Mode matching plays an important role in measuring the continuous variable entanglement. For the signal and idler twin beams generated by a pulse pumped fiber optical parametric amplifier (FOPA), the spatial mode matching is automatically achieved in single mode fiber, but the temporal mode property is complicated because it is highly sensitive to the dispersion and the gain of the FOPA. We study the temporal mode structure and derive the input-output relation for each temporal mode of signal and idler beams after decomposing the joint spectral function of twin beams with the singular-value decomposition method. We analyze the measurement of the quadrature-amplitude entanglement, and find mode matching between the multi-mode twin beams and the local oscillators of homodyne detection systems is crucial to achieve a high degree of entanglement. The results show that the noise contributed by the temporal modes nonorthogonal to local oscillator may be much larger than the vacuum noise, so the mode mis-match can not be accounted for by merely introducing an effective loss. Our study will be useful for developing a source of high quality continuous variable entanglement by using the FOPA.
© 2015 Optical Society of America
Continuous variable (CV) entanglement, whose quadrature-phase amplitudes possess quantum correlation, is not only an important non-classical light source for studying quantum effects, but also a powerful resource for quantum information technologies, such as quantum metrology and quantum imaging [1–4]. Quadrature-amplitude entangled optical fields at different wavelength have been successfully generated by invoking non-degenerate parametric process in a variety of nonlinear media [5–8], or by coherently combining two quadrature-amplitude squeezed states at beam splitters [9, 10]. Nevertheless, constructing a high quality and compact source of CV entanglement is still currently quite challenging.
Fiber-optical parametric amplifiers (FOPA), based on the four-wave mixing (FWM) in optical fibers, provide a convenient and powerful tool to realize optical amplification in broadband and flexible wavelength range. On the one hand, all-optical functional operations, such as wavelength conversion, phase-conjugate wave generation, and ultrahigh speed switching, can be realized by utilizing FWM in fibers . On the other hand, FOPA is also a prominent device to generate non-classical lights. Indeed, a variety of quantum states, such as entangled photon pairs, single photons, and twin beams with intensity difference squeezing, have been generated by FOPAs [12–15]. However, the generation of CV entanglement via an FOPA is less studied . This is partly because, for a FOPA based quantum light source, one usually uses pulsed pump to suppress the background noise, such as guided acoustic wave Brillouin scattering (GAWBS) and Raman scattering [17, 18]. As a result, the temporal mode structure of pulse-pumped FOPAs is much more complicated than those pumped with continuous wave (CW) light [19,20]. A relatively thorough theoretical analysis on the generation of CV entanglement via a CW laser pumped FOPA was presented in Ref. , but the complexity of temporal mode is not considered. In our previous work , we attempted to provide a theoretical model of a pulsed pumped FOPA to analyze the intensity difference squeezing of the twin beams, but the heavy numerical simulations had to be carried out due to the mathematical complexity.
The major complexity involved in the pulse-pumped parametric process is the complicated frequency correlation between the signal and idler twin beams . This leads to mixed inter-mode coupling in frequency domain  and thus the mode mismatch between the pulsed local oscillator and signal/idler fields when homodyne detection is utilized for measuring the quadrature components. The effect of mode mismatch in homodyne detection is usually modeled by an effective loss, which introduces the vacuum noise and hence reduces the degree of quantum entanglement of twin beams. It was argued that a factorized joint spectral function of the signal and idler fields is necessary to decouple the frequency correlation and to describe the process in a single temporal mode fashion [23,24]. Recently, Wasilewski et al. utilize the singular value decomposition (SVD) to decompose the fields generated from a pulse-pumped parametric amplifier into orthogonal temporal modes , which leads to decoupling of the modes. However, the decomposition is on the Green functions that are related to the interaction Hamiltonian in a complicated manner. Therefore, the mode analysis has to resort to numerical methods and does not seem to provide useful insight for the experimental investigation. Nevertheless, the decoupling of the temporal modes significantly simplifies the discussions on the quantum nature (noise reduction) of the signal and idler fields.
