1. Introduction

Quantum entanglement between physically separated systems is a resource that is essential for implementing many of the novel functions in quantum information processing such as database searching, clock synchronization, teleportation, computing, and cryptography [1]. Therefore, the efficient generation and distribution of quantum entanglement is of prime importance. The vast majority of entangled-photon sources that are in use in various laboratories around the world today rely on spontaneous parametric downconversion in χ ⁽²⁾ crystals [2, 3]. However, our group recently demonstrated a new type of source which relies instead on parametric processes in optical fibers utilizing the χ ⁽³⁾, or Kerr nonlinearity [4, 5, 6, 7]. In the experiments conducted by Fiorentino et al. [5] and Li et al. [6, 7], a commercially available dispersion-shifted fiber was used as the χ ⁽³⁾ medium in which photon pairs were generated through nondegenerate four-photon scattering (FPS).

In recent years there has been great interest in using microstructure fibers (MFs) in situations where a large nonlinear coefficient is desirable [8, 9, 10]. Light is guided through small-core MFs in such a way that high intensities can be maintained for interaction lengths of many meters resulting in large nonlinear-optical effects. The “endlessly single mode” behavior of these fibers leads to good spatial overlap between fields of widely differing wavelengths. Additionally, MFs can be fabricated having a short zero-dispersion wavelength, λ₀. The short λ₀ means that FPS can be phase matched in the 800 nm region of the spectrum [11, 12] (i.e., for wavelengths shorter than the zero-dispersion wavelength of pure silica), leading to the possibility of integration with rubidium trapped-atom quantum memory schemes that have been proposed for the storage of engantlement [13].

In this paper, we report on measurements of correlations in the photon-pairs generated through FPS in a MF. The photon-counting experiments described herein provide a convincing demonstration of these correlations. The design of the experiment is similar to that described by Fiorentino et al. [5], which was used to produce correlated photon-pairs in dispersion-shifted fiber near 1550 nm wavelength. The two fundamental differences between the work reported here and the earlier experiments are the wavelength of operation (749 nm versus 1537 nm) and the use of a linear configuration as opposed to a Sagnac-loop configuration [14]. Working at shorter wavelengths provides the advantage of allowing one to use commercially available photon-counting tools. The Sagnac-loop configuration is easily implemented at telecommunication-band wavelengths using a 50/50 fiber coupler, but when using the MF greater flexibility is achieved by using free-space optics. The linear configuration was chosen for these experiments to eliminate the complexity associated with alignment of the Sagnac interferometer in free space.

The remainder of this paper is outlined as follows. Section 2 introduces the conceptual background of FPS and some mathematical relationships needed for understanding the experimental results that are given in Section 4. Section 3 summarizes our characterization of the MF that has been used in these experiments, wherein special attention is paid to the parameters that impact FPS. We conclude in Section 5 by discussing some potential future research directions.

2. Theoretical background

The classical theory of four-wave interactions in optical fibers has been explored fully by Stolen [15] and is summarized along with experimental results by Agrawal [16]. When the two pump waves are degenerate, one obtains the following coupled amplitude equations describing such interactions in the fiber:

In these equations, A _1,2,3 are the field amplitudes of the pump, signal, and idler waves, respectively, and α is the attenuation coefficient of the fiber according to P(z)=P(0)e ^-αz, where P is power and z is propagation distance. The nonlinear coefficient, γ, is related to the nonlinear refractive index, n ₂, by γ=n ₂ ω ₁/A _eff c, where A _eff is the effective mode area of the field, and ω ₁ is the angular frequency of the pump. The wavevector mismatch is given by Δk=2k ₁-k ₂-k ₃, where k ₁, k ₂, and k ₃ are the wavevector magnitudes for the pump, signal, and idler waves, respectively. In Eqs. (1–3) we have assumed that the pump is much stronger than the input signal and that group-velocity mismatch between the three waves is not significant; these are reasonable assumptions for the experiments described in this paper. The effects of linear loss, self-phase and cross-phase modulation are included, however.

