Deterministic reshaping of single-photon spectra using cross-phase modulation

With an ultrafast refractive index change in an optical fiber, frequency entanglement of twin photons is modulated on the fly.


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I. Numerical simulation of XPM interaction between control and signal fields II. Estimating the upper bound of HOM interference visibility from experimental JSI III. Nonlinear polarization rotation IV. Two-photon interference fringes without XPM V. Prospects for larger frequency shifts Fig. S1. Numerically calculated JSA and JSI of the photon pairs. Fig. S2. Singular values and intensity spectra. First we obtain the JSA of photon pairs generated in the PPKTP crystal. Again, the state of a signal and idler photon pair generated in a SPDC process can be written as (51) where we neglect the vacuum and higher-order terms that will be eliminated by coincidence measurement. The JSA S(ω s , ω i ) = α(ω s , ω i )φ(ω s , ω i ) where α(ω s , ω i ) is the envelope function of the SPDC pump pulse and φ(ω s , ω i ) is the phase matching function. In a PPKTP crystal with a length L and a poling period Λ, collinear phase matching between the pump, signal and idler modes with polarizations along the crystallographic y, y, and z axes, respectively, are satisfied where k y(z) (ω, T) = n y(z) (ω, T)ω/c is the wavenumber along the propagation axis, n y(z) (ω, T) is the refractive index of the material for the light with y(z) polarization and T is the crystal temperature. ! , = sinc Δ , , & 2 ⁄ , where the unimportant global phase is omitted.
In Fig. S1 we plot numerically calculated JSA = S(ω s , ω i ) (A) and JSI = |S(ω s , ω i )| 2 (B) as a function of the signal and idler wavelengths λ s and λ i . For the material dispersion we used Sellmeier equations in (52, 53) with the temperature dependence in (54). The temperature expansion coefficient of KTP (54) was also taken into account for Λ. Since these equations are empirical, it was also necessary to additionally multiply with a coefficient of 2.8 for the temperature dependence to reproduce the wavelength of the photon pairs obtained in the experiment.
JSA is expressed by a complete set of orthonormal basis functions that can be obtained by the Schmidt decomposition as follows (55). , = ( )* + + , ,+ , ,+ 3 where A s,j (ω s ) and A i,j (ω i ) are orthonormal functions of the signal and idler wavefunctions, and λ j are non-negative, real numbers satisfying Σ j λ j = 1. The basis functions and the coefficients can be numerically obtained using singular value decomposition (SVD) (51).
In the measurement of the delay-dependent marginal frequency distribution of heralded signal photons (shown in Fig. 3 Therefore, the heralding detection with TBPF2 will eventually filter out the first Schmidt mode of the signal photon A s,1 (ω s ). Based on this approximation, we now only consider the first term A s,1 (ω s ) A i,1 (ω i ) as the initial JSA before the XPM interaction.
Next we numerically simulate the nonlinear pulse evolution between the pump and signal fields.
For the latter A s,1 (ω s ) is used based on the above approximation where the initial JSA is separable. The nonlinear evolution is governed by the following coupled nonlinear Schrödinger equations (NLSEs) (25). Here A c(s) represents the slowly-varying envelope and β c(s),k is the $k$-th order propagation constant at the control and signal frequencies. We use A s,1 for A s . Because the theory is linear in the signal field operators, single photon fields obey the same linear equations of motion as do weak classical fields (21) (see Ref. (57) for a numerical simulation for the entire frequencyentangled JSA). α is the linear attenuation coefficient, which is assumed to be wavelength independent (− 0.016 dB/m from catalog). XPM induced by the signal photon wavepackets is is the time measured in a reference frame of the control pulses. d = β s,1 − β c,1 is the differential group delay between the control pules and signal photons. γ is the nonlinear constant. The catalogue value of γ is 0.011 /W/m, which is used for the value at around the control wavelength. b (0 ≤ b ≤ 1) is the parameter associated with the degraded spatial overlap between mode fields with the greatly separated wavelengths (756 nm and 1512 nm) (58). Terms associated with four-wave mixing, which is not phase matched, are ignored. A symmetric split-step Fourier method was employed for the numerical calculation of the coupled NLSE with up to 6-th order propagation constants (extracted from Fig. 1(B)) taken into account.
In the calculation b and d are the only free parameters.  (Fig. 3(A)), and includes the major blue and red shifts found at the positive and negative delays, respectively. The d value for the fitting is in a reasonably small range considering the wavelength-dependent variation of the group velocity ( Fig. 1(B)). The obtained b value was less than unity. This is mainly due to the wavelength dependence of the effective modal area in the PCF, which reduces the lateral field overlap (58).

