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We demonstrate concurrent frequency up-conversion and down-conversion in a quadratic nonlinear medium for the generation of correlated ultraviolet and visible rainbows with one-to-one angular and spectral correspondence, facilitating generation of continuously tunable and photon-momentum-dependent multi-wavelength UV and visible twin photons.
©2010 Optical Society of America
1. Introduction
Spontaneous parametric processes have been most frequently used to generate highly correlated photon pairs for a broad range of fundamental and practical applications in quantum physics and especially quantum information technology. Parametric down-conversion is one of most practical method to generate correlated photon pairs, where a pump photon spontaneously converts into a pair of highly correlated photons with sum photon energy and momentum conserved [1–4]. However, the spectral region is limited in this method, while four-wave mixing in optical fibers or silicon waveguide offers the possibility to generate photon pairs in different spectral regions [5–8]. Coupled parametric up-conversion occurs as spontaneously down-converted photons interact with pump photons through multiple nonlinear processes under simultaneous phase matching [9]. Recently, tripartite entangled states or blue quantum-correlated light were realized in a bulk non-linear crystal according to this method [10–14]. On the other hand, parametric light scattering manifesting as specific light patterns are merely associated with nonlinear growth of coherent noises [15–18], while nonlinear growth of certain vacuum modes may be induced by modulational instability [19, 20] but typically happens under strong pump and thus hardly applicable to quantum information processing at single-photon level. It is also difficult to generate tunable correlated photons in the UV region for multipartite entanglement experiments [21]. Parametric down-conversion [1–4] still features intrinsic limits in generating broadband multi-wavelength tunable or short-wavelength energetic twin photons. Photonic cascades from atoms [22] or quantum dots [23] may generate highly correlated photon pairs in the UV region that can be used for generating further correlation between three or four photons [24]. Due to the intrinsic limit by nearly resonant transitions of the relevant cascaded emissions, the generated photon pairs exhibit somewhat difficulty in tuning wavelengths.
In this paper, we demonstrate a mechanism to generate correlated photon pairs tunable in the UV and visible regions by means of concurrent frequency up-conversion and down-conversion in a bulk nonlinear optical crystal, which formed correlated UV and visible rainbows with angularly-resolved spectra determined by photon energy and momentum conservation of cascaded parametric couplings. Angularly-resolved spectra were observed for both UV and visible rainbows, and the observed rainbows exhibited quadratic intensity dependence on the pump pulses. Both UV and visible rainbows could be mediated by continuous-wave (cw) or pulsed seeds, and beam-like up-conversion and down-conversion scatterings were observed at the overlapped regions of the rainbows with two seeds at different directions. All these observations suggest cascaded parametric characteristics of the observed rainbows. Such processes may not only provide unique fingerprints for various nonlinear optical phenomena such as parametric scatterings and photo-charge transportation [15–18], but also support the generation of multiple angularly-resolved twin photons in the up-conversion and down-conversion regions with advantageous features inherited from both photonic cascades [22,23] and parametric processes [24], entailing a broadband tunable range and small beam divergence confined by the photon energy and momentum conservation. Multi-wavelength correlated photon pairs could thus be selected at continuously tunable wavelengths along different positions of the rainbows, and a fine wavelength-tuning of twin photons is anticipated to reach some specific atomic or molecular resonances. This may stimulate various experiments and applications in quantum network [25], multipartite entanglement [21,26], quantum memory [27], and photonic interface [28].
