Multicolor multipartite entanglement produced by vector four-wave mixing in a fiber

C. J. McKinstrie; S. J. van Enk; M. G. Raymer; S. Radic

doi:10.1364/OE.16.002720

1. Introduction

Entanglement is an intrinsically quantum-mechanical property of some quantum states that describe two or more separate systems [1]. These states cannot be described in terms of classical correlations between the systems [2]. Entanglement is a resource for quantum communication and computation [3, 4]. Its use was proposed and demonstrated for dense coding [5, 6, 7, 8, 9, 10], key distribution [11, 12, 13, 14, 15], one-way computation [16, 17, 18, 19], teleportation [20, 21, 22, 23, 24, 25] and teleportation networking [26, 27, 28, 29]. Earlier papers focused on two-partite discrete-variable protocols, whereas later papers focused on two-partite continuous-variable and multipartite discrete- or continuous-variable protocols. Optical discrete-variable protocols require the detection of individual photons, whereas optical continuous-variable protocols require the measurement of mode quadratures by balanced homodyne detection.

In some of the aforementioned experiments, two-partite (two-product-mode) entanglement is produced by nondegenerate down-conversion in a crystal [35, 36]. In others, a beam splitter is used to combine the one-product-mode squeezed states produced by degenerate down-conversion [37] in separate crystals. Both types of down-conversion are driven by one pump mode. Three-partite [10, 15, 27, 28, 29, 30] and four-partite [18, 31, 33, 34] entanglement are produced by path-stabilized arrays of beam splitters, which combine the outputs of two or more sources of one-mode squeezed states or two-mode entangled states. Because these methods are restricted to modes (photons) with the same frequency, spatial multiplexing is required to transmit and manipulate the photons. Recently, it was predicted that concurrent difference-and sum-frequency processes in crystals produce entanglement between three or four product modes with different frequencies [38, 39, 40]. Phase matching two or more concurrent processes requires the use of periodic poling [41] or birefringence [42], and is only possible for certain mode frequencies.

Two-product-mode entanglement can also be produced by four-wave mixing (FWM) in a fiber [43, 44, 45, 46]. Three different types of FWM are illustrated in Fig. 1. Modulation interaction (MI) is the degenerate process in which two photons from the same pump mode are destroyed and two different product-mode (sideband) photons are created. Phase conjugation (PC) is the nondegenerate process in which two photons from different pumps are destroyed and two different sideband photons are created. Bragg scattering is the nondegenerate process in which a sideband (signal) photon and a pump photon are destroyed, and different sideband (idler) and pump photons are created.MI and PC produce two-frequency entangled states in the same way that down-conversion produces two-wavevector entangled states, whereas BS combines modes with different frequencies in the same way that a beam splitter combines modes with different wavevectors [47]. Because fibers (and free space) allow frequency multiplexing, it is useful to study the entanglement of modes with different frequencies.

Fig. 1. Frequency diagram for the interaction of two pumps (1 and 2) and four sidebands (1± and 2±). Depending on the fiber dispersion and pump frequencies, six different four-wave mixing (FWM) processes can occur, separately or simultaneously. The red, blue and green dashed lines denote modulation interaction (MI), phase conjugation (PC) and Bragg scattering (BS), respectively.

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FWM is driven by nonlinearity and suppressed by dispersion. By tuning the pump frequencies judiciously, relative to the zero-dispersion frequency (ZDF) of the fiber, one can control whether MI, PC and BS occur separately, or simultaneously [47, 48, 49]. In the latter case, FWM driven by two pumps couples the evolution of four modes with different frequencies, as illustrated in Fig. 1. The classical physics [48, 49] and quantum noise properties [50] of four-mode interactions were studied thoroughly, in the context of classical communication systems. However, the entanglement produced by these interactions was not studied previously, and is the focus of this report.

The entanglement scheme described herein has several potential advantages over existing schemes. First, it produces four-mode entanglement in one step. (A path-stabilized array of beam splitters is not necessary.) Second, the fiber system in which it occurs is simple and compact. Third, fibers can be manufactured with different ZDFs for different applications. (Photons with a variety of frequencies can be generated.) Fourth, because the photons are generated in a fiber, their transverse structure is suitable for transmission through another fiber. (There is no mode-matching problem.) Fifth, because the generated photons have different frequencies, they can be transmitted by the same fiber.

2. Four-mode equations

The interaction of two strong, classical pumps (1 and 2) with four weak, quantum sidebands (1-, 1+, 2- and 2+), is governed by the Hamiltonian

H_{a} = α (a_{1 -}^{†} a_{1 -} + a_{1 +}^{†} a_{1 +}) + α (a_{1 -}^{†} a_{1 +}^{†} + a_{1 -} a_{1 +})

where a_j is the destruction operator of sideband (mode) j, † denotes a Hermitian conjugate, and the nonlinearity coefficients α=γ_KP ₁, β=γ_K(P ₁ P ₂)^1/2 and γ=γ_KP ₂, where γ_K is the Kerr coefficient of the fiber, and P_j is a pump power. Because the phases of modes 1± are measured relative to the phase of pump 1, and the phases of modes 2± are measured relative to the phase of pump 2, the pump phases do not appear explicitly in the Hamiltonian. The terms in the first line of Eq. (1) model the MI of pump 1, in which 2γ ₁→γ _1-+γ ₁+, where γ_j denotes a photon with frequency ω_j. Not only does pump 1 provide nonlinear coupling between modes 1- and 1+, it also imposes cross-phasemodulation (CPM) on them. The terms in the second linemodel the BS processes in which γ _1-+γ ₂→γ ₁+γ _2- and γ ₁₊+γ ₂→γ ₁+γ ₂₊, and the terms in the third line model the PC processes in which γ ₁+γ ₂→γ _1-+γ ₂+ and γ ₁+γ ₂→γ ₁₊+γ ₂-. Both of these processes are driven by pumps 1 and 2. The terms in the fourth line model the MI of pump 2, in which 2γ ₂→γ _2-+γ ₂+. All six processes are illustrated in Fig. 1.

By applying the (spatial) Heisenberg equations

\frac{{da}_{j}}{dz} = i [a_{j}, H_{a}]

to the Hamiltonian (1), where d/dz denotes a distance derivative, one obtains the four-mode equations

\frac{{da}_{1 -}^{†}}{dz} = - i α a_{1 -}^{†} - i α a_{1 +} - i β a_{2 -}^{†} - i β a_{2 +},

\frac{{da}_{1 +}}{dz} = - i α a_{1 -}^{†} + i α a_{1 +} + i β a_{2 -}^{†} + i β a_{2 +},

\frac{{da}_{2 -}^{†}}{dz} = - i β a_{1 -}^{†} - i β a_{1 +} - i γ a_{2 -}^{†} - i γ a_{2 +},

\frac{{da}_{2 +}}{dz} = i β a_{1 -}^{†} + i β a_{1 +} + i γ a_{2 -}^{†} + i γ a_{2 +} .

(The Heisenberg equations describe how the mode operators evolve in time. However, the modes convect at the same speed, so temporal evolution is equivalent to spatial evolution.) Equations (3)–(6) are the standard four-mode equations for orthogonal (perpendicular) pumps [48, 49, 50], with the effects of dispersion neglected.