For efficient interaction, solution of Eqs. (1–3) calls for the following phase-matching condition to be satisfied:

where β is the group-velocity dispersion (GVD) coefficient and ω ₂ is the angular frequency of the signal wave. The GVD parameter, D, is often used instead of β and they are related by D=-2πc/λ ² β. When a strong pump interacts with a weak signal under phase-matching conditions, one observes amplification of the signal wave as well as generation of an idler wave of angular frequency ω ₃ according to ω ₃=2ω ₁-ω ₂. The primary factors influencing the four-wave interaction within the fiber are: the nonlinear coefficient γ, the pump power, the fiber length, and the GVD coefficient β. By operating near the λ ₀ of a fiber, one can achieve β<0 which, in addition to a small amount of pump self-phase modulation, results in κ⋍0 as required by Eq. (4).

In subsequent sections the FPS probability is approximated by calculating the classical power gain, G, for a weak pulsed signal whose spectral width is matched to that of the passband of the detectors. The scattering probability, which is the same as the photon-pair production rate per pump pulse, is taken to be proportional to G-G ₀, where G ₀ is the “gain” when no pump is present. In the case of α=0, G ₀=1, but for lossy fibers G ₀<1.

 

Fig. 1. Experimental measurements of group delay as a function of wavelength for the MF used in our experiments. Second-order polynomial fits accompany the data. The two curves are for the two polarization modes of the MF, clearly showing that there are differences in the GVD behavior for the two modes. Inset shows plots of D vs. wavelength for the same MF and the λ ₀ values are indicated for each polarization mode.

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3. Microstructure fiber properties

The analysis described above applies equally well to four-photon interactions in any type of optical fiber. The advantages of using MFs for demonstrating nonlinear-fiber optical effects arise from several novel properties: the nonlinear coefficient is enhanced in small-core MFs (a few µm²), MFs exist which support a single transverse mode over an extremely broad wavelength range (370 nm–1600 nm), MF design parameters allow one greater flexibility to manipulate the fiber’s GVD properties, and nonlinear interactions are enhanced due to the polarization maintaining properties of MFs, which result from form birefringence present in the core. These four properties combine to allow efficient interactions to occur in MFs which are either much less efficient or not possible at all in standard optical fibers.

Table 1. Various properties of the OFS MF used in these experiments.

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The zero-dispersion wavelength, λ ₀, and the slope of the GVD curve, D _slope, near λ ₀ are determined experimentally by measuring the wavelength dependent group delay in an interferometric configuration [17]. The group delay is then plotted as a function of wavelength and fitted to a low-order polynomial (up to x ² or x ³). The first derivative of the group-delay curve, normalized to the length of the optical fiber under test, then gives the GVD parameter, D.

Results for group delay measurements taken for the MF used in these experiments are shown in Fig. 1. The two curves correspond to the two polarization modes of the MF, revealing that the group delay behavior is quite different between the two polarization modes. The lines given with the data are fits to the second-order polynomial function,

where for λ ₀=749 nm: a=0.321, b=-481, and c=180000; and for λ ₀=743 nm: a=0.338, b=-502, and c=186000. In the inset of Fig. 1 are plots of the first derivative of the fitted curves, revealing the fact that λ ₀ is not the same for the two polarization modes. Other important parameters of the MF used herein pertaining to FPS are summarized in Table 1.

 

Fig. 2. A schematic of the experiment used to generate and detect quantum correlated twin photons generated in MF.

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4. Four-photon scattering experiments

The experimental setup that was used to study photon-pair production in MF is shown in Fig. 2. Linearly polarized, ~3 ps-long pump pulses are launched into a 5.8 m-long MF such that the plane of polarization is aligned along one of the polarization-mode axes of the fiber. The choices of pump wavelength and which MF birefringent axis is used are critical as will be discussed in more detail later. For low pump powers, as the pump pulses propagate through the fiber, there is a small probability that a four-photon parametric scattering event will occur. The resulting signal and idler photon pairs co-propagate along with the pump and emerge from the MF.