2. Experimental observations
In the experiment, cascaded parametric processes were realized in cw or pulse seeded non-collinear optical parametric amplification (NOPA) pumped by fs pulses. As Fig. 1 shows, a 1-kHz regeneratively amplified Ti:sapphire fs laser (Spectra-Physics, Spitfire) was used to produce 45-fs fundamental-wave (FW) pulses near 800 nm with a single-pulse energy of 0.6 mJ, which were telescoped to a transverse beam diameter of 3 mm. The collimated pulses were frequency-doubled with a 0.2-mm-thick 29.2°-cut type-I β-barium borate (β-BBO) crystal (B1) with a conversion efficiency about 35%. After being separated from the FW pulses by a dichroic mirror, the second harmonic (SH) pulses were reflected into a 2-mm-thick 29.2°-cut type-I β-BBO crystal (B2) as the horizontally polarized pump for NOPA. A screen was placed far behind the β-BBO crystal B2 to observe scattering patterns vertical to the SH pump beam where the pattern pictures were taken by a digital camera. As shown in Fig. 2(a), parametric down-conversion was induced as axial colored cones marked as (1) in Fig.2(a), and weak abaxial conical light scatterings (2) were observed with the same color of the pump. The abaxial scattering patterns were originated from degenerate parametric light scatterings by recording gratings [15–18]. As shown in Fig. 2(b), the SH pump incidence was adjusted by properly tilting the B2 so that a strong red component around 632.8 nm appeared in the parametric cone, along which a cw He-Ne laser with the maximum power about 65 mW at 632.8 nm was injected with vertical polarization as the seed signal marked as (3) in Fig. 2(b), non-collinearly interacting with the horizontally polarized pump pulses. Nonlinear amplification of the cw seed was accompanied by the generation of idler pulses at 1087 nm along the non-collinear phase-matched direction, as evidenced by a green beam spot (4) on the observation screen corresponding to their SH pulses. The maximum amplified signal and idler pulses were about a few micro-joules per pulse with the typical pulse duration around 100 fs. The blue abaxial scattering was enhanced in intensity with an intensified central cone, indicating an enhanced nonlinear growth of parametric degenerate scatterings [15–18]. On the other hand, the enhancement of the blue abaxial scatterings in the presence of cw-seeded NOPA as a much brighter and narrower central cone was indicative of multi-photo-ionization induced photo-charge transportation in the NOPA interaction region, where the multi-photo-ionization might involve the absorption of two pump photons and one signal or idler photon that excited asymmetric photo-charge transportation to alter local refractive index, leading to a slightly self-guided parametric degenerate scattering [29].
Fig. 1. Experimental setup of the NOPA. PR: polarization rotator, M1, M2: high-reflection mirrors at 800 nm, FM1-3: concave high-reflection mirrors, L: fused silica lens, M3: dichroic mirror, B1 & B2: type-I β-barium borate (β-BBO) crystals.
Fig. 2. Experimentally observed cascaded parametric scatterings under different pump conditions: (a) parametric down-conversion cone and degenerate scatterings observed with SH pump incident at an angle of 9.4°; (b) NOPA with a cw beam seeding along the 632.8-nm parametric cone; (c) NOPA with two cw beams seeded at different directions along the 632.8-nm parametric cone, (d) NOPA with a synchronized pulse seed.
Interestingly, two multi-color scatterings were observed with the same horizontal polarizations as the pump, which concurred as UV and visible rainbows symmetrically located on both sides of the pump, with gradual but angularly-resolved changes of colors into ultraviolet and red regions off the pump, respectively. The UV and visible rainbows exhibited colors of gradually blue-shifted up-conversion and red-shifted down-conversion as compared with the pump wavelength around 400 nm, respectively. Figure 2(b) presents a typical picture taken as the pump incidence was about 9.4° and the angle between the pump and cw seed was about 11.4° in air. As there existed no seeds in those spatial or spectral regions while degenerate scattering from crystal surfaces or bulk imperfections did not change frequency [15–18], those frequency conversions should be intrinsically caused by parametric amplification of vacuum noises. The multi-color scatterings could be switched off by turning off the seed beam, confirming that the rainbows were closely related with the cw-seeded NOPA, where pump, signal and idler photons were involved in nonlinear couplings, the spontaneously generated photons may have energies larger than the pump photon, and thus frequency up-conversion and down-conversion were simultaneously incorporated in the ultraviolet and visible regions. As indicated by the enhancement of parametric degenerate scattering in the center of the blue abaxial cone, the induced photo-charge transportation established a photo-refractive waveguide channel to slightly confine and thus enhance the cw-seeded parametric interaction. The parametric frequency up-conversion and down-conversion should be also slightly guided in the photo-refractive waveguide, which was confirmed by the quite narrow conical spread (radical width of the cone) of both UV and visible rainbows. As shown in Fig. 2 for the typical experimental observations, the UV and visible rainbows were spread radically with spatial widths comparable to those of the central cone of the enhanced abaxial degenerate scattering, much narrower than the signal and idler beam divergences, as an evident indicative of waveguide-confined parametric interactions.
