ_y, and Ŝ_z denote the x, y, and z projections of the spin operator for corresponding SMM, respectively. $\hat{ℋ}$ _0S and $\hat{ℋ}$ _0F represent the free Hamiltonians of the SMMs and the cavity fields, respectively. 𝒱 represents the interaction Hamiltonian between SMMs and fields.

In general, independently of the form of the transverse anisotropy of SMM, the molecule levels will form a series of doublets split, which are the eigenstates of $\hat{ℋ}$ ₀ due to the dc magnetic field and the transverse anisotropy [48, 49]. The eigenenergies and eigenstates of the Hamiltonian $\hat{ℋ}$ ₀ are denoted as ϵ_n and ∣n〉 (n = 0,1,2,3…), respectively. ∣n_s〉(n_s = 0,2, …) are symmetric functions of the z projection of the molecule spin, whereas ∣n_a〉(n_a = 1,3,…) are antisymmetric functions. As shown in Fig. 1(b), we only need to consider the three lowest energy levels ∣n〉(n = 0-2) relevant for the investigation of the entanglement considered here. For SMM j (j=M, n), the classical field Ω_j dispersively induces transition ∣1_i〉_j →∣2_i〉_j and the cavity field G_j dispersively couples transition ∣0_i〉_j → ∣2_i〉_j. Δ_j is the corresponding single photon frequency detuning and satisfies corresponding two-photon resonance condition, as shown in Fig. 1(b).

Then, in the interaction picture, under the dipole and rotating wave approximation, the interaction Hamiltonian of the SMM-cavity can be written as (ħ = 1)[58, 59, 60, 61]

H_{I}^{s c} = Δ_{M} {∣ 2 〉}_{M} 〈 2 |+ (- Ω_{M} {∣ 2 〉}_{M} 〈 1 ∣ + G_{M} a_{M}^{†} {∣ 0 〉}_{M} 〈 2 ∣ + H . c .)

where the symbol H.c. means Hermitian conjugate; a ^† _M, a ^† _n and a_M a_n are the creation and annihilation operators associating with the corresponding quantized cavity modes. Ω_M, Ω_n and G_M, G_n denote the one-half Rabi frequencies and atom-field coupling constants, respectively.

They are assumed to be real number in this paper, without loss generality, and are defined as [47]

Ω_{M} (t) = \frac{g μ_{B} H_{M} (t)}{2 ħ} 〈 2 ∣ {\hat{S}}_{M y} ∣ 1 〉, G_{M} = \frac{g μ_{B} ℘_{M} (t)}{ħ} 〈 2 ∣ {\hat{S}}_{M x} ∣ 0 〉,

Ω_{n} (t) = \frac{g μ_{B} H_{n} (t)}{2 ħ} 〈 2 ∣ {\hat{S}}_{n y} ∣ 1 〉, G_{n} = \frac{g μ_{B} ℘_{n} (t)}{ħ} 〈 2 ∣ {\hat{S}}_{n x} ∣ 0 〉,

where n=1,2,…N. In the above expressions, we have considered the selection rules for coupling corresponding transitions of SMM, and chosen the polarization of classical fields and cavity fields along the y- and x-axis, respectively.

By applying standard quantum optical techniques [62], under the large-detuning condition, i.e., ∣Δ_M∣, ∣Δ_n∣ ≫ ∣Ω_M∣, ∣Ω_n∣, ∣G_M∣, ∣G_n∣, the excited states of SMMs ∣2〉_M and ∣2〉_n are only virtually excited in the process of SMM-field interaction. So, we can adiabatically eliminate the excited states of SMMs and obtain the effective Hamiltonian [63, 64, 65]

H_{eff}^{s c} = [Ω_{e M} a_{M}^{†} ∣ {1 〉}_{M} 〈 1 ∣ + \sum_{n = 1}^{N} Ω_{e n} a_{n} {∣ 1 〉}_{n} 〈 0 ∣ + H . c .],

where $Ω_{e M} = \frac{G_{M} Ω_{M}}{Δ_{M}}$ and $Ω_{e n} = \frac{G_{n} Ω_{n}}{Δ_{n}}$ are the effective Rabi frequencies for the corresponding Raman transitions ∣1〉_M → ∣e〉_M → ∣0〉_M and ∣0〉_n → ∣e〉_n →∣1〉_n. In the process of deriving the effective Hamiltonian (4), we have safely neglected the terms of cavity- and laser-induced level shifts, which can be compensated for quite straightforwardly via using corresponding second lasers which couple corresponding SMM’s levels nonresonantly with additional levels farther up in the level scheme [63].

