High-flux photon-pair source from electrically induced parametric down conversion after second-harmonic generation in single optical superlattice

We present here a possible high-flux photon-pair source constructed by single lithium niobate optical superlattice (OSL) with a combined quasi-periodically and periodically poled structure, which is from the principle of electrically induced parametric down conversion (PDC) after second-harmonic generation (SHG), predicted by the united theory developed in this paper, in which SHG, PDC and electro-optic (EO) effect are comparably treated as two-order nonlinear effects. In the OSL, the epolarized fundamental frequency photons are first converted to double frequency ones with the same polarization; then the PDC process is triggered by EO effect when the fundamental frequency photons are almost exhausted; finally, the double frequency photons are converted again to a series of two-photon pair of fundamental wave. It is demonstrated that at 100 C, in a mm 2 . 20 long OSL with a mm V / 30 applied electric field, a 100MW/cm, 1080 nm laser beam can be translated to a flux of high-purity two-photon pairs with a conversion efficiency close to 100%; and for a longer OSL the pump intensity can be further lowered. The device can also act as an ultra-low field electro-optic switch. ©2007 Optical Society of America OCIS codes: (190.2620) Frequency conversion; (190.4410) Nonlinear optics, parametric processes; (270.0270) Quantum optics. References and links 1. A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991). 2. C. H. Bennett and S. J. Wiesner, “Communication via oneand two-particle operators on Einstein-PodolskyRosen states,” Phys. Rev. Lett. 69, 2881 (1992). 3. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. 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Introduction
Quantum entanglement plays a key role in quantum information science and has attracted lots of attention.The entanglement light is one of important resources in this realm, which is with potentially useful applications from quantum communications, including cryptography [1] and dense coding [2], to quantum teleportation [3], "entanglement swapping" [4] and quantum computation [5].Spontaneous [6][7][8] and seed injected [9,10] PDC with type I [11] or type-II [12] phase matching are very useful techniques to generate entanglement and were widely used in experiments.It is a pity that spontaneous PDC is usually so weak, especially from the non-collinearly phase matching in which only small amount of photons in the output cone is collectible [13], being inconvenient for coupling the emitted photons into optical fibers [14], that it is unsuitable for long distance communications.Besides, to get entanglement beams in traditional methods, a double frequency pump laser beam [13][14][15] or two separated sets for SHG and PDC [16,17] were required.However, a simple high-flux source of entangled photons, without two separated sets but with the same frequency of pump and signal beam, would be expected for some practical implementation of applications.
The optical parametric processes in periodically or/and quasi-periodically poled ferroelectric materials may be hopeful for our intention since it is with reasonably high efficiency and flexibility controllable.In fact, some efficient SHG [18][19][20], sum-frequency generation (SFG) [21], and optical parametric oscillation (OPO) [22,23], notably thirdharmonic generation [24][25][26] in these materials have been reported.Recently, cascaded frequency conversion and signal tuning in these materials received much interest [27][28][29][30][31][32][33][34].A quasi-phase matched (QPM) crystal composed of an unpoled lithium niobate (LN) dispersion section sandwiched between two periodically poled LN (PPLN) sections was constructed, in which amplitude modulation and frequency conversion can be obtained simultaneously by using the EO effect to control the relative phase among mixing waves [30].The two functions were integrated in single OSL [31] subsequently.Very recently, a way different from those reported by Refs.[30] and [31], using an applied electric field to control both polarization and magnitude of the second harmonic, was presented [34].Besides, QPM materials have also been used for efficient generation of photon pairs [13,16,35].In this paper, we present a principle to integrate two functions including SHG and PDC into single LN OSL therefore yield squeezed light that could be turned into entangled light beams by using a 50/50 Y-type single mode fiber optic beam splitter [35].The basic idea of the principle is as follows: construct such an OSL with an applied electric field, in which the double frequency process is performed at first and then the PDC process is turned on when the fundamental frequency photons are almost converted to double frequency ones; finally, the double frequency photons are converted again to a series of two-photon pair of fundamental wave.The key step is utilizing EO effect to trigger the PDC process.