The weak-dispersion approximation is valid in at least two cases. In the first case, the pump and sideband frequencies are all near the ZDF of the fiber [51, 52], whereas in the second, the sideband frequencies ω _1± and ω_2± are comparable to the pump frequencies ω ₁ and ω ₂, respectively [48, 49, 50]. The first configuration, which is illustrated in Fig. 2(a), provides phase-sensitive amplification of a signal polarized at 45° to the pumps [53], unimpaired by the generation of secondary pumps and idlers [51, 52], and has been used to generate cross-polarized photon pairs [54]. The second configuration,which is illustrated in Fig. 2(b), has been used to generate photons in a polarization-independentmanner [55], and to wavelength-convert signals between the low-loss windows near 1310 and 1550 nm [56]. Most fibers have one ZDF. In such fibers, the range of sideband frequencies for which dispersion can be neglected is narrow. However, some fibers have two ZDFs [57, 58]. If the pump frequencies are near the ZDFs, the range of sideband frequencies for which dispersion can be neglected is broad.

Fig. 2. Polarization diagram for the four-sideband interaction driven by perpendicular pumps. (a) Special case in which the pump-pump frequency difference is twice the pump-sideband difference. (b) General case in which the pump-pump difference is (much) larger than the pump-sideband difference.

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Equations (3)–(6) describe photon generation by MI and PC, and photon exchange by BS. The associated Manley–Rowe–Weiss (MRW) equations [59, 60], which relate the photon numbers of the modes, are derived in Appendix A.

Despite their complexity, Eqs. (3)–(6) have the simple solutions

a_{1 -}^{†} (z) = (1 - i α z) a_{1 -}^{†} (0) - i α {za}_{1 +} (0) - i β {za}_{2 -}^{†} (0) - i β {za}_{2 +} (0),

a_{1 +} (z) = i α {za}_{1 -}^{†} (0) + (1 + i α z) a_{1 +} (0) + i β {za}_{2 -}^{†} (0) + i β {za}_{2 +} (0),

a_{2 -}^{†} (z) = - i β {za}_{1 -}^{†} (0) - i β {za}_{1 +} (0) + (1 - i γ z) a_{2 -}^{†} (0) - i γ {za}_{2 +} (0),

a_{2 +} (z) = i β {za}_{1 -}^{†} (0) + i β {za}_{1 +} (0) + i γ {za}_{2 -}^{†} (0) + (1 + i γ z) a_{2 +} (0),

which are valid for all distances z. In the absence of dispersion, the modes grow linearly with distance, rather than exponentially, because the effects of CPM (wave-number shifts) balance those of nonlinear coupling (amplification and frequency conversion).

3. Quadrature correlations

Equations (7)–(10) show that each output mode depends on all of the input modes: The output modes are correlated. It is common to quantify these correlations in terms of the mode quadratures, which can be measured by balanced homodyne detection [61, 62]. For each mode j, the quadrature

q_{j} (θ_{j}) = \frac{(a_{j}^{†} e^{{i θ}_{j}} + a_{j} e^{- i θ_{j}})}{2^{\frac{1}{2}}},

where θ _j is the phase of the local oscillator used in the detection process. The conjugate quadrature p_j(θ_j)=q_j(θ_j+π/2). These quadratures satisfy the canonical commutation relations. The quadrature deviation

δ q_{j} (θ_{j}) = q_{j} (θ_{j}) - 〈 q_{j} (θ_{j}) 〉,

where 〈〉 denotes an expectation value. The output deviations depend on the input deviations (quantum fluctuations), but not on the input quadratures (signal amplitudes).

In [63], detailed studies were made of the quantum noise properties of multiple-mode interactions characterized by the input–output equations

a_{j} (z) = \sum_{k} [μ_{j k} (z) a_{k} (0) + ν_{j k} (z) a_{k}^{†} (0)] .

Formulas were derived for the means and variances of the quadratures and photon numbers of the modes. Equations (7)–(10) can be written in the form of Eq. (13). For example, if j=1- and k=1-, then µ=1+iαz and ν=0. If j=1- and k=1+, then µ=0 and ν=iαz. By extending the analysis of [63], one finds that the quadrature correlations

〈 δ q_{j} (θ_{j}) δ q_{k} (θ_{k}) 〉 = \sum_{l} \frac{(μ_{j l} e^{- i θ_{j}} + ν_{j l}^{*} e^{i θ_{j}}) (μ_{kl}^{*} e^{i θ_{k}} + ν_{k l} e^{- i θ_{k}})}{2} .

The input correlations are real by construction. Because the quadratures commute, 〈δ_qjδ_qk〉=〈δ_qkδ_qj〉. It follows from these facts and Eq. (14) that the output correlations are also real. For the case in which j=k, Eq. (14) reduces to Eq. (40) of [63]. (For reference, the quadratures defined above are larger than those defined in [63] by a factor of 2^1/2. Both normalizations appear in the literature [61, 62].)

By combining Eqs. (7)–(10) with Eq. (14), one finds that the quadrature variances

〈 δ q_{1 \pm}^{2} (θ_{1 \pm}) 〉 = \frac{[1 + 2 (α^{2} + β^{2}) z^{2}]}{2},

〈 δ q_{2 \pm}^{2} (θ_{2 \pm}) 〉 = \frac{[1 + 2 (β^{2} + γ^{2}) z^{2}]}{2}

and the quadrature correlations

〈 δ q_{1 -} (θ_{1 -}) δ q_{1 +} (θ_{1 +}) 〉 = α z \sin (θ_{1 -} + θ_{1 +}) - (α^{2} + β^{2}) z^{2} \cos (θ_{1 -} + θ_{1 +}),

〈 δ q_{1 -} (θ_{1 -}) δ q_{2 -} (θ_{2 -}) 〉 = β (α + γ) z^{2} \cos (θ_{1 -} - θ_{2 -}),

〈 δ q_{1 -} (θ_{1 -}) δ q_{2 +} (θ_{2 +}) 〉 = β z \sin (θ_{1 -} + θ_{2 +}) - β (α + γ) z^{2} \cos (θ_{1 -} + θ_{2 +}),

〈 δ q_{1 +} (θ_{1 +}) δ q_{2 -} (θ_{2 -}) 〉 = β z \sin (θ_{1 +} + θ_{2 -}) - β (α + γ) z^{2} \cos (θ_{1 +} + θ_{2 -}),

〈 δ q_{1 +} (θ_{1 +}) δ q_{2 +} (θ_{2 +}) 〉 = β (α + γ) z^{2} \cos (θ_{1 +} - θ_{2 +}),

〈 δ q_{2 -} (θ_{2 -}) {δ q_{2 +}}_{} (θ_{2 +}) 〉 = γz \sin (θ_{2 -} + θ_{2 +}) - (β^{2} + γ^{2}) z^{2} \cos (θ_{2 -} + θ_{2 +}) .

Although the variances are phase independent, the correlations are phase dependent.

For the case in which P ₁≫P ₂(α≫β≫γ) and the phases are equal, Eqs. (15) and (17) reduce to

〈 δ q_{1 \pm}^{2} (θ) 〉 \approx \frac{[1 + 2 {(α z)}^{2}]}{2},

〈 δ q_{1 -} (θ) δ q_{1 +} (θ) 〉 \approx (α z) \sin (2 θ) - {(α z)}^{2} \cos (2 θ) .