On the other hand, in the short fiber limit (2L v̄)=(2πc)≪1, where L is the length of fiber and v̄ is the decay rate of the cavity field into a continuum of fiber modes, only the resonant mode of the fiber will interact with the cavity modes. For this case, the interaction Hamiltonian of cavity-fiber can be written as [52]

H_{I}^{c f} = \sum_{n = 1}^{N} [η_{n} a_{M} b_{n}^{†} + η_{n} b_{n} a_{n}^{†} + H . c .],

where b_n is the annihilation operator of resonant mode of the fiber; η_n denotes the corresponding coupling strength.

Summing up the above discussion, we can obtain the total Hamiltonian of this cavity-fiber-cavity system in the interaction picture

H_{I} = H_{e f f}^{s c} + H_{I}^{c f}

3. The generation of entanglement

In this section, we begin to discuss the generation of N-qubit W state of spatially separated SMMs. First, let us qualitatively describe the process of generating N-qubit W state, i.e., ∣Ψ_W〉_N = $\frac{1}{\sqrt{N}}$ [∣1〉₁∣0〉₂…∣₀〉_N + ∣0〉₁∣₁〉₂…∣0〉_N + ….∣0〉₁∣0〉₂…∣1〉_N] based on our scheme.

As shown in Fig. 1(b), consider that at the initial time the SMM M is in state ∣1〉_M, the other N SMM all in state ∣0〉_n (n=1,2…N), and all the field modes in vacuum state $\overset{N + 1}{{\overset{︷}{∣ 00 \dots 00 〉}}_{c}} \overset{N}{{\overset{︷}{∣ 00 \dots 0 〉}}_{f}}$ . Dominated by Hamiltonian (6), SMM M will go though the Raman transition ${∣ 1 〉}_{M} \overset{Ω_{M}}{\to} {∣ 2 〉}_{M} \overset{a_{M}^{†}}{\to} {∣ 0 〉}_{M}$ and emit a photon into the cavity M. Then, the photon will traverse one of N fibers with probability amplitude $\frac{1}{\sqrt{N}}$ , and reach to the corresponding cavity n. At last, the corresponding SMM n trapped in cavity n will absorb the photon and go though the Raman transition ${∣ 0 〉}_{n} \overset{a_{n}}{\to} {∣ 2 〉}_{n} \overset{Ω_{n}}{\to} {∣ 1 〉}_{n}$ . Summing up the above processes, the system will evolve into state $∣ ψ (t) 〉 = {∣ 0 〉}_{M} \otimes {∣ Ψ_{W} 〉}_{N} \otimes \overset{N + 1}{{\overset{︷}{∣ 00 \dots 00 〉}}_{c}} \overset{N}{{\overset{︷}{∣ 00 \dots 0 〉}}_{f}}$ at a proper time from $∣ Ψ (0) 〉 = {∣ 1 〉}_{M} \otimes ({∣ 0 〉}_{1} {∣ 0 〉}_{2} \dots {∣ 0 〉}_{N}) \otimes \overset{N + 1}{{\overset{︷}{∣ 00 \dots 00 〉}}_{c}} \overset{N}{{\overset{︷}{∣ 00 \dots 0 〉}}_{f}}$ . Then, via switching off the interaction between the SMMs and fields at the proper time, we can obtain the N-qubit W state of spatially separated SMMs, ∣Ψ_W〉_N, which is completely separated from the states of SMM M, cavity fields and fiber modes. Before starting following discussion, we would like to point out the flexibility of our scheme. In the present scheme, the N entangled qubits connect together with a center cavity via N fibers, and hence the distribution of N entangled qubits can be expediently designed according to our needs via properly choosing the positional relation between N fibers, which may be useful for the multi-node quantum communications network.

Fig. 2. (Color online) The four-qubitWstate in the configurations of square [panel (a)] and regular tetrahedron [panel (b)].

Download Full Size | PDF

Fig. 3. (Color online) The fidelity F_W of realizing four-qubit W state ∣Ψ_W〉 versus time Ω_M t. The parameters are chosen as Ω_en=0.5Ω_eM, ηn = 25Ωe_M (n=1,2,3,4).

Download Full Size | PDF

Fig. 4. (Color online) The fidelity F_W of realizing four-qubit W state ∣Ψ_W〉 versus time ΩeMt and proportional coefficient s [panel (a)] and versus s when t= $\frac{4.95}{Ω_{e M}}$ [panel (b)].