The united theory of SHG, PDC and EO effect
To discuss our principle we first develop a wave coupling theory describing the united effect of EO, SHG and PDC in a PPLN or/and quasi-periodically poled LN (QPPLN) OSL following the idea involved in Ref.s [36,37] instead of that including the use of refractive index ellipsoid theory in Ref. [34].In the united theory SHG, PDC and (EO) effect are comparably treated as two-order nonlinear effects.Figure 1 shows a schematic diagram of SHG and PDC in the OSL controlled by an applied electric field, where the propagation direction and the polarization of pump wave are along the x -axis and z -axis of the OSL, respectively.And the applied dc electric field 0 E is along the y -axis of Section 1 of the OSL; F is a filter, only allowing e-polarized fundamental wave to pass.Section 1 and 2 represent QPPLN and PPLN, respectively.When the pump wave is injected into the OSL, multi-effects including EO coupling [38] and frequency conversions [39] appear simultaneously if there exits an external electric field.We take EO effect as the second-order nonlinear one as did Ref.[36,37] for LN, the coupling equations for present case (EO effect: ; SHG and/or PDC: ) can be deduced from Maxwell's equations as follows: where  are the double frequency and EO coefficients, respectively; c is the speed of light in vacuum; the asterisk denotes complex conjugation; and is the structure function that is +1 or -1 for the positive or negative domains of the OSL, respectively.The right side of each equation includes two parts: the bracketed stands for SHG and PDC; and the others refer to EO effect.When the external electric field is absent, the coupling equations (1)-( 4) reduce to the familiar wave coupling equations describing SHG or PDC.

High-flux photon-pair source from electrically induced parametric down conversion after second harmonic generation in single OSL
We now use Eqs.( 1)-( 4) to find a OSL that satisfies our requirement.As well known, the key to QPM in OSL is to design a structure that provides a set of reciprocal vectors to compensate for the mismatches of wave vectors owing to the dispersion of refractive index.In general, the one-component periodic structure can provide one reciprocal vector to compensate for one mismatch of wave vector, but in special case, the first-order and third-order reciprocal vectors of the structure can simultaneously compensate for the phase mismatches of second-harmonic and sum-frequency effects, respectively [41].The two-component quasi-periodic OSL providing two reciprocal vectors has been proved useful to some coupled parametric processes such as the direct third-harmonic generation [25,26].One can see, from Eqs. ( 1)-( 4), that in general, there exist six mismatches of wave vectors relative to SHG, PDC and EO effects.For our purpose, however, we need only to take account of two compensations for , respectively.Therefore the structure of QPPLN can be determined [42] to be ABABABABABA…, in which block A or B is further composed of a positive ferroelectric domain and a negative one, respectively, i.e., k can be determined [39], which are 10

− + + =
, respectively.For the PPLN, the fundamental block A' is further composed of one positive and one negative ferroelectric domain, whose length is . Then we expand ) (x f as a Fourier series such as .Therefore, by ignoring those terms with nonzero mismatches of wave vectors, Eqs. ( 1)-( 4) can be simplified as We fix the pump intensity of e-polarized fundamental wave at   One can see from Fig. 2 that the applied electric field can effectively control the parametric processes in OSL.For the case of zero electric field, OSL as a whole plays a role of frequency doubler and only the SHG takes place since in this case three coupling Eqs. ( 8) -( 10) reduce to two ones describing SHG (please note that the coupling equations describing PDC are formally the same as those of SHG).The e-polarized fundamental frequency (pump) photons are only converted to second harmonic ones with the same polarization [Fig.2(A)].But for a properly weak electric field, for example, 30V/mm, due to the compensations of a k Δ and f k Δ two processes including SHG and EO effect take place synchronously in the first section of OSL and the PDC process occurs in the second section of OSL after SHG and EO effect [Fig.2(B)].In the first section of OSL a large portion of e-polarized fundamental frequency photons are converted to double frequency ones with the same polarization and a very small portion of them are turned into o-polarized fundamental frequency photons by EO effect.At  the double frequency photons are almost fully converted to a series of fundamental two-photon pairs still with e-polarization through PDC.Actually, in the OSL the second harmonic only plays a role of intermedium.When further increasing the electric field, for example, , the EO effect becomes stronger, which might obstruct the generation of pure two-photon pairs of e-polarized fundamental wave, because the e-polarized fundamental seeds for PDC is much more than that at lower electric field.For the process of degenerate PDC described above, the Heisenberg equations of motion are [44] which lead immediately to where , are the photon number operators of double frequency and fundamental lights, respectively.Besides, we have the frequency relation Equation ( 15) tells us that a pair of fundamental photons is consequentially created when a double frequency photon disappears.And Eq.( 14) tells us that at any moment, the increase of photon number of fundamental light is always the double of the decrease of photon number of double frequency one.Since at the beginning of parametric downconversion, the seed of epolarized fundamental one is so weak relative to the double frequency one that it can be ignored; the e-polarized double frequency photons and residual o-polarized fundamental ones can be conveniently removed by a band-polarizing filter F, the fundamental output will consist mainly of photon pairs.The output can be coupled to a 50/50 Y-type single mode fiber optic beam splitter to separate the twin photons [35].Two output beams of the splitter are then directed into two single mode fibers that are so long enough that their output beams become very weak, in which the photons do not overlap with each other.The two weak are introduced to passively quenched LN 2 cooled germanium avalanche photodiodes (Ge-APDs) operating in Geiger mode for detecting [35].The coincidence rate can be obtained by using two onechannel analyzers [45].In a word, applying a properly weak electric field on an OSL with appropriate structure and length, pumped by a laser beam, we can get a source of high-flux and high-purity two-photon pairs.The further calculation indicates that if a longer OSL is used the intensity of pump beam can be lowered for performing the same function.For example, for an ).The second harmonic also only plays a role of intermedium in the device.