We performed several further experimental studies on the observed rainbows. Firstly, we used two cw seed beams impinging the B2 instead of one while kept the pump unchanged. The He-Ne laser was split by a beam-splitter with a fixed ratio to form two cw seeds marked as (5, 6) in Fig. 2(c), which were then steered inside the B2 to overlap the pump beam along the 632.8-nm parametric cones at different propagation directions. Two different cw-seeded NOPAs occurred, and two groups of UV and visible rainbows were observed as shown in Fig. 2(c), with enhanced beam-alike scatterings (7, 8) along the respectively overlapped directions. The enhancement was caused by coherent accumulation of the corresponding UV and visible scatterings, dependent on the intensity ratio and seeding directions of the two cw seeding beams. Secondly, the cw seed was replaced by a pulse seed with a broad spectrum around 632.8 nm from temporally synchronized parametric super-fluorescence which was generated from an additional β-BBO crystal under the pump of a small split part of the 400-nm fs pulse. As shown in Fig. 2(d), enhanced rainbows were observed due to increased number of signal photons initially seeded within the pump duration. By changing the delay of the pulse seed, ultrafast switches of both UV and visible rainbows were demonstrated, indicating transient dynamics of no cascaded delays.
3. Discussions
In the NOPA, the pump, signal, and idler waves fulfill the non-collinear phase matching as schematically shown in Fig. 3(a): ωp = ωs + ωi and k⃗ep = k⃗oi + k⃗oi where ωp,s,i and k⃗ep,s,i represent the angular frequencies and wave-vectors of the pump, signal, and idler waves, and superscripts e and o refer to the extraordinarily and ordinarily polarized waves, respectively. It is well-known that the pump photons could bring about spontaneous down-conversion, originating from nonlinear growth of certain vacuum modes determined by the photon energy and momentum conservation: ωp = ω 1 + ω 2 and k⃗ep = k⃗o 1 + k⃗o 2, where the spontaneously down-converted photons (ω 1,k⃗o 1) and (ω 2,k⃗o 2) form correlated photon pairs. In the presence of the signal and idler photons in the configuration of NOPA, the (ω 1,k⃗o 1) and (ω 2,k⃗o 2) photons participate in cascaded quadratic nonlinear couplings with the signal ((ωs,k⃗os) and idler (ωi,k⃗is) photons through the processes
Fig. 3. Schematic diagram of cascaded parametric scattering to generate correlated up-conversion and down-conversion photons in cw or pulse seeded NOPA with non-collinear couplings among pump, signal, and idler pulses: (a) under phase-matched condition k⃗ep = k⃗os + k⃗oi and through spontaneous generation of down-conversion photon pairs k⃗ep = k⃗o 1 + k⃗o 2; (b) spontaneous down-conversion photon pairs interact in cascade with the signal and idler pulses of the NOPA to generate ultraviolet and visible rainbows with the photon momentum conservation k⃗ep + (k⃗os + k⃗oi) = k⃗eb + k⃗eg.