As a consequence of choosing the linear configuration, spectral filtering is quite difficult. Photon pairs emerge from the MF accompanied by a very strong pump field. Given a pump pulse containing ⋍10⁷ photons, numerically solving Eqs. (1–3) in the presence of loss predicts that the FPS probability is about 17%. Therefore, the output of the MF consists of a large number of pump photons and an extremely small number of spontaneously scattered signal and idler photons that are separated in wavelength from the pump by ~10 nm. For efficient detection of the signal-idler pairs, pump-photon suppression by greater than 90 dB is required. We obtain this suppression by use of three diffraction gratings (Richardson Grating Laboratories, 600 lines/mm, nominally blazed for 800 nm) arranged so that two different spectral regions can be detected on separate photon-counting modules (PCMs). The PCMs are commercially available devices (Perkin-Elmer, model SPCM-AQR-16), which count individual photons at 750 nm wavelength with a quantum efficiency of ⋍75%. Each time a photon is detected by the PCM it generates a 5V pulse of 30 ns duration. Counts are registered using either a gated photon counter (Stanford Research Systems, model SR-400) or a multi-channel scaler/averager (Stanford Research Systems, model SR-430). The dark-count levels for the two detectors are typically ~25 counts/s for the signal detector and ~140 counts/s for the idler detector.

 

Fig. 3. Graphs showing the spectral response of the detection filter in comparison with the bandwidth of the injected pump. Note that the passband centers of the detection filter for the signal and idler are located symmetrically with respect to the pump’s center wavelength, and that the detection bandwidth (2.1 nm FWHM) is considerably larger than the pump bandwidth (0.3 nm FWHM).

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The output from the MF is directed onto the first diffraction grating, G₁. A mirror (M1 in Fig. 2) is used to direct the long wavelength (signal) portion of the spectrum to a second grating, G₂, while the short wavelength (idler) portion passes the mirror and hits a third grating, G₃. As such, the signal and idler are diffracted twice each. The diffracted waves are then directed through interference filters (IFs) whose purpose is twofold: to provide additional filtering of the pump and to block the ambient room light. The IFs are specified to have 10 nm passbands centered at either 740 nm or 760 nm. The passband center of these IFs can be tuned in wavelength by slightly rotating them such that the incident wave is no longer normal to the surface. Finally, the signal and idler each pass through an aspheric lens (f=11 mm) and a 10µm slit to achieve the desired spectral filtering. The detection filter optics are arranged so that the strong pump wave passes without being scattered by any surface until it is dumped onto a black surface. In addition, the number of optical elements that are used in the filter system is minimized in order to reduce unnecessary scattering of light from surfaces and imperfections.

The transfer characteristic of the detection filter is shown in Fig. 3. Here the relative count rate (in dB) on each photon counter is shown as a function of the wavelength for an injected signal. The spectrum of the input pump pulses is included to show that the center wavelength of the pump is placed symmetrically in between the passbands of the detection filter. The 3 dB bandwidth of each of the two passbands is 2.1 nm, which is large compared with the spectral width of the pump (~0.3 nm). The peak total detection efficiency at the center of each passband of the filter (including the quantum efficiency of the respective PCM as well as propagation losses from the fiber to the PCM) is about 4±1%.

Even if, prior to detection, one is able to perfectly filter away photons at the pump wavelength, one also has to take great care to ensure that there are no photons injected into the MF whose wavelengths correspond to that of the signal and/or the idler. Use of a pulsed pump implies that there are photons in the spectral sidebands of the pump at the input to the MF. These extra photons could cause stimulated FPS rather than the spontaneous FPS we are seeking. Therefore, to achieve suppression of the pump sidebands, we pass the pump wave through a double-grating monochromator before injecting it into the MF. This yields a pump sideband suppression in excess of 100 dB. The double-grating monochromator also filters the pump pulses, yielding a narrow pump bandwidth of ~0.3 nm (see Fig. 3). Such narrow pump bandwidth is desirable because the system then more closely approximates what is expected for a CW pump.