These results, which characterize the MI of pump 1, are consistent with the results of [63]. Although modes 1+ and 1- are not squeezed separately (their quadrature variances are phase-independent), they are strongly correlated (so superpositions of modes 1+ and 1- have phase-dependent variances). Hence, the MI of pump 1 is a two-mode squeezing interaction [61, 62]. Similar formulas characterize the MI of pump 2 (α is replaced γ).

For the case in which P ₁=P ₂ (α=β=γ) and the phases are equal, Eqs. (15) and (16) reduce to

〈 δ q_{1 \pm}^{2} (θ) 〉 = \frac{[1 + 4 {(z')}^{2}]}{2},

〈 δ q_{2 \pm}^{2} (θ) 〉 = \frac{[1 + 4 {(z')}^{2}]}{2},

where z′=βz, and Eqs. (17)–(22) reduce to

〈 δ q_{1 -} (θ) δ q_{1 +} (θ) 〉 = z' \sin (2 θ) - 2 {(z')}^{2} \cos (2 θ),

〈 δ q_{1 -} (θ) δ q_{2 -} (θ) 〉 = {2 (z')}^{2},

〈 δ q_{1 -} (θ) δ q_{2 +} (θ) 〉 = z' \sin (2 θ) - 2 {(z')}^{2} \cos (2 θ),

〈 δ q_{1 +} (θ) δ q_{2 -} (θ) 〉 = z' \sin (2 θ) - 2 {(z')}^{2} \cos (2 θ),

〈 δ q_{1 +} (θ) δ q_{2 +} (θ) 〉 = 2 {(z')}^{2},

〈 δ q_{2 -} (θ) δ q_{2 +} (θ) 〉 = z' \sin (2 θ) - 2 {(z')}^{2} \cos (2 θ) .

The variances and correlations associated with the MI of pump 1, and the four-mode interaction driven by pumps with equal powers, are illustrated in Fig. 3. The input modes have (vacuum-level) variances of 1/2, and are uncorrelated.As distance increases, so also do the variances and correlations. Although the MI and four-mode results are similar, the latter process is noisier than the former [Eqs. (23), (25) and (26)].

Fig. 3. Quadrature variances and correlations, normalized to the input variance 1/2 and measured in dB, plotted as functions of distance. (a) MI of pump 1, which involves modes 1- and 1+. The solid curve denotes the variance of either mode, whereas the dashed curve denotes the correlation between the modes [Eqs. (23) and (24)]. The local-oscillator phase θ=π/2 and the distance parameter is γ_KP₁z. Similar results apply to the interaction between the superposition modes b ₊ and c ₊, for which the distance parameter is γ_K(P ₁+P ₂)z [Eqs. (37)–(39)]. (b) Four-mode interaction driven by pumps with equal powers. The solid curve denotes the variance of any mode, whereas the dashed curve denotes the correlation between any pair of modes [Eqs. (25)–(32)]. The phase is π/2 and the distance parameter is γ_KPz.

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Equations (27)–(32) show that mode 1- has the same correlation properties as mode 2-, and mode 1+ has the same properties as mode 2+. These results prompt consideration of the sum and difference modes

b_{\pm} = \frac{(a_{1 -} \pm a_{2 -})}{2^{\frac{1}{2}}},

c_{\pm} = \frac{(a_{1 +} \pm a_{2 +})}{2^{\frac{1}{2}}},

which satisfy the canonical commutation relations. Because the quadrature deviations are linear functions of the mode operators and their conjugates, one can deduce the correlations between the superposition modes b _± and c _± from the correlations between their constituent modes. For example,

2 〈 δ q_{b +} δ q_{c \pm} 〉 = 〈 δ q_{1 -} δ q_{1 +} 〉 \pm 〈 δ q_{1 -} δ q_{2 +} 〉 + 〈 δ q_{2 -} δ q_{1 +} 〉 \pm 〈 δ q_{2 -} δ q_{2 +} 〉,

2 〈 δ q_{b -} δ q_{c \pm} 〉 = 〈 δ q_{1 -} δ q_{1 +} 〉 \pm 〈 δ q_{1 -} δ q_{2 +} 〉 - 〈 δ q_{2 -} δ q_{1 +} 〉 \mp 〈 δ q_{2 -} δ q_{2 +} 〉 .

By proceeding in this way, one finds that

〈 δ q_{b +}^{2} (θ) 〉 = \frac{[1 + 2 {(2 z')}^{2}]}{2},

〈 δ q_{c +}^{2} (θ) 〉 = \frac{[1 + 2 {(2 z')}^{2}]}{2},

〈 δ q_{b +} δ q_{c +} (θ) 〉 = (2 z') \sin (2 θ) - {(2 z')}^{2} \cos (2 θ) .

Equations (37)–(39) are similar to Eqs. (23) and (24), which describe the MI of pump 1 (αz is replaced by 2z′). Hence, the sum modes participate in a two-mode squeezing interaction. One also finds that

〈 δ q_{b -}^{2} (θ) 〉 = \frac{1}{2},

〈 δ q_{c -}^{2} (θ) 〉 = \frac{1}{2},

〈 δ q_{b -} (θ) δ q_{c -} (θ) 〉 = 0 .

Furthermore, the correlations between modes b _- or c _-, and modes b ₊ or c ₊, are all zero. These results show that the difference modes are inert. (Their variances and correlations are characteristic of vacuum fluctuations.)

4. Superposition modes

For the special case in which the pump powers are equal, one can simplify the four-mode equations by rewriting them in terms of two sum modes, which participate in a two-mode squeezing interaction, and two difference modes, which do not interact with any other mode (sum or difference). One can also simplify these equations for the general case in which the pump powers are unequal. Define the distance parameter z′=βz and the normalizedHamiltonian H′_a=H_a/β. Then da_j/dz′=i[a_j,H′_a]. Now define the superposition modes

b_{+} = ε (σ a_{1 -} + a_{2 -}), b_{-} = ε (a_{1 -} - σ a_{2 -}),

c_{+} = ε (σ a_{1 +} + a_{2 +}), c_{-} = ε (a_{1 +} - σ a_{2_{+}}),

where the pump-strength parameter σ=α/β=β/γ and the normalization coefficient ε=1/(1+σ²)^1/2. Then b _± and c _± satisfy the canonical commutation relations. The + modes are coupled to each other, but not the - modes:

b_{+} (z') = [1 + i (σ + \frac{1}{σ}) z'] b_{+} (0) + i (σ + \frac{1}{σ}) z' c_{+}^{†} (0),

c_{+}^{†} (z') = - i (σ + \frac{1}{σ}) z' b_{+}^{†} (0) + [1 - i (σ + \frac{1}{σ}) z'] c_{+} (0) .

In contrast, the - modes are inert:

b_{-} (z') = b_{-} (0),

c_{-} (z') = c_{-} (0) .

Now let z″=(σ+1/σ)z′. Then Eqs. (45) and (46) describe solutions of the two-mode equations

\frac{{db}_{+}}{dz ″} = {ib}_{+} + {ic}_{+}^{†},

\frac{{dc}_{+}^{†}}{dz ″} = - {ib}_{+} - {ic}_{+}^{†},

which produce two-mode squeezed states. [Had we used definitions (43) and (44) in the previous section, instead of definitions (33) and (34), we would have obtained the variance and correlation equations (37)–(39), with 2z′ replaced by z″.]