Download Full Size | PDF

Next, we will quantitatively discuss the generation of N-qubit W state based on our scheme. As an example, we choose the case of N=4 without loss of generality. Its extension to the case of N > 4 is just a repetitive work. For the case of N=4, the connective manner between SMMs M and n (n=1-4) is depicted in Fig. 2, which shows that four entangled SMMs can form conveniently the framework of square [panel (a)] and regular tetrahedron [panel (b)] based on the present scheme. At the initial time, the system is considered to be in the state ∣ψ(0)〉_W = ∣1〉_M∣0〉₁∣0〉₂∣0〉₃∣0〉₄∣00000〉_c∣0000〉_f. Then, it will evolve in the domination of the Schrödinger equation (ħ = 1)

i \frac{\partial}{\partial t} ∣ ψ (t) 〉 = H_{I} ∣ ψ (t) 〉,

where H_I is given by the Hamiltonian (6) and the condition of N=4. ∣ψ(t)〉 denotes the state of system at time t, and are restricted to the subspaces spanned by the following basis vectors

∣ ϕ_{1} 〉 = {∣ 1 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{2} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 10000 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{3} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 1000 〉}_{f},

∣ ϕ_{4} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 01000 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{5} 〉 = {∣ 0 〉}_{M} {∣ 1 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{6} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0100 〉}_{f},

∣ ϕ_{7} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00100 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{8} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{9} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0010 〉}_{f},

∣ ϕ_{10} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00010 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{11} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 1 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{12} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00000 〉}_{c} {∣ 0001 〉}_{f},

∣ ϕ_{13} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 0 〉}_{4} {∣ 00001 〉}_{c} {∣ 0000 〉}_{f},

∣ ϕ_{14} 〉 = {∣ 0 〉}_{M} {∣ 0 〉}_{1} {∣ 0 〉}_{2} {∣ 0 〉}_{3} {∣ 1 〉}_{4} {∣ 00000 〉}_{c} {∣ 0000 〉}_{f},

Fig. 5. (Color online) The fidelity F_W of realizing four-qubit W state ∣Ψ_W〉versus time Ω_eMt and coupling constant η [panel (a)] and versus η when t= [panel (b)].

Download Full Size | PDF

where n_cM, n _c1, n _c2, n _c3, n _c4, and n _f1, n _f2, n _f3, n _f4 in ∣n_cM n _c1 n _c2 n _c3 n _c4〉_c∣n _f1 n _f2 n _f3 n _f4〉_f denote the photon numbers in the cavities M, n, and fiber n (n=1-4), respectively.

Then, in the above subspace, we numerically solve the Schrödinger equation (7), and plot the evolution of fidelity of realizing the four-qubit W state (F_W) in Fig. 3. The F_W is defined as F_W = ∣_f 〈0000∣_c〈00000∣_M〈0∣〈Ψ_W∣ψ(t)〉∣². In the above calculation, we have set Ω_e1 = Ω_e2 = Ω_e3 = Ω_e4 and η ₁ = η ₂ = η ₃ = η ₄ for simplicity. From the Figs. 3, it is shown that the W state can be deterministically generated at appropriate time, which is consistent with our qualitative discussions. In addition, the plot of F_W changes smoothly as a function of the dimensionless time Ω_eMt and this characteristic is useful for switching off the interaction between the SMMs and the fields in time for obtaining the steady W state.

In Figs. 3, we have chosen the appropriate conditions of parameter, i.e., Ω_en = 0.5Ω_eM, η_n = 25Ω_eM for deterministically generating W state. However, the above conditions of parameter may not be satisfied exactly in practical situations. In order to study the influences of the parameter deviations on the fidelity of realizing W states, we present the three-dimensional plots of the dependence of F_W on time Ω_eMt and the proportional coefficient s (coupling constant h), as shown in Fig. 4 (Fig. 5). The proportional coefficient s satisfies relationship Ω_en = sΩ_eM, and η = η_n. It is clearly shown from Figs. 4(a) and 5(a) that the F_W is very steady with respect to the fluctuations of system parameters s and η. In order to further show this point explicitly, we also plot the two-dimensional curves of F_W with respect to s and η at a fixed time, as shown in Figs. 4(b) and 5(b). It is noticed that there is just a little decrease of F_W compared to F_W = 1 when the ratio coefficient s changes from s=0.47 to s=0.53 and the coupling constant η from η = 24.5 to η = 25.5. As a result, based on our scheme, the four-qubit W states of SMMs also can be realized with high fidelity, even that the corresponding conditions of parameter could not be satisfied accurately in practical situations. Here, it should be pointed out that the present conclusion is also suit for the case of N>4. Because, there is not qualitative difference between the cases of N=4 and N>4.