 

Fig. 4. Graphs showing the idler count rate for FPS with different pump wavelengths and a given detection filter setting. Data points are accompanied by theoretical scattering curves, which were fitted to the data using Eqs. (1–3). The λ ₀ is assumed to be 748 nm for these calculations. One sees that the optimum response is achieved for a pump wavelength of 749 nm. The dashed curve in the inset shows the measured filter response for the idler passband in comparison with the theoretical variation of FPS efficiency versus wavelength.

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Proper choices of the pump, signal, and idler wavelengths, and careful alignment of the system are critical to obtaining the desired result. As can be seen in Fig. 4, the choice of pump wavelength and the detection filter passband wavelengths with respect to the fiber’s λ ₀ have a dramatic effect on the FPS phase-matching bandwidth and, consequently, the signal-idler pair production rate. The filter and pump wavelengths must be chosen so that the passbands of the detection filter are near the peaks of the phase matching curve, as is the case for λ_p=749 nm in the example shown in the inset of Fig. 4.

We made FPS measurements for the idler photon counts as a function of the pump power, and the results are shown in Fig. 4. The fiber’s polarization mode corresponding to λ ₀⋍749 nm was excited with pump pulses whose wavelength varied from 747 nm to 751 nm. It is evident that the count rate is dramatically reduced for pump pulses of wavelengths detuned from λ ₀ by more than ±2 nm. In practice, detuning the pump wavelength by more than ±1 nm from the optimum value reduces the pair production rate so dramatically that the background noise level due to leaked light and dark counts is dominant. Recall that the fiber’s λ ₀ is polarization dependent due to birefringence in the MF (see Fig. 1). This means that the polarization of the pump also dramatically affects the signal/idler count rate in the experiment.

The data in Fig. 4 has been fitted to theory by numerically solving Eqs. (1–3) to calculate the power gain, G, and then scaling the numerical results according to

where R is the total detected photon scattering efficiency per pump pulse, G ₀ is the gain when the injected number of photons per pump pulse (P _pump) is equal to zero. The fitting parameter a ₁=0.027 counts/pulse is the total photon detection efficiency, which agrees within the margin of error with the independently measured value of 4±1%. The fitting parameter b ₁=8.7×10^-9 counts/photon is the leakage through the detection filter and/or the contribution of spontaneous Raman scattering. In this experiment, FPS, Raman scattering, and linear loss are all of the same order of magnitude. As a result, it is difficult to quantify the contribution due to Raman scattering without a more comprehensive theory [18, 19]. The count rate on the vertical axis in Fig. 4 is obtained by multiplying R by the laser repetition rate (75 MHz).

 

Fig. 5. Histogram data gathered using the multichannel scaler where one counter is used to trigger the acquisition and the other is taken to be the signal. One bin, the coincident bin, has many more counts than the others indicating that the coincident events are registered at the PCMs with a higher probability than non-coincident events. The variation in the counts in the non-coincident bins is due to asynchronous photon detection relative to the arrival of pump pulses.

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To detect temporally coincident counts using the multichannel scaler, the output from one of the PCMs is used as trigger input to the scaler while the pulses from the other PCM are measured at the signal input of the scaler. The instrument then generates a histogram of the arrival time of signal pulses with respect to the trigger pulses. A typical set of data recorded by the scaler is shown in Fig. 5, where it can be seen that one of the peaks is much taller than the others. The peak is evidence of a higher probability of registering a count on the signal PCM in the same time window as the count registered on the trigger PCM. The other peaks are spaced from each other by 13 ns (modulo 5 ns due to the bin resolution limitation on the scaler) and arise due to the fact that pulsed pump at 75-MHz repetition rate is used. One can clearly see the 13 ns repetition period on the histogram.