By making an appropriate change of basis, one can convert a two-mode squeezed state into two one-mode squeezed states [61, 62]: The degree of correlation (entanglement) between two modes depends on the basis that defines them [64]. It follows from Eqs. (49) and (50) that the alternative superposition modes

r = \frac{(b_{+} + c_{+})}{2^{\frac{1}{2}}},

s = \frac{(b_{+} - c_{+})}{2^{\frac{1}{2}}}

satisfy the uncoupled one-mode equations

\frac{dr}{d z ″} = ir + i r^{†},

\frac{ds}{d z ″} = is - i s^{†},

which have the solutions

r (z ″) = (1 + i z ″) r (0) + i z ″ r^{†} (0),

s (z ″) = (1 + i z ″) s (0) - i z ″ s^{†} (0) .

The preceding analysis shows that one can think of a two-mode squeezing interaction as two separate one-mode squeezing interactions followed by a beam-splitter-like process, which combines the two outputmodes. One can also think of the four-mode interaction as a two-mode squeezing interaction followed by two separate beam-splitter-like processes, in which the two output modes are combined with two separate vacuum modes.

5. Photon-number correlations

Not only does the use of superposition modes elucidate the nature of two- and four-mode interactions, it also facilitates the derivation of number-state expansions for the state vectors. By rewriting H′_a in terms of the first set of superposition modes, which were defined in Eqs. (43) and (44), one (eventually) obtains the transformed Hamiltonian

H_{bc}^{'} = (σ + \frac{1}{σ}) (b_{+}^{†} b_{+} + c_{+}^{†} c_{+} + b_{+}^{†} c_{+}^{†} + b_{+} c_{+}) .

Consistent with Eqs. (47) and (48), H′_bc does not depend on b _- or c _-. The form of Eqs. (2) and (57) prompts the definitions z″=(σ+1/σ)z′ and H″_bc=H′_bc/(σ+1/σ). By rewriting H″_bc in terms of the second set of superposition modes, which were defined in Eqs. (51) and (52), one obtains the alternative transformed Hamiltonian

H_{rs}^{″} = r^{†} r + \frac{[{(r^{†})}^{2} + r^{2}]}{2} + s^{†} s - \frac{[{(s^{†})}^{2} + s^{2}]}{2} .

Consistent with Eqs. (53) and (54), the r and s terms in H″_rs are separate.We denote them by H″_r and H″_s, respectively. In the rest of this section, the superscript ″ will be omitted for simplicity. It follows from Eq. (2) that the aforementionedHamiltonians are constant operators. Hence, the input and output states are related by the equation |ψ(z)〉=exp(iHz)|ψ(0)〉.

Hamiltonians (57) and (58) differ from the standard two- and one-mode Hamiltonians [61, 62] because of CPM. One can write H_r=K ₊+K _-+2K ₃-1/2, where the operators K ₊=(r^†)²/2, K _-=r ²/2 and K ₃=(r ^† r+rr ^†)/4. These operators satisfy the angular-momentum-like commutation relations [K ₊,K _-]=-2K ₃ and [K ₃,K _±]=±K _±. By using a standard operator-ordering theorem [61], which is proved in Appendix B, one finds that

\exp (i H_{r} z) = \exp (γ_{+} K_{+}) \exp (γ_{3} K_{3}) \exp (γ_{-} K_{-}),

where γ _±=iz/(1-iz), γ ₃=-2ln(1-iz) and the phase factor exp(-iz/2) was omitted. If the input is the one-mode vacuum state |0〉, then the output is the squeezed state

∣ r 〉 = \frac{1}{{(1 - iz)}^{\frac{1}{2}}} \sum_{n = 0}^{\infty} {(\frac{iz}{1 - iz})}^{n} \frac{{[(2 n)!]}^{\frac{1}{2}}}{2^{n} n!} ∣ 2 n 〉,

where the basis vectors |2n〉=(r ^†)²ⁿ|0〉/[(2n)!]^1/2. The number-state expansion of |s〉 is similar. [In the numerator of Eq. (60), iz is replaced by -iz.]

One can also write H_bc=K ₊+K _-+2K ₃-1, where the operators K+=b ^† ₊ c ^† ₊,K _-=b ₊ c ₊ and K ₃=(b ^† ₊ b ₊+c ₊ c ^† ₊)/2. These operators satisfy the commutation relations stated in the previous paragraph. By using the same operator-ordering theorem, one finds that exp(iH_bcz) can also be written in the form of Eq. (59), where γ _±=iz/(1-iz), γ ₃=-2ln(1-iz) and the phase factor exp(-iz) was omitted. If the input is the two-mode vacuum state |0,0〉, then the output is the squeezed state

∣ b_{+}, c_{+} 〉 = \frac{1}{1 - i z} \sum_{n = 0}^{\infty} {(\frac{i z}{1 - i z})}^{n} ∣ n, n 〉,

where the basis vectors $∣ n, n 〉 = \frac{{(b_{+}^{†} c_{+}^{†})}^{n} ∣ 0, 0 〉}{n!}$ . These vectors are consistent with the two-mode MRW equation $〈 n_{b_{+}} 〉 = 〈 n_{c_{+}} 〉$ , which is a limit of the four-modeMRW equations derived in Appendix A. By using the inverses of Eqs. (51) and (52), one can show that formula (61) is equivalent to the direct product of formula (60) and its analog for |s〉.

The properties of the two-mode state described by Eq. (61) are illustrated in Fig. 4. The probability that there are n photons in each of superposition modes b ₊ and c ₊ is

A (n, z) = \frac{z^{2 n}}{{(1 + z^{2})}^{n + 1}} .

This probability distribution (PD) is plotted as a function of photon number in Fig. 4(a), for short, intermediate and long distances. As distance increases, so also do the probabilities of many-photon states. The related probability that there are k photons in mode b ₊ and l photons in mode c ₊ is

Q (k, l, z) = A (k, z) δ (k, l),

where δ(k, l) is the Kronecker delta. This joint PD is plotted in Fig. 4(b) for the intermediate distance. Figure 4(b) illustrates the simple photon-number correlation that exists between the superposition modes. These results are similar to the standard results for unstable two-mode interactions [61, 62], such as MI and PC.

The four-mode formula follows from Eq. (61) and the binomial expansions of ${(b_{+}^{†})}^{n} = ε^{n} {(σ a_{1 -}^{†} + a_{2 -}^{†})}^{n}$ and ${(c_{+}^{†})}^{n} = ε^{n} {(σ a_{1 +}^{†} + a_{2 +}^{†})}^{n}$ . The result is

∣ ψ (z) 〉 = \frac{1}{1 - i z} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \sum_{l = 0}^{n} {(\frac{i z}{1 - i z})}^{n} \frac{σ^{k + l} n! ∣ k, l, n - k, n - l 〉}{{(1 + σ^{2})}^{n} {[k! l! (n - k)! (n - l)!]}^{\frac{1}{2}}},

where the basis vectors $∣ k, l, n - k, n - l 〉 = \frac{{(a_{1 -}^{†})}^{k} {(a_{1 +}^{†})}^{l} {(a_{2 -}^{†})}^{n - k} {(a_{2 +}^{†})}^{n - l} ∣ 0, 0, 0, 0 〉}{{[k! l! (n - k)! (n - l)!]}^{\frac{1}{2}}} .$ . These vectors are consistent with the four-mode MRW equations (79)–(81). If σ≫1, the only terms that contribute significantly are those for which k=n and l=n. In this case,

∣ ψ (z) 〉 \approx \frac{1}{1 - i z} \sum_{n = 0}^{\infty} {(\frac{i z}{1 - i z})}^{n} ∣ n, n, 0, 0 〉,

Fig. 4. (a) Probability (in dB) that there are n photons in each of modes b+ and c+ [Eq. (62)]. The dashed, dot-dashed and solid lines represent the distance parameters γ_K(P ₁+P ₂)z=0.3, 1.0 and 3.0, respectively. (b) Joint probability distribution (PD) of modes b+ and c+ [Eq. (63)] for the intermediate distance. These modes are correlated.