4. Effects of SMM’s spontaneous decay and photon leakage

In this section, we will study the influences of SMM’s spontaneous decay and photon leakage out of the cavities and fibers on the generation of N-qubit W state ∣Φ_W〉_N based on the original Hamiltonians (2) and (5). Similar to the section 3, here we still choose the case of N=4. Using the density-matrix formalism, the master equation for the density matrix of whole system can be expressed as:

\dot{ρ} w = - i [H_{I}^{sc} + H_{I}^{cf}, ρ] - \frac{κ_{M}}{2} (a_{M}^{†} a_{M} ρ - 2 a_{M} ρ a_{M}^{†} + ρ a_{M}^{†} a_{M})

where γ^ei_aj (j = M,n) denotes the spontaneous decay rate of SMM from level ∣2〉_j to ∣i〉_j; κ_j and γ_fn denote the decay rates of cavity fields and fiber modes, respectively; σ^j_pq = ∣p〉_j〈q∣ are the usual Pauli matrices. By numerically solving the Eq. (9) in the subspace spanned by basis vectors (8) and ∣ϕ ₁₅〉, ∣ϕ ₁₆〉, ∣ϕ ₁₇〉, ∣ϕ ₁₈〉, ∣ϕ ₁₉〉, we present the effects of the decay rates Γ_a, κ and γ_f on the fidelity of generating four-qubit W state ∣Ψ_W〉, as shown in the Figs. 6 and 7. In the above calculation, ∣ϕ ₁₅〉 = ∣2〉_M∣0〉₁∣0〉₂∣0〉₃∣0〉₄∣00000〉_c∣0000〉_f, ∣ϕ ₁₆〉 = ∣0〉M∣2〉₁∣0〉₂∣0〉₃∣0〉₄∣00000〉_c∣0000〉_f, ∣ϕ ₁₇〉 = ∣0〉_M∣0〉₁∣₂〉₂∣0〉₃∣0〉₄∣00000〉_c∣0000〉_f, ∣ϕ ₁₈〉=∣0〉_M∣0〉₁∣0〉₂∣2〉₃∣0〉₄∣00000〉_c∣0000〉_f, ∣ϕ ₁₉〉=∣0〉_M∣0〉₁∣0〉₂∣0〉₃∣2〉₄∣00000〉_c∣0000〉_f. In addition, for simplicity, we also have chosen κ_M = κ_n = κ (n=1-4), γ^ei_an = γ^ei_aM = γ/2, γ_fn = γ_f without loss of generality. It is shown from the Fig. 6 that the influences of system spontaneous decay on the fidelity F_W is very little when κ =γ_a= γ_f > 0.01γ, and F_W is still larger than 97% when Γ = 0.01γ. To show more clearly about the influences of system decay on F_W, we also plot respectively the function curves for F_W versus κ,γ_aand γ_f in Fig. 7 when γ_t=3.16. A comparison of the main and inserted parts of Figs. 7(a) and (b) clearly shows that the influences of SMM’s spontaneous decay rate γ_a and fiber decay rate γ_f on F_W are much smaller than that of cavity field decay rate Γ, and hence γ_aand γ_f can be neglected safely in our scheme. This numerical results can be qualitatively explained as follows. Under the conditions ∣Δ_M∣, ∣Δ_n∣ ≫ ∣Ω_M∣, ∣Ω_n∣, ∣Ω_M∣, ∣G_n∣, and ∣η∣ ≫ ∣Ω_eM∣, ∣Ω_en∣, the SMM’s excited states ∣2〉_M, ∣2〉_n, and fiber modes bn are only virtually excited in the whole interaction process, and hence the effects of SMM’s decay rate γ_a and fiber decay rate γ_f are suppressed strongly when ∣Δ_M∣, ∣Δ_n∣, ≈ 10∣Ω_M∣, ∣Ω_n∣, ∣Ω_M∣, ∣G_n∣, and ∣η∣ ≈ 25∣Ω_eM∣, ∣Ω_en∣, as shown in Fig. 7. In addition, it is also shown from the Fig. 7 that low properly decay rate of cavity field (κ > 0.04γ) is still needed for obtaining W state with high fidelity (F_W ≥ 95%) in our scheme.

Fig. 6. (Color online) The fidelity F_W of realizing the four-qubit W state ∣Ψ_W〉 versus time γ_t for different decay rates κ (γ_a = γ_f = κ). The corresponding system parameters are chosen as: γ_n = 8γ (n=1-4), g_M = 16γ, Ω_n = 10γ, Ω_M = 10γ, η = 25γ, and Δ_M = Δ_n = 100γ.