Data from the scaler can be used to make a comparison between the rate of coincidences versus non-coincidences. Figure 6 shows a plot of coincident counts registered and non-coincidence counts registered as a function of the number of pump photons per pulse. This plot clearly demonstrates that more coincident photons are generated at the signal/idler wavelengths than the accidental coincidences that result from the background photons.

The coincidence counting results shown in Fig. 6 do show the existence of correlated photons, but only on top of a large background of accidental coincidence counts. The “true coincidence counts” are obtained by subtracting the accidental coincidence counts from the total coincidence counts (see the inset in Fig. 6). While we expect a quadratic variation, we observe that the true coincidence count rate varies slightly sub-linearly with pump power. The origin of this discrepancy is unclear at this time. The accidental coincidences may arise from mechanisms such as pump leakage due to imperfect filtering, but it is highly likely that the bulk of these accidental counts arise from spontaneous Raman scattering. Recently, we have presented a quantum theory of the four-wave interactions in fibers with systematic inclusion of the Raman effect [18, 19]. A quantitative comparison with this theory is underway and will be presented elsewhere. If one makes the comparison between the results presented here and those of Fiorentino et al. [5] and Li et al. [6, 7] we expect the coincidence-to-background ratio to be much worse for the choice of wavelengths used here because the signal wavelength falls much closer to the peak of the Raman gain curve.

 

Fig. 6. Plots of total coincidence counts (triangles) and accidental coincidence counts (boxes) as a function of the number of pump photons per pulse with the photon counters aligned to detect at 737 nm and 761 nm wavelengths, respectively. At low pump powers there is a quadratic dependence of the counts on pump power, but as the power is increased the increasing signal and idler count rates start saturating the photon counters. The inset shows true coincidences (the difference between the total coincidence counts and the accidental coincidence counts) as a function of the number of pump photons per pulse.

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5. Future considerations

The quality of the correlated photons generated by FPS in the MF is limited by the high rate of accidental coincidences. Accidental coincidence counts due to imperfect filtering as well as from spontaneous Raman scattering could be dramatically reduced if the signal and idler photons are generated with greater wavelength separation from the pump. In a recent paper, Harvey et al. [20] describe measurements of modulation instability in highly nonlinear MFs where the signal- and idler-to-pump detuning exceeds 100 nm. This wavelength separation would result in photon pairs being generated away from the peak of the Raman gain, which can then be efficiently filtered from each other as well as from the pump photons.

When starting the experiment, we had hoped to create polarization entanglement by pumping both polarization modes of the MF simultaneously [6, 7]. However, it is clear from the data presented in Fig. 4 that optimum pair production depends upon choosing a pump wavelength almost exactly equal to the λ ₀ of the fiber. Since the two polarization modes of the fiber have λ ₀ differing by about 5 nm, this approach will not work because the efficiency of the FPS process must be the same in both polarizations. However, it is still possible to generate polarization entanglement by using a counter-propagating pump configuration wherein one polarization mode of a MF is pumped in both directions at the same time. By including a 90-degree twist in the fiber orientation, and mixing the two outputs (orthogonally polarized because of the twist) using a polarization beam splitter, polarization entanglement can be produced.

In conclusion, we have studied the coincident nature of photon-pairs created in MF due to FPS. We have shown that in order to detect photon correlations one must understand the group-velocity dispersion and polarization properties of the fiber being used. Polarization dependent dispersion behavior in MF provides unique challenges for the generation of polarization entanglement.

This research was supported in part by the U.S. Army Research Office under a collaborative MURI grant (DAAD19–00–1–0177) and the associated MURI Fellowship (DAAD19–00–1–0469) for J. E. Sharping.