In this paper, we follow the same idea of Wasilewski et al  for mode decomposition. However, instead of decomposing the Green functions, we apply the SVD method to the so called joint spectral function (JSF) that is directly related to the interaction Hamiltonian and can be directly determined by the experimental parameters of pump and nonlinear medium. We first extend our multi-frequency model in Ref.  and derive the evolution of the decomposed temporal modes, showing that the temporal modes are completely decoupled. We then theoretically study the CV entanglement properties of the twin beams by numerically calculating the degree of entanglement when the experimental parameters of the FOPA and the local oscillators are varied. The result shows that the influence of the temporal mode mismatch on the homodyne detection process are far more complicated than the simple model of an effective detection loss due to the coherent superposition of the temporal modes of twin beams. Therefore, controlling the mode-matching between the temporal modes of the signal/idler beam and the local oscillator of homodyne detection systems is crucial for developing a high quality source of pulsed CV entanglement.
The rest of the paper is organized as follows: In Sec. II, after briefly reviewing our previous work  on the evolution of the output fields generated from a pulse-pumped FOPA, we characterize the temporal mode structure of the signal and idler fields after applying the SVD to JSF and derive the input-output relation of the FOPA in each temporal mode. In Sec. III, we analyze the entanglement properties of signal and idler beams in the decomposed temporal modes. In Sec. IV, we discuss the influence of mode matching when the quadrature entanglement is measured by the homodyne detection system. Finally, we briefly conclude in Sec. V.
2. Temporal mode property of the signal and idler fields generated from a pulse-pumped FOPA
2.1. Multi-frequency mode model
We start by introducing the multi-frequency mode theory developed in Ref. . For the parametric process of four wave mixing (FWM) in a pulse pumped FOPA (see the area in dashed frame in Fig. 1), two pump photons at angular frequencies ωp1 and ωp2 are coupled via χ(3) of the nonlinear fiber to simultaneously create signal and idler photons at frequencies ωs and ωi, respectively, such that ωp1 +ωp2 = ωs +ωi. The Hamiltonian of this process isEqs. (2) and (3) into Eq. (1) and changing dV in Eq. (1) to dV = Aeff dz, we carry out the integral over the whole fiber length L (from 0 to L) and arrive at the Hamiltonian in the time dependent form
Because of the broadband nature of the pulsed pump fields, the output signal (idler) field is a superposition of many amplified input signal (idler) frequency modes , and a general input-output relationships of the operators are Eq. (4), where G ∝ γPpL is the gain coefficient of FWM, and the joint spectrum function (JSF)
2.2. Singular value decomposition and multi-temporal mode model
When the JSF is factorizable, i.e., F(ωs,ωi) = ϕ (ωs)ψ(ωi), the Green functions in Eqs. (8–11) can be significantly simplified since the signal and the idler fields can be described by a single temporal mode model . In general, however, the JSF is not factorizable. In this case, after applying SVD method to JSF, we arrive at 
Accordingly, the input-output relation of the FOPA in Eq. (5) can be simplified as
The operators in Eqs. (19)–(22) can be viewed as the generalized annihilation operators for individual SVD modes described by ϕk(ωs) and ψk(ωi), because they always satisfy the standard commutation relationships for bosons: and . In fact, the single-photon state of , for example, describes a single-photon wave packet having the temporal shape of . Notice that in Eqs. (17) and (18), different modes are decoupled from each other. The operators of signal (idler) field at the input and output of FOPA, Âks(i) and , have the same temporal profile determined by ϕk(ωs) (ψk(ωi)).