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which is the two-mode state produced by the MI of pump 1. A similar result applies to the case in which σ≪1. (Number states of the form |0,0,n,n〉 are produced by the MI of pump 2.) For the intermediate case in which σ≈1,

∣ ψ (z) 〉 \approx \frac{1}{1 - i z} \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \sum_{l = 0}^{n} {(\frac{i z}{1 - i z})}^{n} \frac{n! ∣ k, l, n - k, n - l 〉}{2^{n} {[k! l! (n - k)! (n - l)!]}^{\frac{1}{2}}} .

In this parameter range, the functions σ+1/σ and σ ^k+l/(1+σ ²)ⁿ depend only weakly on σ, so the pump powers need only be comparable, not equal: Four-mode correlations are robust.

The properties of the four-mode state described by Eq. (66) are illustrated in Figs. 5–7. The total probability that there are m photons in mode 1- is

P_{t} (m, z) = \sum_{n = m}^{\infty} A (n, z) B (n, m, n - m),

where A(n, z) was defined in Eq. (62) and

B (n, k, l) = \frac{n!}{(2^{n} k! l!)} .

B(n,k,n-k) is a binomial distribution, which has the property ∑ⁿ _k=0 B(n,k,n-k)=1. PD (67) is plotted as a function of photon number in Fig. 5, for short, intermediate and long distances. As distance increases, so also do the probabilities of many-photon states. Although there are qualitative similarities between Figs. 4(a) and 5, there are also quantitative differences.

The conditional probability that there are k photons in mode 1_- and l photons in mode 1₊, given that there are n photons in each of the superposition modes, is

Q_{c} (k, l) = B (n, k, n - k) B (n, l, n - l) .

This joint PD is plotted in Fig. 6(a) for the intermediate distance. If the MI of pump 1 were to occur in isolation, the probability would be nonzero only if k=4 and l=4. It is clear from the figure that the couplings to modes 2- and 2+, which are enabled by pump 2, have significant effects. Because formula (66) depends symmetrically on l and n-l, the joint PD of modes 1-and 2+ is the same as the PD for modes 1- and 1+. The conditional probability that there are k photons in mode 1- and l photons in mode 2-, given that there are n photons in each of the superposition modes, is

Fig. 5. Total probability (in dB) that there are n photons in mode 1-[Eq. (67)]. The dashed, dot-dashed and solid lines represent the distance parameters 2γ_KPz=0.3, 1.0 and 3.0, respectively. The PDs of modes 1+, 2- and 2+ are identical.

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R_{c} (k, l) = B (n, k, n - k) δ (n - k, l) .

This joint PD is plotted in Fig. 6(b) for the intermediate distance. Modes 1- and 2- are anti-correlated.

Fig. 6. (a) Joint PD (in dB) of modes 1- and 1+ [Eq. (69)] for the n=4 state and the intermediate distance-parameter 2γ_KPz=1.0. The joint PD of modes 1- and 2+ is identical. (b) Joint PD of modes 1- and 2- [Eq. (70)] for the same state and distance. These modes are anti-correlated.

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The total probability that there are k photons in mode 1- and l photons in mode 1+ is

Q_{t} (k, l, z) = \sum_{n = max (k, l)}^{\infty} A (n, z) B (n, k, n - k) B (n, l, n - l) .

This joint PD is plotted in Fig. 7(a) for the intermediate distance. The total probability that there are k photons in mode 1- and l photons in mode 2- is

R_{t} (k, l, z) = A (k + l, z) B (k + l, k, l) .

This joint PD is plotted in Fig. 7(b) for the intermediate distance. A comparison of Figs. 4(b) and 7(a) shows that the couplings to modes 2- and 2+ changes significantly the correlation properties of modes 1- and 1+.

Fig. 7. (a) Joint PD (in dB) of modes 1- and 1+ [Eq. (71)] for the intermediate distance-parameter 2γK_Pz=1.0, which should be compared to the PD shown in Fig. 4(b). The joint PD of modes 1- and 2+ is identical. (b) Joint PD of modes 1- and 2- [Eq. (72)] for the same distance.

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6. Entanglement

Consider the pure two-mode state (61). From a mathematical standpoint, the absence of the off-diagonal terms c_mn|m〉|n〉 prevents |b ₊,c ₊〉 from being written as the direct product |b ₊〉|c ₊〉. Hence, it is an entangled state. From a physical standpoint, if one measures the number of photons in mode c ₊, one determines the number of photons in mode b ₊: Measuring the second mode affects the result of a subsequent measurement of the first mode. For any state vector |ψ〉, the associated density operator (matrix) ρ=|ψ〉〈ψ|. The reduced density matrix (RDM) ρ _b+=Tr_c+(ρ), where Tr denotes a trace, characterizes the properties of mode b+, without regard to mode c+. (The effects of the different states of mode c+ are ensemble averaged.) By following a standard procedure, which is described in Appendix C, one finds that

ρ_{b +} (z ″) = \sum_{n = 0}^{\infty} A (n, z ″) ∣ n 〉 〈 n ∣,

where A(n, z″) is the probability that there are n photons in mode b+ at the distance z″=γ_K(P ₁+P ₂)z. The formula for ρ _c+ is identical. Because the two-mode state (61) is pure, the fact that both RDMs describe one-mode mixed states confirms that state (61) is entangled. For the MI of pump 1, which involves modes 1- and 1+, ρ _1- is given by the same formula, with z″=γ_KP₁z (because pump 2 is absent): This MI also produces a two-mode entangled state.

Now consider the pure four-mode state (66). To prove that it is fully four-partite entangled, one must prove that every partition of it is entangled. Let S denote a subset of the four modes and T denote the complimentary subset. Because the state vector depends symmetrically on k and l, one can interchange the subscripts 1- and 1+, and 2- and 2+, without changing the properties of a partition. Hence, the independent partitions {S|T} are {1-|1+,2-,2+}, {2-|2+,1-,1+}, {1-,1+|2-,2+}, {1-,2-|1+,2+} and {1-,2+|1+,2-}. For the case in which the pump powers are equal, the state vector depends symmetrically on k and n-k, and l and n-l, so one can interchange the subscripts 1- and 2-, or 1+ and 2+, without changing the properties of the partition. In this case, the second partition is equivalent to the first, and the fifth is equivalent to the third. Consider the three independent partitions in turn. For the first partition, if one measures the photon numbers of modes 1+, 2- and 2+, one determines l, n and k, and, hence, the photon number of mode 1-. For the third partition, if one measures the photon numbers of modes 2- and 2+, one determines the difference k-l and, hence, the difference between the photon numbers of modes 1- and 1+. For the fourth partition, if one measures the photon numbers of modes 1+ and 2+, one determines the sum of the photon number of modes 1- and 2-. In each case, the stated measurement on T affects a subsequent measurement on S. Hence, state (66) is fully four-partite entangled. To determine the RDM ρ _1-, one has to trace out modes 1+, 2- and 2+. By following a standard procedure, which is described in Appendix C, one (eventually) finds that

ρ_{1 -} (z ″) = \sum_{m = 0}^{\infty} P_{t} (m, z ″) ∣ m 〉 〈 m ∣,

where P_t(m, z″) is the total probability that there are m photons in mode 1- at the distance z″=2γ_KPz. Equation (74) describes a one-mode mixed state, which confirms that state (66) is entangled. The RDMs associated with the other partitions, which are determined in Appendix C, also describe mixed states. These results confirm that state (66) is fully four-partite entangled.