Download Full Size | PDF

Before ending this section, let us briefly discuss the experimental feasibility of our scheme. First, the three-level SMM can be realized via choosing the general SMM (e.g., Mn₁₂ and Fe₈) subjected a dc magnetic field perpendicular to its easy axis [47, 48]. For instance, in the case of Fe₈ molecule subjected to dc magnetic field H ₀=22.4KOe [47], the three-level configuration of SMM can be realized by applying a classical field with y-axis polarization and the cavity field with x-axis polarization couple the corresponding molecule transitions, as shown in Fig. 1(b). Secondly, the dissipation of system is always an important issue for realizing entangled state experimentally. In the present scheme, the effects of SMM’s spontaneous decay and photon leakage out of the fiber can be suppressed effectively via choosing large detuning condition for the SMM-field interaction and strong coupling strength for the cavity-fiber interaction, as shown in Figs. 6 and 7. The dissipation of optical cavity is surely a problem in our scheme for re-alizing a high-fidelityWstate since the cavity modes are excited in the evolution process. However, the transition frequencies for the general SMMs are usually in the range of microwaves (For example, the transition frequencies of Fe₈, we 0 ≈ 3.56×10¹¹, we 1 ≈ 3.27×10¹¹ [47]), and hence, the cavity with resonant frequency in the range of microwave can be used to trap SMMs in our scheme. Based on the recent experiments about microwave cavity, the cavity lifetime with a photon T ≈ 1ms has been realized [65]. Therefore, we can choose κ=2π ≈ 1 kHz Ω_M=2π =Ω_n=2π ≈1 MHz, Δ_M=2π = Δ_n=2π ≈10 MHz, as the basal system parameters of our schemes. Then, the condition κ ≤ 0.04γ that is required for realizing four-qubit W states with high fidelity can be satisfied with these system parameters. Of course, the above discussion is also suit for the case of N>4. The only difference is that the required conditions for realizing entangled state with high-fidelity may be more strict than the case of N=4. Lastly, it should be pointed out that there also exist some obstacles for implementing our scheme experimentally. For example, how to trap a SMM in a microwave cavity, and how to couple N cavities to a common cavity through fibers. However those obstacles can be overcame in principle along with the progress of fabricated technology. As a theoretical research, our scheme is still potentially feasible in the near future.

Fig. 7. (Color online) Fidelity of realizing the W state ∣Ψ_W〉versus κ and γ_a [panel (a)]; versus κ and γ_f [panel (b)]; when γ_t=3.16. The other system parameters are the same as in Fig. 6

Download Full Size | PDF

In addition, we also pin our hope on the coupled fiber-microcavity (i.e., microtoroid or microdisk resonators) system [67] for realizing our scheme experimentally in the near future. Of course, those microcavity systems can satisfy the low dissipation condition of cavity modes (i. e., κ ≤ 0.04γ), which is required for implementing our scheme, since it has ultrahigh Q-factor. On the one hand, along with the progress of fabricated technology [68, 69], the microcavity with fundamental cavity mode in the range of microwaves may be realized in the near future. Then, in our scheme, the coupling between SMM and microcavity can be realized via locating SMM near the microcavity surface, which is similar to the method of coupling atom with microcavity [70, 71]. On the other hand, peoples have realized experimentally the coupling between microcavity and fiber [67, 72, 73, 74] in recent years. Therefore, we believe that the coupling between N microcavity and a common microcavity through N fibers will be possibly realized in the near future.

5. Conclusion

In conclusion, we have proposed a scheme for realizing N-partite W state of SMMs, which are trapped into spatially separated cavities connected by fibers. Through numerically simulating the case of N=4, we show that the influences of SMM’s spontaneous decay and fiber loss on entanglement generation can be effectively suppressed, since the SMM’s excited states and fiber modes are only virtually excited in our scheme. In addition, we also have demonstrated that the present scheme is robust with respect to the deviations of experimental parameters. Lastly, the experimental feasibility of our scheme is discussed and as a result, the present scheme is considered as a promising scheme for realizing N-partite W states of distant SMMs with high fidelity, which may be useful for the progress of multi-node quantum information science.

Acknowledments

The research was supported in part by the Natural Science Foundation of China (Grants No. 10575040, No. 10634060, No. 10704017, and No. 10874050), by the National Basic Research Program of China (Contract No. 2005CB724508), and by the Foundation from the Ministry of the National Education of China (Grant No. 200804870051), by the graduate innovation Fund of Huazhong University of Science and Technology (Grant No. HF-06-010-08-012).We would like to thank Professor Ying Wu for helpful discussion and his encouragement.