2.3. Temporal/spectral property of twin beams generated by the FOPA with broad gain bandwidth in telecom band
Aiming at developing an all-fiber source of CV entanglement by using the commercially available fiber components, which is compact and low cost, we will analyze the temporal/spectral structure of the twin beams generated by a pulse-pumped FOPA in telecom band. We assume the central wavelength of pump is very close to the zero dispersion wavelength of the nonlinear fiber and the gain bandwidth of FWM in the fiber is very broad. In this case, for the sake of convenience, we rewrite Eq. (7) as26] Eq. (24) counterbalance each other by satisfying the condition . So we will adapt this condition in the numerical analysis presented hereinafter.
We first study the temporal mode property of the twin beams by calculating the JSF. For the sake of brevity, all the frequency scales are in the unit of the pump bandwidth σp. In the calculation, the dispersion coefficients and are substituted in Eqs. (23) and (24). Figure 2(a) plot the normalized amplitude of JSF |F(Ωs,Ωi)/F(0,0)|. One sees that the frequencies of signal and idler fields are anti-correlated. The distribution ranges of frequency in the direction perpendicular and parallel to the line of Ωs + Ωi = 0 are confined by the the pump envelope term and phase matching term , respectively. Figure 2(b) plots the phase of JSF, arctan . One sees that the phase is within the range [−π,π]. The discontinuity of the phase values comes from the π phase jump, which is occurred when the value of changes from positive to negative or vice versa.
We then study the decomposed temporal modes of the signal and idler twin beams. In the simulation, the normalized amplitude and phase of the JSF in Figs. 2(a) and 2(b) are digitized with a 1000×1000 matrix , and the matrix is decomposed into the form , where matrix is diagonalized with the diagonal elements serving as the mode amplitudes rk, and is the matrix representing mode profiles ϕk(Ωs) (ψk(Ωi) of the signal (idler) by the discrete relation . Figure 2(c) shows the relative mode strength rk/r1 for the decomposed mode with the index k. It is clear that the amplitudes of higher order modes attenuate quickly, therefore, for the signal and idler twin beams, most of the contribution is included in the modes with k ≤ 10.
Figures 2(d) and 2(e) show the temporal mode profiles of individual signal beam ϕk(Ωs), in which only the first three dominant modes are plotted. From the mode intensity |ϕk(Ωs)|2 in Fig. 2(d), one sees that the spectral bandwidth of the mode increase with the mode index k, and the number of predominant peaks is the same as the mode index k, although the small peaks which are less than 10% of the predominant peaks may exist due to the oscillation of the phase matching term . From the phase distribution Arg[ϕk(Ωs)] in Fig. 2(e), one sees that the phase varies linearly except in some frequency points at which a π phase jump exists. Comparing Figs. 2(d) and 2(e), we find that the π phase jump happens at the frequencies at which the values of the intensity |ϕk(Ωs)|2 are zero.
The mode profiles of the intensity and phase distributions of individual idler beam, ψk(Ωi) and Arg[ψk(Ωi)], are shown in Figs. 2(f) and 2(g), respectively. We find that the difference between the intensity distributions of signal (|ϕk(Ωs)|2) and idler (|ψk(Ωi)|2) modes is very small. This is because the absolute value of the JSF in Fig. 2(a) is only slightly asymmetric with respect to the line of Ωs + Ωi = 0. On the contrary, the phase distribution of the ψk(Ωi) in Fig. 2(g) differs significantly from that of ϕk(Ωs) in Fig. 2(e). The notable difference is in consistent with the phase distribution of the JSF in Fig. 2(b), where the asymmetry with respect to the line of Ωs + Ωi = 0 is more obvious than that in Fig. 2(a) because the dispersion for the signal and idler field is different.
3. Generation of continuous variable entanglement
After applying SVD to JSF (see Eq. (12)), the decomposed temporal modes of the signal and idler fields, defined by ϕk(Ωs) and ψk(Ωi), respectively, are decoupled from each other. Therefore, we can study the CV entanglement of the signal and idler fields by utilizing the single-mode method .