II. Estimating the upper bound of HOM interference visibility from experimental JSI
In the HOM interference experiment, the coincidence count rate after the photons have passed through the NPBS can be described as follows (59): where t s and t i are the detection timings of the signal and idler photons. Accordingly, for a known S(ω s , ω i ) we can obtain a two-photon interference fringe by plotting R as a function of the arrival time difference between photons δt = t i -t s . However, we cannot reconstruct the phase information of JSA from the experimental JSI. Nonetheless, we can obtain the lower bound visibility of the two photon interference fringe from |JSA| = NJSI; any complex phase terms in JSA will induce an increase in R and thus decrease in the visibility (59). Substituting S(ω s , ω i ) in Eq. (7) with the square root of the experimental JSI after XPM reshaping ( Fig. 4(C), right panel), we obtain dip visibility to be 91%. The value well describes the experimental value (87 ± 1)% without accidental coincidence counts. The imperfection of the estimated visibility mainly originates from the asymmetric component of JSI with respect to the zero-detuning axis ω s = ω i , which results in the distinguishability in the single count spectra.

III. Nonlinear polarization rotation
The two photon interference fringes shown in Fig. 4(B) and (E) exhibited a reduction in coincidence counts when the photons interacted with the control pulses. This is due to the nonlinear polarization rotation (25) of the photons induced by the control pulses originating from the fact that the PCF that was used was slightly birefringent. This is confirmed as follows.
First, using the setup shown in fig. S3(A), we recorded the coincidence count as a function of signal-to-control delay ∆T for the cases with and without the polarizer (PL) in the signal channel in front of the SPCM. These results are shown in fig. S3(B) as closed and open blue dots. It is clear that the coincidence loss occurred when PL was used. This indicates the polarization rotation of the signal photons, which was also detectable in the two-photon interference setup with polarizers in place. For comparison we also plot coincidence counts R classical obtained for the experiment shown in Fig. 4(B) as a function of ∆T (red closed squares). The behavior is explained by the closed blue dots. This polarization rotation can be simply eliminated by matching the birefringent axis of the PCF with the polarization axis of the control and signal fields, as in previously reported experiments such as (30).

V. Prospects for larger frequency shifts
We explore how the finite fiber dispersion affects the frequency shifts using the numerical simulation introduced in Sec. I. In fig. S5(A), we plot the evolution of the signal pulse wavepacket A s,1 under the experimental conditions described in Sec. I. Then, we set d = 0 ps/m, which can be realized by tuning the wavelengths of the signal and control fields. Also, by optimizing the temporal width of the Gaussian-shaped control pulses (t 0c = 2.5 ps), we obtained the result shown in fig. S5(B). The data indicate that a frequency shift larger than the bandwidth of the signal field is available. Furthermore, we plot the results obtained when both the group velocity dispersions β c,2 and β s,2 were 50% and 20% of those of the PCF used in figs. S5(C) and (D), respectively. Hence, the reduced GVD leads to larger spectral shifts. We could also further tailor these results by modifying the temporal shape of the control pulses. These are obtained from the Schmidt decomposition of JSA with and without TBPF2 in the idler channel. Note that both A s,1 and A' s,1 are real.