References and links

1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, U.K., 2000)

2. C. H. Bennett and D. P. DiVincenzo, “Quantum information and computation,” Nature (London) 404, 247–255 (2000). [CrossRef]

3. D.P. DiVincenzo, “Quantum Computation,” Science 270, 255–261 (1995). [CrossRef]

4. D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–579 (1997). [CrossRef]

5. Hiroki Takesue, “Long-distance distribution of time-bin entanglement generated in a cooled fiber,” Optics Express , 14, 3453–3460 (2006). [CrossRef] [PubMed]

6. A. K. Ekert, “Quantum cryptography based on Bells theorem,” Phys. Rev. Lett. 67, 661–663 (1991). [CrossRef] [PubMed]

7. A. Barenco, D. Deutsch, A. Ekert, and R. Jozsa, “Conditional Quantum Dynamics and Logic Gates,” Phys. Rev. Lett. 74, 4083–4086 (1995). [CrossRef] [PubMed]

8. W.-X. Yang, Z.-M. Zhan, and J.-H. Li, “Efficient scheme for multipartite entanglement and quantum information processing with trapped ions,” Phys. Rev. A 72, 062108(1-6) (2005). [CrossRef]

9. I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Phys. Rev. A 52, 3489–3496 (1995). [CrossRef] [PubMed]

10. A. Karlsson and M. Bourennane, “Quantum teleportation using three-particle entanglement,” Phys. Rev. A 58, 4394–4400 (1998). [CrossRef]

11. M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” Phys. Rev. A 59, 1829–1834 (1999). [CrossRef]

12. G. A. Durkin, C. Simon, and D. Bouwmeester, “Multiphoton Entanglement Concentration and Quantum Cryptography,” Phys. Rev. Lett. 88, 187902(1-4) (2002). [CrossRef] [PubMed]

13. N. Gisin and S. Massar, “Optimal Quantum Cloning Machines,” Phys. Rev. Lett. 79, 2153–2156 (1997). [CrossRef]

14. D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990). [CrossRef]

15. W. Dür, G. Vidal, and J. I. Cirac, “Three qubits can be entangled in two inequivalent ways,” Phys. Rev. A 62, 062314(1-12) (2000). [CrossRef]

16. M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter, “Experimental Realization of a Three-Qubit Entangled W State,” Phys. Rev. Lett. 92, 077901(1-4) (2004). [CrossRef] [PubMed]

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

18. J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter, and A. Zeilinger, “Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement,” Nature (London) 403, 515–519 (2000); [CrossRef] [PubMed] D. Bouwmeester, J.-W. Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, “Observation of Three-Photon Greenberger-Horne-Zeilinger Entanglement,” Phys. Rev. Lett. 82, 1345–1349 (1999).

19. R. J. Nelson, D. G. Cory, and S. Lloyd, “Experimental demonstration of Greenberger-Horne-Zeilinger correlations using nuclear magnetic resonance,” Phys. Rev. A 61, 022106(1-5) (2000). [CrossRef]

20. X. B. Zou, K. Pahlke, and W. Mathis, “Conditional generation of the Greenberger-Horne-Zeilinger state of four distant atoms via cavity decay,” Phys. Rev. A 68, 024302(1-4) (2003). [CrossRef]

21. C. Yu, X. X. Yi, H. Song, and D. Mei, “Robust preparation of Greenberger-Horne-Zeilinger and W states of three distant atoms,” Phys. Rev. A 75, 044301(1-4) (2007). [CrossRef]

22. D. Gonta, S. Fritzsche, and T. Radtke, “Generation of four-partite Greenberger-Horne-Zeilinger and W states by using a high-finesse bimodal cavity,” Phys. Rev. A 77, 062312(1-13) (2008). [CrossRef]

23. T. Yamamoto, K. Tamaki, M. Koashi, and N. Imoto, “Polarization-entangled W state using parametric downconversion,” Phys. Rev. A 66, 064301(1-4) (2002). [CrossRef]

24. M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter, “Experimental Realization of a Three-Qubit Entangled W State,” Phys. Rev. Lett. 92, 077901(1-4) (2004). [CrossRef] [PubMed]

25. H. Mikami, Y. Li, and T. Kobayashi, “Generation of the four-photon W state and other multiphoton entangled states using parametric down-conversion,” Phys. Rev. A 70, 052308(1-7) (2004). [CrossRef]

26. G.-C. Guo and Y.-S. Zhang, “Scheme for preparation of the W state via cavity quantum electrodynamics,” Phys. Rev. A 65, 054302(1-3) (2002). [CrossRef]

27. B. Yu, Z.-W. Zhou, and G.-C. Guo, “The generation of multi-atom entanglement via the detection of cavity decay,” J. Opt. B: Quantum Semiclass. Opt. 6, 86–90 (2004). [CrossRef]

28. Y.-F. Xiao, Z.-F. Han, J. Gao, and G.-C. Guo, “Generation of multi-atom Dicke states through the detection of cavity decay,” J. Phys. B: At. Mol. Opt. Phys. 39, 485–491 (2006). [CrossRef]

29. Y.-F. Xiao, X.-B. Zou, and G.-C. Guo, “Generation of atomic entangled states with selective resonant interaction in cavity quantum electrodynamics,” Phys. Rev. A 75, 012310(1-5) (2007). [CrossRef]