The degrees of two-partite entanglement of a pure state are measured by the entropies E=-Tr(ρ_Slogρ_S), where ρ_S is the RDM associated with the subset S and logρ_S is defined by its Taylor series. The RDMs (73) and (74) are both written in the diagonal form ρ=∑^∞ _n=0 p_n|n〉〈n|, for which the associated entropy E=-∑^∞ _n=0 p_n log(p_n). Two-and fourmode entropies are plotted as functions of distance in Fig. 8. For all distances, the two-mode entropy of mode b ₊ is higher than the four-mode entropy of 1-. It follows from Eqs. (43) that modes 1- and b ₊ are related by the beam-splitter-like equation a _1-=ε(σ b ₊ +b_-), where mode b _- is inert. Combining an entangled state with a vacuum (unentangled) state dilutes the entanglement, so it makes sense that the entropy of mode 1- is lower than that of b ₊. However, the entropy of mode 1- is higher when it participates in the four-mode interaction than when it participates in the MI of pump 1, because the nonlinear coupling for the four-mode interaction is stronger than the coupling for the MI (pump 2 is absent).

Fig. 8. Entanglement (entropy) of mode 1- plotted as a function of the distance parameter 2γ_KPz. The dashed and solid curves denote the two-mode interaction with mode 1+ [Eq. (73)] and the four-mode interaction with 1+, 2- and 2+ [Eq. (74)], respectively. For comparison, the dot-dashed curve denotes the entropy of mode b ₊, which interacts with mode c ₊ [also Eq. (73)].

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7. Summary

Multipartite entanglement is a resource for quantum communication and computation. Its use has been demonstrated in dense coding, key distribution, one-way computation and teleportation networking. The standard way to produce such entanglement is to combine several sources of one-color, two-product-mode entanglement,which are based on parametric down-conversion in crystals, and combine their outputs using a path-stabilized array of beam splitters. Three-color, pump- and product-mode entanglement is also being studied [65].

Vector four-wave mixing (FWM) in a fiber, driven by two strong pumps, couples the evolution of four weak sidebands (modes). This four-mode interaction is a combination of six two-mode interactions (modulation instability, phase conjugation and Bragg scattering), which occur simultaneously when dispersion is weak. Two of the mode frequencies are similar to the frequency of pump 1, and two are similar to the frequency of pump 2. A fiber with one zero-dispersion frequency (ZDF), supports an interaction between modes with similar frequencies (typically separated by about 1 THz), whereas a fiber with two ZDFs supports an interaction between modes with dissimilar frequencies (typically separated by more than 10 THz).

In this report, the discrete- and continuous-variable entanglement produced by vector FWM was studied in detail. Formulas were derived for the variances of, and correlations between, the mode quadratures [Eqs. (25)–(32)]. These formulas showed that the modes are strongly correlated, and prompted the reformulation of the interaction in terms of superposition (sum and difference) modes. The sum modes participate in a two-mode interaction [Eqs. (37)–(39)], whereas the difference modes are inert [Eqs. (40)–(42)]. This result allows one to interpret the four-mode interaction as a two-sum-mode interaction followed by two beam-splitter-like processes, which mix the output (sum) modes with vacuum (difference) modes. The number-state expansion of the state vector was also derived [Eq. (66)]. This formula showed that the interaction produces four-partite entanglement. It also enabled the derivation of formulas for the unconditional and conditional photon-number distributions of the modes [Eqs. (67) and (69)–(72)].

In summary, vector FWM in a fiber produces four-color, four-partite entanglement naturally: Multiple sources of two-partite entanglement and path-stabilized arrays of beamsplitters are not necessary. In future work, we will analyze the van Loock-Furasawa inequalities for multipartite entanglement of Gaussian states [66] to determine the best types of measurements for verifying and characterizing the entanglement produced in experiments. (Losses cause the output states to be mixed, not pure.) Further work is also required to determine the practicality of four-color entanglement in schemes for quantum communication and computation.

Acknowledgments

The research of MR and SR was supported by the National Science Foundation under contracts ECS-0621723 and ECS-0406379, respectively.

A: Manley–Rowe–Weiss equations

Let n_j=a ^† _j a_j be the photon-number operator of mode (sideband) j. (In the main text, n was used to denote a photon number.) Then Eqs. (3)–(6) imply that

d_{z} n_{1 -} = i α (a_{1 -}^{†} a_{1 +}^{†} - a_{1 -} a_{1 +}) + i β (a_{1 -}^{†} a_{2 -} - a_{1 -} a_{2 -}^{†})

d_{z} n_{1 +} = i α (a_{1 -}^{†} a_{1 +}^{†} - a_{1 -} a_{1 +}) + i β (a_{1 +}^{†} a_{2 -}^{†} - a_{1 +} a_{2 -})

d_{z} n_{2 -} = iγ (a_{2 -}^{†} a_{2 +}^{†} - a_{2 -} a_{1 +}) + i β (a_{1 -} a_{2 -}^{†} - a_{1 -}^{†} a_{2 -})

d_{z} n_{2 +} = iγ (a_{2 -}^{†} a_{2 +}^{†} - a_{2 -} a_{2 +}) + i β (a_{1 -}^{†} a_{2 +}^{†} - a_{1 -} a_{2 +})

By combining Eqs. (75)–(78), one obtains the Manley–Rowe–Weiss equations

d_{z} (n_{1 -} - n_{1 +}) = d_{z} (n_{2 +} - n_{2 -}),

d_{z} (n_{1 -} + n_{2 -}) = d_{z} (n_{1 +} + n_{2 +}),

d_{z} (n_{1 -} - n_{2 +}) = d_{z} (n_{1 +} - n_{2 -}) .

Equations (79)–(81) are not independent. They are different ways of expressing the same constraint. The terms on the left side of Eq. (79) pertain to the MI of pump 1, in which 2γ ₁→γ ₁- +γ ₁₊, whereas the terms on the right side pertain to the MI of pump 2, in which 2γ ₂→γ _2-+γ ₂₊. If these processes were to occur in isolation, 〈n _1-〉-〈n ₁₊〉 and 〈n _2-〉-〈n ₂₊〉would be constants, because sideband photons are produced in pairs. The left side of Eq. (80) pertains to the BS process in which γ _1- +γ ₂ →γ ₁ +γ _2-, whereas the right side pertains to the process in which γ ₁₊+γ ₂→γ ₁+γ ₂₊. If these processes were to occur in isolation, 〈n _1-〉+〈n _2-〉 and 〈n ₁₊〉+〈n ₂₊〉 would be constants, because photons are exchanged between the sidebands. The left side of Eq. (81) pertains to the PC process in which γ ₁+γ ₂→γ _1-+γ ₂₊, whereas the right side pertains to the process in which γ ₁+γ ₂ →γ ₁₊+γ _2-. If these processes were to occur in isolation, 〈n _1-〉-〈n ₂₊〉 and 〈n ₁₊〉-〈n _2-〉 would be constants, because sideband photons are produced in pairs. In the presence of four-sideband coupling, the aforementioned combinations of photon numbers are not constant. They evolve, subject to constraints (79)–(81).