where ωb,g and k⃗eb,g represent angular frequencies and wave-vectors of photons generated by cascaded parametric processes. Such parametric cascades transform the single-photon fields a 1 at ω 1 (1 = 1,2,b,g) as
where gs and gi denote the coupling coefficients of the nonlinear interactions expressed in Eqs. (1&2) with classical signal and idler beams of electric amplitudes Es and Ei, respectively. As the parametric down-conversion photon pairs (ω 1,k⃗o 1) and (ω 1,k⃗o 2) may independently couple with the signal (ωs,k⃗os) and idler (ωi,k⃗oi) photons, respectively, the newly generated photons (ωb, k⃗eb) and (ωg,k⃗eg) may form correlated photon pairs in the ultraviolet and visible regions only when the cascaded parametric processes expressed in Eqs. (1&2) have sufficiently high conversion efficiencies. In principle, the photon correlation between (ω 1,k⃗o 1) and (ω 1,k⃗o 2) pairs can be completely transferred to the UV and visible photon pairs (ωb,k⃗eb) and (ωg,k⃗eg) as unity conversion efficiencies are reached for complete quantum up-conversions ab,(L) = a 1(0) and ag(L) = a 2(0) in the cascaded parametric processes. For an interaction length of 2 mm in β-BBO crystal, we estimate that such complete quantum up-conversions require signal and idler peak intensities about 1.0 and 5.0 kW, corresponding to 0.1 and 0.5 μJ per signal and idler pulse of ~ 100 fs in duration, respectively. Such requisite signal and idler pulse energies fell within the typical output signal and idler pulse energies of the cw-seeded NOPA used in the experiment. Under this circumstance, the parametric cascades mentioned above could be represented by the equivalent hyper-parametric processes as indicated in Fig. 3(b):
Fig. 4. Spectra of selected points on the observation screen. (a) Schematic positions of the selected scattering points; (b) typical spectra of the enhanced degenerate scattering; (c) spectra of some spots selected from the up-conversion rainbows; (d) spectra of some spots selected from the down-conversion rainbows.
Interestingly, the correlated photon pairs (ωb, keb) and (ωg, keg) along different wavelengths and positions consist of correlated rainbows, with photon-momentum-dependent spectra in a broadband tunable range and small beam divergences confined by photon energy and momentum conservation. This is consistent well with the experimental observation of up-conversion and down-conversion rainbows of gradually changed colors non-degenerate with the pump beam.
The photon-momentum-dependent spectra were confirmed by angularly-resolved spectral measurements of the rainbow scatterings. Figure 4(a) illustrates that both UV and visible rainbows exhibit angularly-resolved spectra, i.e., different spots in the observed rainbows exhibit different scattering spectra. As shown in Fig. 4(b), the enhanced blue abaxial scatterings possessed the same wavelength with the pump pulses at 400 nm, confirming that the scattering was originated from parametric processes degenerate in frequency [15–18]. Spectra at various scattered spots in Figs. 4(c) and (d) clearly indicate photon-momentum dependence and continuous tunability of both UV and visible rainbows. As consistent with color changes observed by naked eye, from spot to spot off the pump beam in the UV and visible rainbows, the up-conversion and down-conversion wavelengths were gradually blue-shifted from 400 to 300 nm and red-shifted from 500 to 650 nm, respectively, corresponding to frequency up-conversion
Fig. 5. Power dependence and temporal control of the converted scattering beam: the up-converted scattering intensity versus the input pump power (a) and the cw seed power (b); (c)&(d): ultrafast control of up-conversion and down-conversion rainbows with a pulsed seed along the parametric cone.
According to the Eqs. (5&6), the NOPA can be seen as a four-wave mixing process, in which the generated modes exhibit a quadratic pump power dependence [30]. As shown in 5Figs. 5(a) and (b), the intensities of the observed rainbows had a quadratic dependence on the pump intensity, and a nearly linear dependence on the cw seed power or pulsed seed intensity. The rainbows could be observed under a small peak-intensity pump down to 0.1 GW/cm2, and an extremely weak cw seed down to a few micro-watts, corresponding to a seed of only a few photons within the pump duration. In the case of pulse-seeded NOPA, the rainbows exhibited a delay dependence with typical full-widths at half-maximum (FWHMs) of 136 and 95 fs, as shown in Figs. 5(c) and (d) for up-conversion and down-conversion rainbows, respectively. This is consistent with our estimate that the signal and idler pulses were about 100 fs in duration. The up-conversion scattering exhibited a longer FWHM duration than the down-conversion one, which could be ascribed to different dispersions in the corresponding spectral regions. Such ultrafast switches further confirmed the transient features induced by nonlinear couplings between the pump, signal, and idler pulses. In addition, the sensitive dependence on the seed beams down to a few average photon numbers within the pump pulse duration suggests a perspective to realize an ultrafast all-optical control at few-photon or even single-photon level [31].