30. N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter, “Experimental Analysis of a Four-Qubit Photon Cluster State,” Phys. Rev. Lett. 95, 210502(1-4) (2005). [CrossRef] [PubMed]

31. X. L. Zhang, K. L. Gao, and M. Feng, “Efficient and high-fidelity generation of atomic cluster states with cavity QED and linear optics,” Phys. Rev. A 75, 034308(1-4) (2007). [CrossRef]

32. Y.-K. Bai and Z. D. Wang, “Multipartite entanglement in four-qubit cluster-class states,” Phys. Rev. A 77, 032313(1-6) (2008). [CrossRef]

33. M. Yukawa, R. Ukai, P. van Loock, and A. Furusawa, “Experimental generation of four-mode continuous-variable cluster states,” Phys. Rev. A 78, 012301(1-6) (2008). [CrossRef]

34. Z.-R. Lin, G.-P. Guo, T. Tu, F.-Y. Zhu, and G.-C. Guo, “Generation of Quantum-Dot Cluster States with a Superconducting Transmission Line Resonator,” Phys. Rev. Lett. 101, 230501(1-4) (2008). [CrossRef] [PubMed]

35. Gao-xiang Li, S. Ke, and Z. Ficek, “Generation of pure continuous-variable entangled cluster states of four separate atomic ensembles in a ring cavity,” Phys. Rev. A 79, 033827(1-9) (2009). [CrossRef]

36. C. F. Roos, M. Riebe, H. Haffner, W. Hansel, J. Benhelm, G. P. T. Lancaster, C. Becher, F. Schmidt-Kaler, and R. Blatt, “Control and Measurement of Three-Qubit Entangled States,” Science 304, 1478–1480 (2004). [CrossRef] [PubMed]

37. C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano, and C. M. D. J. Wineland, “Experimental entanglement of four particles,” Nature (London) 404, 256–259 (2000). [CrossRef] [PubMed]

38. A. Rauschenbeutel, G. Nogues, S. Osnaghi, P. Bertet, M. Brune, J.-M. Raimond, and S. Haroche, “Step-by-step engineered multiparticle entanglement,” Science 288, 2024–2028 (2000). [CrossRef] [PubMed]

39. F. Bodoky and M. Blaauboer, “Production of multipartite entanglement for electron spins in quantum dots,” Phys. Rev. A 76, 052309(1-7) (2007). [CrossRef]

40. L.-M. Duan and H. J. Kimble, “Efficient Engineering of Multiatom Entanglement through Single-Photon Detections,” Phys. Rev. Lett. 90, 253601(1-4) (2003). [CrossRef] [PubMed]

41. R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak, “Magnetic bistability in a metal-ion cluster,” Nature (London) 365141–143 (1993). [CrossRef]

42. L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, “Macroscopic quantum tunnelling of magnetization in a single crystal of nanomagnets,” Nature (London) 383145–147 (1996). [CrossRef]

43. K. Petukhov, S. Bahr, W. Wernsdorfer, A.-L. Barra, and V. Mosser, “Magnetization dynamics in the single-molecule magnet Fe8 under pulsed microwave irradiation,” Phys. Rev. B 75064408(1-12) (2007). [CrossRef]

44. M. Misiorny and J. Barna, “Magnetic switching of a single molecular magnet due to spin-polarized current,” Phys. Rev. B , 75134425(1-5) (2007). [CrossRef]

45. W. Wernsdorfer and R. Sessoli, “Quantum Phase Interference and Parity Effects in Magnetic Molecular Clusters,” Science 284133–135 (1999). [CrossRef] [PubMed]

46. M. N. Leuenberger and D. Loss, “Quantum computing in molecular magnets,” Nature (London) 410789–793 (2001). [CrossRef] [PubMed]

47. A. V. Shvetsov, G. A. Vugalter, and A. I. Grebeneva, “Theoretical investigation of electromag- netically induced transparency in a crystal of molecular magnets,” Phys. Rev. B 74054416(1-6) (2006). [CrossRef]

48. X.-T. Xie, W. Li, J. Li, W.-X. Yang, A. Yuan, and X. Yang, “Transverse acoustic wave in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 75184423(1-6) (2007). [CrossRef]

49. Y. Wu and X. Yang, “Four-wave mixing in molecular magnets via electromagnetically induced transparency,” Phys. Rev. B 76054425(1-6) (2007). [CrossRef]

50. Y. Wu and X. Yang, “Giant Kerr nonlinearities and solitons in a crystal of molecular magnets,” Appl. Phys. Lett. 91094104(1-3) (2007). [CrossRef]

51. X.-Y. Lü, Ji-Bing Liu, Yü Tian, Pei-Jun Song, and Zhi-Ming Zhan, “Single molecular magnets as a source of continuous-variable entanglement,” Europhys. Lett. 8264003(1-6) (2008). [CrossRef]