Conclusion
In conclusion, a united wave coupling theory describing SHG, PDC and EO effect in PPLN or/and QPPLN OSL has been developed in this letter to the best of our knowledge for the first time, which shows a principle for constructing a high-flux photon-pair source from electrically induced PDC after second-harmonic generation in single OSL.It is demonstrated that at 100 o C, in a mm 2 .20 long OSL with appropriate structure and a mm V / 30 field, a 100MW/cm 2 , 1080 nm laser beam can be translated to a flux of high-purity twophoton pairs with a conversion efficiency close to 100% ; and for a longer OSL the pump intensity can be further lowered.The device can also act as an ultra-low field electro-optic switch, which requires much lower field ( mm V / 10 ~) than that of traditional one with halfwave field ( mm V / 10 10 ~3 2 − ) [38].It can be used as Q switch [46,47] or low-driving-power and high-speed optical switch in optical communication network or optical signal-processing system.

Fig. 1 .
Fig. 1.Schematic diagram of the SHG, PDC in the OSL controlled by an applied electric field.x , y and z represent three axes of the crystal.The arrows indicate the polarized directions.F is a filter, only allowing e-polarized fundamental wave to pass.Section 1 and 2 represent QPPLN and PPLN, respectively.

.
in the first section of OSL and one compensation for f k Δ (PDC) in the second section of OSL, respectively.So we can choose a hybrid OSL consisting of a two-component QPPLN (section 1) and a one-component PPLN (section 2).Suppose that a nm 1080 extraordinary wave is used as a pump one and the temperature is at According to Ref.[42], the reciprocal vector conditions of QPM in section 1 are

Δ
can be compensated by 1 G in section 2. Under these conditions,

#(
82250 -$15.00USD Received 19 Apr 2007; revised 1 Jun 2007; accepted 1 Jun 2007; published 18 Jun 2007 more suitable.The numerical results reflecting the dependences of normalized intensities of e-polarized pump fundamental wave, e-polarized second harmonic and o-polarized fundamental wave respectively on the length of the crystal and electric field are shown in Fig.2, in which (A)-(C) corresponds to electric fields 0, 30 and 100V/mm, respectively.In the calculations the Sellmeier equation[43] for LN and 0

Fig. 2 .
Fig. 2. Dependence of normalized intensities of e-polarized pump fundamental wave, epolarized second harmonic and o-polarized fundamental wave respectively on the length of the crystal with different external electric field.The total length of the OSL is mm 2 .20 , and the pump intensity is

;.
e-polarized fundamental frequency photons are converted to the double frequency ones and the remnant ratio is about while the o-polarized fundamental wave from EO effect is with a normalized intensity of In the first section of OSL there also exists another process: the o-polarized fundamental frequency photons are partially turned, again by EO effect, into a part of the seeds of e-polarized fundamental frequency photons for PDC, which is the key turning PDC in the second section of OSL.At

30 .
One can see in fact that the device can also act as an ultralow field electro-optic switch, which requires much lower field ( . So, corresponding to each monochromatic component of light i