The photon-momentum conservation was verified by the angular distributions and angularly-resolved spectra of the UV and visible rainbows. From the experimentally measured angular and spectral distributions of the UV rainbow, we calculated angularly-resolved spectra of the visible rainbow on the basis of the cascaded parametric processes expressed in Eqs.(3&4) and compared them with the corresponding experimental data. As shown in Figs. 6(a) and (b), the theoretical fit with the experimental data confirms origin of the observed up-conversion and
Fig. 6. Angular position and angular-resolved spectra of the rainbow. (a)Experimentally measured angular positions of the up-conversion and down-conversion rain-bows(circles)with the numerically calculated down-conversion angles (squares); (b) Angularly-resolved spectra of the visible rainbow (circles) with its corresponding numeric calculations (squares).
We attenuated the SH pump so that the hyper-parametric scatterings occurred at the single-photon level. A time-interval analyzer was then used to record the time interval between the detected photons. We measured coincidence counts of the generated UV and visible photons by spectrally filtering the rainbows and detecting the corresponding photons with coincidence gates of 5 ns properly delayed from the pump pulses. Single-photon counting modules used for detecting the UV and visible photons were based on photomultiplier and silicon avalanche photodiode, respectively. Time-interval analysis was implemented by recording coincidence counts among the pump pulse sequence, which gave equivalently the time-coincidence counts. Figure 7(a) presents the coincidence-to-accidental ratio of photon pairs selected from the rainbows, which clearly demonstrates enhanced coincidence counts for the corresponding up-conversion and down-conversion photons from the same pump pulse (equivalent to zero delay in the time-interval analysis). The detected photon pair rate was about 2 × 10-3 per gate. This low value was mainly induced by the large loss of the UV filters in our experiment. The limited coincidence to accidental-coincidence ratio might come from the imperfect up-conversions of spontaneously down-conversion photon pairs in the nonlinear crystal, while the coincidence detection was interfered by parametric super-fluorescence and other scattering noises. As the SH pump was attenuated to about 0.5 μJ per pulse, we could still observe correlation between the UV and visible photons. Under this circumstance, the signal and idler pulse energies were dramatically reduced, much less than the requisite pulse energies for complete quantum conversion determined by Eqs.(3&4). If the UV or visible fields were in vacuum states before parametric interaction, even in the case of incomplete quantum conversion, the up-converted UV and visible photon numbers at ωb or ωg, according to the relationship expressed in Eqs.(3&4), were proportional to ⟨nb⟨ ∝= ⟨n 1⟨ sin2(∣gsEs∣L) and ⟨ng⟨ ∝= (n 2) sin2 (∣giEi∣L), respectively. While their correlation was proportional to ⟨nbng⟨ ∝ ⟨n 1 n 2⟨ sin2 (∣gsEs∣L) sin2 (∣gsEs∣L). We thus reached ⟨nb,ng⟨/⟨nb⟨(ng) = (n 1 n 2)/(n 1)(n 2), where nb,g and n 1,2 denote photon number operators at nb,g and n 1,2, respectively. This implies that the UV and visible photons could still be paired in the same way as the spontaneously down-conversion photon pairs even in the incomplete quantum conversion, which serves as the intrinsic origin of the observed correlation between UV and visible photons under extremely weak SH pump powers. However, this quantum pair conversion was broken if there existed some UV and visible background noise photons, lead-
Fig. 7. The coincidence of the UV and visible photon pairs. (a) coincidence-to-accidental ratio of up-conversion and down-conversion photons at 319 and 557 nm under a pump wavelength at 405.6 nm; (b) coincidence (C0) to accidental-coincidence (C1) ratio for the hyper-parametric down-conversion photon selected with a bandwidth of 3 nm and central wavelength tuned within 551–561 nm as the up-conversion photon fixed at 319 nm.