B: Operator-ordering theorem

The main results of this report, Eqs. (60), (61) and (64), were obtained by the use of an operator-ordering theorem (OOT). Although such theorems are common in the quantum-optics literature [61], they are not common in the optical-communications literature. Consequently, in this appendix the OOT (59) will be proved from first principles.

The proof of this OOT relies on the Baker–Campbell–Hausdorff (BCH) lemma

\exp (a) b \exp (- a) = \sum_{n = 0}^{\infty} \frac{{[a, b]}_{n}}{n!},

where a and b are operators, and the nth-order commutator [a,b]n is defined recursively: [a,b]₀=b, [a,b]₁=[a,b] and [a,b]_n=[a, [a,b]_n-1]. By expanding the exponentials on the left side of Eq. (82) in Taylor series, one finds that

\exp (a) b \exp (- a) = \sum_{n = 0}^{\infty} \sum_{m = 0}^{n} \frac{a^{m} b {(- a)}^{n - m}}{m! (n - n)!} .

Equations (82) and (83) are equivalent if and only if the commutator

{[a, b]}_{n} = \sum_{m = 0}^{n} \frac{{n! a}^{m} b {(- a)}^{n - m}}{m! (n - m)!} .

It is easy to verify that Eq. (84) is valid for n=0 and n=1 (the first non-trivial order). Suppose that it is valid for some order n≥1. Then the next-order commutator

{a [a, b]}_{n} - {[a, b]}_{n} a = \sum_{m = 0}^{n} \frac{n! [a^{m + 1} {b (- a)}^{n - m} - {a^{m} b (- a)}^{n - m} a]}{m! (n - m)!},

= \sum_{m = 1}^{n + 1} \frac{{{n! a}^{m} b (- a)}^{n + 1 - m}}{(m - 1)! (n + 1 - m)!} + \sum_{m = 0}^{n} \frac{{{n! a}^{m} b (- a)}^{n + 1 - m}}{m! (n - m)!} .

There are three cases to consider. First, for m=0 the second series on the right side of Eq. (86) contributes the term a ⁰ b(-a)ⁿ⁺¹=(n+1)!a ⁰ b(-a)ⁿ⁺¹/[0!(n+1)!]. Second, for m=n+1 the first series contributes the term a ⁿ⁺¹ b(-a)⁰=(n+1)!a ⁿ⁺¹ b(-a)0/[0!(n+1)!]. Third, for 1≤m≤n, the combined contribution is

\frac{{{n! a}^{m} b (- a)}^{n + 1 - m}}{(m - 1)! (n - m)!} [\frac{1}{n + 1 - m} + \frac{1}{m}] = \frac{{(n + 1) {! a}^{m} b (- a)}^{n + 1 - m}}{m! (n + 1 - m)!} .

In each case, the mth-term in the expansion of the commutator [a,b]_n+1 has the form required by Eq. (84). This result proves the BCH lemma (82).

Equation (59) provides a normally-ordered formula for the Schrödinger evolution-operator exp(iHz), where H is a Hamiltonian and z is a distance variable. In this report H=K ₊+2K ₃+K _-, where the operators K _± and K ₃ satisfy the commutation relations [K ₊,K _-]=-2K ₃ and [K ₃,K _±]=±K _±. (Formulas for these operators were stated in the main text.) Define the function

F (z) = \exp [i (K_{+} + {2 K}_{3} + K_{-}) z] .

Because the K-operators form a closed set under commutation, one can rewrite Eq. (88) in the normally-ordered form

F (z) = \exp [ip (z) K_{+}] \exp [iq (z) K_{3}] \exp [ir (z) K_{-}],

where p, q and r are functions of z (to be determined). It follows from Eq. (88) that

F' = i (K_{+} + {2 K}_{3} + K_{-}) F,

where F′=dF/dz. Likewise, it follows from Eq. (89) that

F' = (i p' K_{+} + I q' e^{i p K_{+}} K_{3} e^{- i p K_{+}} + i r' e^{i p K_{+}} e^{i q K_{3}} K_{-} e^{- i q K_{3}} e^{- i p K_{+}}) F .

By using lemma (82) and the aforementioned commutation relations, one finds that

e^{i p K_{+}} K_{3} e^{- i p K_{+}} = K_{3} - i p K_{+},

e^{i q K_{3}} K_{-} e^{- i q K_{3}} = K_{-} e^{- i q},

e^{i p K_{+}} K_{-} e^{- i p K_{+}} = K_{-} - 2 i p K_{3} - p^{2} K_{+} .

By using these results to simplify Eq. (91), and equating the coefficients of K ₊, K ₃ and K _- in Eqs. (90) and (91), one obtains the differential equations

p' - i p q' - p^{2} (r' e^{- i q}) = 1,

q' - 2 i p (r' e^{- i q}) = 2,

r' e^{- i q} = 1,

respectively. For the boundary (initial) conditions p(0)=0, q(0)=0 and r(0)=0, the solutions of Eqs. (95)–(97) are

i p (z) = \frac{i z}{(1 - i z)},

i q (z) = - 2 \log (1 - i z),

i r (z) = \frac{i z}{(1 - i z)} .

Equations (98)–(100) are consistent with the formulas for γ _± and γ ₃ stated after Eq. (59).

C: Reduced density operators

The two-mode state vector in Eq. (61) can be written in the compact form

∣ ψ 〉 = \sum_{n = 0}^{\infty} a_{n} {∣ n 〉}_{b} {∣ n 〉}_{c},

where a_n(z)=(iz)ⁿ/(1-iz)ⁿ⁺¹, and b and c are abbreviations for b ₊ and c ₊, respectively. The associated density operator (matrix) ρ=|ψ〉〈ψ|. By combining this definition with Eq. (101), one finds that

ρ = \sum_{n = 0}^{\infty} \sum_{n' = 0}^{\infty} a_{n} a_{n'}^{*} {∣ n 〉}_{b} {∣ n 〉}_{c} {〈 n' ∣}_{b} {〈 n' ∣}_{c} .

The reduced density matrix (RDM) ρ_b=Tr_c(ρ)=∑^∞ _n″=0〈n″|cρ|n″|c characterizes the properties of mode b, without regard to mode c.When one calculates this partial trace, one encounters the summations ∑^∞ _n′=0∑^∞ _n″=0〈n″|n〉c〈n′|n″〉c=∑^∞ _n′=0∑^∞ _n″=0〈n′|n″〉c〈n″|n〉c=∑^∞ _n′=0 δ _n′n, where δ_ij is the Kronecker delta. By applying this result to Eq. (102), one obtains the RDM

ρ_{b} = \sum_{n = 0}^{\infty} {∣ a_{n} ∣}^{2} {∣ n 〉}_{b} {〈 n ∣}_{b} .