Subsequently, it is straightforward to deduce the variances of the correlated quadrature-phase amplitudes of the twin beams, and :27]. Equation (29) clearly shows that Ik always decreases with the gain coefficient G and trends to zero as G approaches to infinity. However, for a fixed coefficient G, the value of Ik increases with the mode index k because the mode amplitude rk decreases with the increase of k. It is worth noting that if the JSF of the FOPA is spectrally factorable, i.e., F(ωs,ωi) = ϕ(ωs)ψ(ωi), we have rk = δk,1. In this case, only the fundamental mode exists, and Ik in Eq. (29) is the same as that in Ref. , describing a CW pumped FOPA operated as a single mode parametric amplifier.
4. Detection of quadrature entanglement
Having demonstrated that the entangled signal and idler twin beams can be decomposed into many pairs of SVD modes, in this section, we will formulate a homodyne detection (HD) process and analyze how to improve the measured degree of entanglement. For simplicity, we assume the polarization states of the individual signal (idler) field and its local oscillator of the HDs (HDi), LOs (LOi), are identical. So the optical fields can be represented as scalars.
The principle of measuring the quadrature components of signal and idler field by using homodyne detectors, HDs and HDi, is shown in the area framed by the dash-dotted line in Fig. 1. The HDs/HDi is comprised of a 50/50 beam splitter (BS) and two photodiodes (PD). Because the signal and idler twin beams are pulsed fields, we take the local oscillators, LOs and LOi, as transform-limited pulses in the form of
When the response times of the HDs and HDi are much longer than the pulse durations of signal, idler, LOs, and LOi fields, the result of the homodyne detection can be treated as a time integral of the optical fields. The photocurrent out of HDs and HDi are expressed as :
Since the decomposed modes functions, ϕk(ωs) and ψk(ωi), form a complete and orthogonal set in the frequency domain of signal and idler beams, the spectra of LOs and LOi (see Eq. (30)) can be expanded into the Fourier series:Eqs. (30) and (35), the photocurrents in Eq. (33) can be rewritten as Eq. (25). Note that in order to clearly demonstrate the mode mismatching effect on measured degree of entanglement, in the analysis hereinafter, we will assume the transmission efficiency of twin beams and detection efficiency of detectors are ideal, i.e., ηs = ηi = 1.
4.1. The spectrum of the LO is matched to a specified SVD mode
When the spectra of LOs and LOi are the same as a pair of decomposed temporal mode, say the kth order modes, Eq. (35) is simplified asEq. (36) becomes ξsk = ξik = 1 and ξsl = ξil = 0 (l ≠ k) because of the orthogonality. In this case, only the signal and idler fields described by the mode functions φk(ωs) and ψk(ωi), respectively, will contribute to the pho-tocurrents in Eq. (37). The evolution of the detected signal and idler fields is described by Eqs. (17) and (18) with an effective gain of Geff = rkG, and the measured inseparability Ik is given by Eq. (29), which goes to zero as G becomes infinitely large.
Among the decomposed SVD modes, the fundamental mode (k = 1) has the largest mode amplitude r1, so its effective parametric gain Geff = r1G is the highest. Hence, we can obtain the highest degree of entanglement when the spectra of LOs and LOi are shaped to satisfy the conditions ALs(ωs) = φ1(ωs) and ALi(ωi) = ψ1(ωi). In particular, when the JSF of the FOPA is factorable, i.e., F(ωs,ωi) = φ (ωs)ψ(ωi), and the spectra of LOs and LOi satisfy the conditions ALs(ωs) = φ (ωs) and ALi(ωi) = ψ(ωi), we will obtain the maximum degree of entanglement for a given gain parameter G because r1 = 1 and rk = 0 (k ≠ 1), which means all the energy of twin beams is concentrated in the fundamental mode. However, it is worth pointing out that the factorable JSF is not a necessary condition for obtaining the entanglement with a high degree. As we have seen, we can always obtain the high quality entanglement characterized by inseparability Ik → 0 as G → ∞ by shaping the spectrum of LOs and LOi to a pair of decomposed modes and by increasing the gain coefficient G.