The correlation and angularly-resolved spectral measurements confirmed one-by-one coincidence of UV and visible photons. Along different propagation directions in the UV and visible rainbows, continuously tunable or multi-wavelength twin photons can be constructed. Under fs pump, the generated UV and visible photons are correlated within a relatively broad bandwidth. As shown in Fig. 7(b) ρ = ∑A(ωb,ωg)∣(ωb,ωg⟨⟩ωb,ωg∣, where the angularly resolved pairs are summed at different wavelengths with the corresponding probability A(ωb,ωg), and ∣ ωb, ωg represents a correlated pairs of UV and visible photons at angular wavelength ωb and ωg, respectively. The angularly-resolved features of the correlated rainbows make it easy to select multi-wavelength pairs of UV and visible photons by distilling different wavelengths along the corresponding directions. An interesting phenomenon occurs as the pump intensity decreases to generate only a single pair of correlated photons along the overlapped directions of UV and visible rainbows caused by two seeds, the corresponding single-photon scatterings may be possibly induced by either seedings. A entangled state could be established with superposition closely related with the seeding signal ∣ψ⟩ = ∣ωb, ωg)1 + ∣ωb, ωg⟩2 where ∣ωb, ωg⟩1 and ∣ωb, ωg⟩2 represent correlated photon pairs originated from the relevant NOPA with different seed signal, seed 1 or seed 2 as shown in Fig. 1(c).
The correlated rainbows may provide a solution for broadband tunable, multi-wavelength and even multipartite entanglements. Multiple correlated pairs of UV and visible photons can be distilled from the rainbows at different directions and wavelengths. As the ultraviolet rainbow was observed within 302 – 368 nm, with a typical spectral window of 1 nm width selected by using wavelength-division multiplex technology, at least 67 individual spectral windows could be angularly filtered to correlate with the corresponding visible rainbow (592 – 439 nm), which could be quite useful in quantum network [25]. Moreover, the UV rainbow can be used to generate twin photons in 604 – 736 nm by further parametric down-conversions, which may provide a solution for broadband tunable, multi-wavelength and multipartite entanglements [21,24,26]. These UV-visible twin photons can be further improved in spatial divergence by beam-alike parametric photonic cascades realized with two seed beams (Fig. 2(c)), and seed beam mediation offers readiness in ultrafast switch and linear control of multi-wavelength twin photons.
4. Conclusion
In summary, parametric photonic cascades have been demonstrated to support broadband correlated UV and visible rainbows with angularly-resolved spectra determined by photon energy and momentum conservation of cascaded parametric interactions, entailing correlated pairs of UV and visible photons with various advantages, such as angularly-resolved tunable spectra in broadband ranges. As a consequence, broadband tunable twin photons are generated by a technique intrinsically different from those hitherto available, supporting the generation of correlated photon pairs beyond the spectral ranges of spontaneous down-conversion and photonic cascade. It also permits ready multi-wavelength selection by easily selecting different positions in the UV and visible rainbows. In addition, we can easily generate twin photons of a broadband tunability. The cw or pulsed seed mediation of the observable rainbows may provide unique linear control of twin photons. Beam-alike parametric cascades generated by using two seeding beams can be also used to form twin photons of improved brightness or small beam divergence. And even different kinds of superposition may be constructed at single-photon level by controlling the ratio of seeding beams. The seeding beam control offer another interesting prospect of single-photon control, as the observed rainbows are sensitive to the initial seeding beam down to a few photons within the pump pulse duration. The transient features of the observed rainbows provide a unique technique of ultrafast switch down to few-photon level. We emphasize some important applications of our experiments. It is desirable for a broad range of fundamental and practical applications, particularly in quantum information science and technology, to have correlated or entangled photon sources of features to support multi-wavelength selection and broadband spectral tunability especially near some atomic or molecular resonances, and to generate energetic photons in the UV region, which are nevertheless un-realizable by using the currently available techniques. The technique demonstrated here supports the generation of correlated rainbows with broadband angularly-resolved spectra in the UV and visible regions. It can be used to generate correlated photon sources with many advantageous features, such as no cascade delays, continuous tuning by angular dispersion, photon-momentum-sensitive correlation between UV and visible photons, small beam divergence at phase-matched directions, broadband rainbows with angularly-resolved tunable spectra, multi-wavelength twin photons of well-dispersed spectra, and so on. This may eventually stimulate various experiments and applications in quantum information processing, such as quantum network, multipartite entanglement, quantum memory, and photonic interface.
Acknowledgement
This work was funded in part by National Natural Science Fund of China (10525416, 107740445 and 10990101), National Key Project for Basic Research (2006CB921105 and 2006CB806005), and Shanghai Leading Academic Discipline Project (B408).
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