Equation (103) is equivalent to Eq. (73). It describes a mixed state, because there is no coupling between the eigenstates associated with different photon numbers n. For reference, the preceding analysis shows that the identity Tr(| ϕ〉〈ϕ′|)=〈ϕ′|ϕ〉 is valid even when |ϕ〉 and |ϕ′〉 are parts of a higher-dimensional DM and the operation is a partial trace.

The four-mode state vector in Eq. (66) can be written in the compact form

∣ ψ 〉 = \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \sum_{l = 0}^{n} a_{n} b_{nk} b_{nl} {∣ k 〉}_{1} {∣ l 〉}_{2} {∣ n - k 〉}_{3} {∣ n - l 〉}_{4},

where b_nk=[n!/2ⁿ k!(n-k)!]^1/2 and modes 1-, 1+, 2- and 2+ were relabeled 1, 2, 3 and 4, respectively. This state vector depends symmetrically on k and n-k, and l and n-l. Interchanging either pair of indices causes the same set of terms to appear in reverse order. Hence, one can interchange the mode subscripts 1 and 3, and 2 and 4, separately. The state vector also depends symmetrically on k and l, so one can interchange the subscripts 1 and 2, and 3 and 4, simultaneously. The associated DM

ρ = \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \sum_{l = 0}^{n} \sum_{n' = 0}^{\infty} \sum_{k' = 0}^{n'} \sum_{l' = 0}^{n'} a_{n} b_{nk} b_{nl} (a_{n'} b_{n' k'} b_{n' l'}) *

Define the RDMs ρ_ijk=Tr_l(ρ), ρ_ij=Tr_kl(_ρ) and ρ_i=Tr_jkl(ρ). Then it follows from the aforementioned symmetries that there are only two distinct twice-reduced DMs, ρ ₂₄ and ρ ₃₄, and there is only one distinct thrice-reduced DM, ρ ₄. These RDMs are all reductions of ρ ₂₃₄=Tr₁(ρ). By rewriting the first set of summations in Eq. (105) as ∑^∞ _k=0∑^∞ _n=k∑ⁿ _l=0, rewriting the second set in similar way and tracing out mode 1, one finds that

ρ_{234} = \sum_{k = 0}^{\infty} \sum_{n = k}^{\infty} \sum_{l = 0}^{n} \sum_{n' = k}^{\infty} \sum_{l' = 0}^{n'} a_{n} b_{nk} b_{nl} {(a_{n'} b_{n' k} b_{n' l'})}^{*}

The RDM ρ ₂₄=Tr₃(ρ ₂₃₄). By rewriting the k-, n- and n′-summations in Eq. (106) as ∑^∞ _n=0∑^∞ _n′=0∑^k+ _k=0, where k ₊=min(n,n′), and tracing out mode 3, one finds that

ρ_{24} = \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \sum_{l = 0}^{n} \sum_{l' = 0}^{n} {∣ a_{n} ∣}^{2} {∣ b_{nk} ∣}^{2} b_{nl} b_{nl'}^{*} {∣ l 〉}_{2} {∣ n - l 〉}_{4} {〈 l' ∣}_{2} {〈 n - l' ∣}_{4} .

The binomial identity ∑ⁿ _k=0 |b_nk|²=1 allows one to rewrite Eq. (107) in the simpler form

ρ_{24} = \sum_{n = 0}^{\infty} \sum_{l = 0}^{n} \sum_{l' = 0}^{n} {∣ a_{n} ∣}^{2} b_{nl} b_{nl'}^{*} {∣ l 〉}_{2} {∣ n - l 〉}_{4} {〈 l' ∣}_{2} {〈 n - l' ∣}_{4} .

Let V_ij be the product vector-space of modes i and j. Then Eq. (108) shows that ρ ₂₄ projects subspaces of V ₂₄, in which the total photon number of modes 2 and 4 is n, onto themselves: There is no coupling between the subspaces associated with different values of n. Hence, ρ ₂₄ describes a mixed state.

The RDM ρ ₄=Tr₂(ρ ₂₄). By rewriting the summations in Eq. (108) as ∑^∞ _l=0∑^∞ _l′=0 ∑^∞ _n=n-, where n _-=max(l, l′), and tracing out mode 2, one finds that

ρ_{4} = \sum_{l = 0}^{\infty} \sum_{n = l}^{\infty} {∣ a_{n} ∣}^{2} {∣ b_{nl} ∣}^{2} {∣ n - l 〉}_{4} {〈 n - l ∣}_{4},

which also describes amixed state. By rewriting the summations in Eq. (109) as ∑^∞ _n-l=0 ∑^∞ _n=n-l, and interchanging l and n-l, one obtains the alternative equation

ρ_{4} = \sum_{l = 0}^{\infty} \sum_{n = l}^{\infty} {∣ a_{n} ∣}^{2} {∣ b_{nl} ∣}^{2} {∣ l 〉}_{4} {〈 l ∣}_{4},

which is equivalent to Eq. (74).

The RDM ρ ₃₄=Tr₂(ρ ₂₃₄). By rewriting the k-, n- and n′-summations in Eq. (106) as ∑^∞ _n=0∑^∞ _n′=0∑^k+ _k=0, where k+=min(n,n′), rewriting the l- and n- summations as ∑^∞ _l=0∑^∞ _n=l, rewriting the l′- and n′- summations in a similar way, and tracing out mode 2, one finds that

ρ_{34} = \sum_{l = 0}^{\infty} \sum_{n = l}^{\infty} \sum_{n' = l}^{\infty} \sum_{k = 0}^{k_{+}} (a_{n} b_{nk} b_{nl}) {(a_{n'} b_{n' k} b_{n' l})}^{*} {{∣ n - k 〉}_{3} ∣ n - l 〉}_{4} {〈 n' - k ∣}_{3} {〈 n' - l ∣}_{4} .

Equation (111) shows that ρ ₃₄ projects subspaces of V ₃₄, in which the photon numbers of modes 3 and 4 differ by d=k -l, onto themselves: There is no coupling between the subspaces associated with different values of d. Hence, ρ ₃₄ describes a mixed state.

The RDM ρ ₄=Tr₃(ρ ₃₄). By rewriting the l-, n- and n′-summations in Eq. (111) as ∑^∞ _n=0∑^∞ _n′=0∑^l+ _l=0, where l ₊=min(n,n′), and tracing out mode 3, one finds that

ρ_{4} = \sum_{n = 0}^{\infty} \sum_{l = 0}^{n} \sum_{k = 0}^{n} {∣ a_{n} ∣}^{2} {∣ b_{nk} ∣}^{2} {∣ b_{nl} ∣}^{2} {∣ n - l 〉}_{4} {〈 n - l ∣}_{4} .

By using the binomial identity, interchanging l and n-l, and rewriting the n- and l-summations as ∑^∞ _l=0∑^∞ _n=l, one can rewrite Eq. (112) in the form of Eq. (110). Hence, the RDM ρ ₄ does not depend on the order in which the reductions are made.

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Multicolor multipartite entanglement produced by vector four-wave mixing in a fiber

Abstract

1. Introduction

2. Four-mode equations

3. Quadrature correlations

4. Superposition modes

5. Photon-number correlations

6. Entanglement

7. Summary

Acknowledgments

A: Manley–Rowe–Weiss equations

B: Operator-ordering theorem

C: Reduced density operators

Cited By

Figures (8)

Equations (121)

Optics Express