4.2. The spectrum of the LO is not matched to any particular SVD mode
In general, the LOs and LOi are not matched to any pair of the decomposed modes. The photocurrent out of HDs/HDi is contributed by all the temporal modes non-orthogonal to the spectrum of LOs/LOi (the mode-matching efficiency , see Eq. (37)). Now let us analyze how the multi-mode nature of the twin beams affect the experimentally measured inseparabilityEq. (37), we have Eq. (44b) is originated from the π/2-phase difference between two quadrature components.
From Eqs. (40)–(44), one sees that the key to minimize the inseparability Iexp is to maximize the correlation terms in Eqs. (44a) and (44b) by adjusting the phase of the local oscillators θLs and θLi. It is straightforward to maximize the individual terms CXk and CYk for a given mode index k in Eqs. (44a) and (44b), however, it is difficult to maximize the correlation term for all the terms with |ξks|2 ≠ 0/|ξki|2 ≠ 0, because the phase θks and θki may varies with the mode index k. This will generally result in a decrease in the measured degree of entanglement.
Having understood the measurement principle of entanglement, we are ready to study the parameters that will influence the degree of the measured quadrature amplitude entanglement generated by the FOPA analyzed in Sec. 2.3, which has a broad gain bandwidth in telecom band. Assuming the local oscillators LOs and LOi have the same bandwidth σL, but their central frequencies are the same as the corresponding signal and idler fields, the spectrum of LOs/LOi, which is Gaussian shaped and transform limited, can be expressed as:
We first analyze the mode matching of the homodyne detection systems. Using the twin beams with mode structure shown in Fig. 2 and substituting Eq. (45) into Eq. (36), we calculate the mode matching coefficient for each pair of the decomposed SVD modes. Figure 3 shows the calculated mode-matching efficiency and the phase for the kth order decomposed signal/idler mode ϕk(ωs)/ψk(ωi) when the bandwidths of LOs and LOi are σL = 0.6σp, σL = 2σp, and σL = 3σp, respectively. For each case, one sees that the phase θks(i) of the signal (idler) mode varies with the index k, indicating that it is impossible to simultaneously obtain the maximized correlation terms CXk and CYk for each pair of decomposed modes. On the other hand, since only the modes with the non-zero mode matching efficiencies contribute to the measurement of HDs and HDi, Fig. 3 shows that the main contribution is from the modes with index number k < 5. Moreover, for the mode with a fixed index number, the mode matching coefficient varies with the bandwidth σL. For the case of σL = 0.6σp (Figs. 3(a)–(d)), the sum of mode-matching efficiency for the first- and third-order modes are about 90%, and the mode matching efficiency for the other modes are too small to be obviously observed; for the case of σL = 3σp (Figs. 3(i)–(l)), the mode matching efficiency for the first order is about 90%. While for the case of σL = 2σp (Figs. 3(e)–(h)), the mode matching is obviously better than the other two cases because the mode-matching efficiency for the first order mode is very close to 1.
We then study the dependence of the measured degree of entanglement upon the mode matching of HDs and HDi by numerically calculating the inseparability Iexp in different conditions. Figure 4 shows the calculated Iexp as a function of the gain coefficient G when the bandwidths of LOs and LOi are σL = 0.6σp, σL = 2σp, and σL = 3σp, respectively. In the calculation, for each gain coefficient, we deduce the corresponding mode structure and mode matching for different spectrum of LOs and LOi, which is similar to the procedure of obtaining the plots in Figs. 2 and 3. The results corresponding to the JSF in Fig. 2 are marked by cross points between the data and the dashed line. Additionally, as a comparison, the inseparability I1 for LOs and LOi with the spectra the same as the corresponding fundamental modes ϕ1(ωs) and ψ1(ωi) is also plotted in Fig. 4 as a function of G. Obviously, for a fixed gain coefficient G, the value of I1 is always smaller than that of Iexp.