52. S. Mancini and S. Bose, “Distributed Quantum Computation via Optical Fibers,” Phys. Rev. A 70, 022307(1-4) (2004). [CrossRef]

53. L.-B. Chen, M.-Y. Ye, G.-W. Lin, Q.-H. Du, and X.-M. Lin, “Generation of entanglement via adiabatic passage,” Phys. Rev. A 76062304(1-7) (2007). [CrossRef]

54. P. Peng and F.-L. Li, “Entangling two atoms in spatially separated cavities through both photon emission and absorption processes,” Phys. Rev. A , 75, 062320(1-7) (2007). [CrossRef]

55. S.-Y. Ye, Z.-R. Zhong, and S.-B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A , 77, 014303(1-4) (2008). [CrossRef]

56. X.-Y. Lü, J.-B. Liu, C.-L. Ding, and J.-H. Li, “Dispersive atom-field interaction scheme for three-dimensional entanglement between two spatially separated atoms,” Phys. Rev. A , 78, 032305(1-6) (2008); [CrossRef] X.-Y. Lü, L.-G. Si, M. Wang, S.-Z. Zhang, and X. Yang, “Generation of entanglement between two spatially separated atoms via dispersive atom-field interaction,” J. Phys. B: At. Mol. Opt. Phys. 41, 235502(1-6) (2008).

57. Z. Yin and F. Li, “Multiatom and resonant interaction scheme for quantum state transfer and logical gates between two remote cavities via an optical fiber,” Phys. Rev. A 75, 012324(1-11) (2007). [CrossRef]

58. Y. Wu, J. Saldana, and Y. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67013811(1-5) (2003). [CrossRef]

59. Y. Wu and L. Deng, “Achieving multifrequency mode entanglement with ultraslow multiwave mixing,” Optics Letters 29, 1144–1146 (2004). [CrossRef] [PubMed]

60. Y. Wu and X. Yang, “Carrier-envelope phase-dependent atomic coherence and quantum beats,” Phys. Rev. A 76013832(1-4) (2007). [CrossRef]

61. Y. Wu and X. Yang, “Strong-Coupling Theory of Periodically Driven Two-Level Systems,” Phys. Rev. lett. 98013601(1-4) (2007). [CrossRef] [PubMed]

62. C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991).

63. T. Pellizzari, “Quantum Networking with Optical Fibres,” Phys. Rev. Lett. 79, 5242–5245 (1997). [CrossRef]

64. Y. Wu, “Effective Raman theory for a three-level atom in the configuration,” Phys. Rev. A 54, 1586–1592 (1996). [CrossRef] [PubMed]

65. Y. Wu and P. T. Leung, “Lasing threshold for whispering-gallery-mode microsphere lasers,” Phys. Rev. A 60, 630–633 (1999). [CrossRef]

66. J. M. Raimond, M. Brune, and S. Haroche,“Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. , 73565–582 (2001). [CrossRef]

67. K. J. Vahala,“Optical microcavities,” Nature , 424839–846 (2003). [CrossRef] [PubMed]

68. B. Min, L. Yang, and K. Vahala, “controlled transition between parameteric and Raman oscillations in ultrahigh- Q silica toroidal microcavities,” Appl. Phys. Lett. 87181109(1-3) (2005). [CrossRef]

69. M. Hossein-Zadeh and K. Vahala, “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Optics Express 15166–175 (2007). [CrossRef] [PubMed]

70. D. W. Vernooy and H. J. Kimble, “Quantum structure and dynamics for atom galleries,” Phys. Rev. A 551239 (1997). [CrossRef]

71. Y.-F. Xiao, Z.-F. Han, and G.-C. Guo, “Quantum computation without strict strong coupling on a silicon chip,” Phys. Rev. A 551239 (1997).

72. P. E. Barclay, K. Srinivasan, O. Painter, B. Lev, and H. Mabuchi, “Integration of fiber-coupled high-Q SiNx microdisks with atom chips,” Appl. Phys. Lett. 89, 131108(1-3) (2006). [CrossRef]

73. P.-B. Li, Y. Gu, Q.-H. Gong, and G.-C. Guo, “Quantum-information transfer in a coupled resonator waveguide,” Phys. Rev. A 79, 042339(1-4) (2009). [CrossRef]

74. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. 91, 043902(1-4) (2003). [CrossRef] [PubMed]

N-qubit W state of spatially separated single molecule magnets

Abstract

1. Introduction

2. Model and Hamiltonian

3. The generation of entanglement

4. Effects of SMM’s spontaneous decay and photon leakage

5. Conclusion

Acknowledments

References and links

Cited By

Figures (7)

Equations (32)

Optics Express