In Fig. 4, when the spectra of LOs and LOi are not matched to the fundamental modes, one sees that for a fixed value of G, the lowest and highest Iexp respectively correspond to the case of σL = 2σp and σL = 0.6σp. If we compare the mode matching efficiency in Fig. 3, it is straightforward to see that the measured degree of entanglement increase with the mode matching efficiency between the local oscillators and the fundamental modes, |ξ1s|2 and |ξ1i|2, and the departure between Iexp and I1 will increase when the spectra of LOs and LOi are more evenly distributed among the decomposed othogonal modes.
From Fig. 4, one also sees that different from I1, which always decreases with the increase of G, the measured Iexp may increase with G when G is larger than a certain value. For the case of σL = 3σp, it is obvious that Iexp start to increase with G for G > 1.4. For the case of σL = 2σp, this kind variation trend is also observable: Iexp increases with G for G > 1.9. We think the reason is because the the correlation term CXk/CYk (see Eqs. (44a) and (44b)) for different k can not simultaneously achieve the maximum for the LOs/LOi with a given phase. Although the mode matching efficiency of the first order mode is the highest (see Fig. 3), the noise variance of Iexp contributed by the signal and idler beams in higher order SVD modes may exponentially increase with G, which will results in the unusual trend of Iexp. Moreover, it is worth noting that the influence of mode mismatching effect on measured entanglement is not equivalent to an effective detection loss because the noise contributed by higher order modes with |ξks|2 ≠ 0/|ξki|2 ≠ 0 will become significant, particularly in the high gain regime of FOPA.
In summary, we theoretically analyzed the temporal mode structure and the degree of measured CV entanglement of the twin beams generated from a pulse-pumped FOPA by applying the SVD to the JSF. We are able to successfully decouple different temporal modes and derive the input-output relation for each temporal mode. The results indicate that the temporal mode structures are highly sensitive to the dispersion of the nonlinear fiber and the gain coefficient of FOPA. While for the measurement of CV entanglement, when the temporal modes of LOs and LOi are the same as one pair of decomposed modes, ϕk(ωs) and ψk(ωi), the measurement result of the homodyne detection systems is only contributed by the signal and idler fields in the modes ϕk(ωs) and ψk(ωi), which is similar to case of pulsed pumped parametric amplifier having a factorizable JSF; when the modes of LOs and LOi can not match any pair of the decomposed modes, the measurement result of the homodyne detection systems is contributed by all the modes non-orthogonal to the spectra of local oscillators, leading to a poor value of the inseparability. Therefore, in order to obtain the high degree CV entanglement, making the JSF factorable is not necessary, but matching the spectra of local oscillators to one pair of decomposed modes is crucial. Moreover, for the FOPA with broad gain bandwidth of FWM in telecome band, we numerically studied the temporal mode functions of the twin beams, and calculated its corresponding degree of CV entanglement when the spectra of LOs and LOi of HD systems are varied. The results demonstrate the detailed temporal mode structure of this kind of FOPA as well as the strategy for optimizing the spectra of the local oscillators in the detection process. Hence, our study is useful for developing a high-quality source of pulsed CV entanglement by using the FOPA.
Our investigation indicates that the determination of the JSF, including the absolute value and the phase term arctan , is utmost important for the mode analysis of a pulse-pumped parametric process. So far, the measurements of the absolute value of JSF have been demonstrated [29–32]. If there is a practical scheme to realize the measurement of the phase term of JSF, which has not been reported yet, it will be straightforward to realize the required mode matching by shaping the spectra of LOs and LOi .
This work was supported in part by the National Natural Science Foundation of China (NSFC) (No. 11527808, No. 11304222), the State Key Development Program for Basic Research of China (No. 2014CB340103), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120032110055), PCSIRT and 111 Project B07014.
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