Q uantum frequency conversion1,2 is a nonlinear process transducing an input beam of light with a given

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1 OPEN SUBJECT AREAS: THEORETICAL PHYSICS QUANTUM INFORMATION Received 5 October 3 Accepted December 3 Published 9 December 3 Correspondence nd requests for mterils should be ddressed to H.F. (hfn@iphy.c.cn) Tunble single-photon frequency conversion in Sgnc interferometer Wei-Bin Yn, Jin-Feng Hung,3 & Heng Fn Beijing Ntionl Lbortory for Condensed Mtter Physics, Institute of Physics, Chinese Acdemy of Sciences, Beijing 9, Chin, Stte Key Lbortory of Theoreticl Physics, Institute of Theoreticl Physics, Chinese Acdemy of Sciences, Beijing 9, Chin, 3 Deprtment of Physics nd Institute of Theoreticl Physics, The Chinese University of Hong Kong, Shtin, Hong Kong Specil Administrtive Region, People s Republic of Chin. Quntum informtion crriers like photons might be mnipulted, stored nd trnsmitted in different quntum systems. It is importnt to integrte those systems efficiently. The cpbility of converting photons from one wvelength to nother wvelength is key requirement for combining the photons in telecommunictions bnd for quntum trnsmission nd the photons in ner-visible bnd for quntum storge. Here, we investigte the tunble single-photon frequency conversion in the five-level emitter-sgnc interferometer system. We show tht the efficient single-photon conversion cn be chieved in this scheme, t the sme time, the frequencies of the input nd output photons cn be tuned in lrge scle by controlling the frequencies nd Rbi frequencies of the externl driving fields. The reliztion of this scheme my led to the efficient combintion of quntum storge system with the quntum communiction system. Q untum frequency conversion, is nonliner process trnsducing n input bem of light with given frequency into n output bem of light with different frequency. It hs mny criticl pplictions in quntum communiction nd quntum informtion processing 3 9. The highly efficient photon frequency conversion cn be chieved in the lrge-flux limit,. Recently, the uthors in Ref.,3 pointed out tht the highly efficient photon frequency conversion t low light levels hs not been chieved. To chieve this conversion, they proposed relizble scheme tht Sgnc interferometer 4 7 coupled to three-level quntum emitter. They show tht the highly frequency conversion t single photon level cn be chieved due to the interference when the coupling strengths between the different tomic trnsitions to the wveguide loop of the Sgnc interferometer re equl, nd dditionlly with resonnt condition which hs fixed the photon frequencies. We notice tht in the highly efficient photon frequency conversion, both t the lrge-flux limit or low-light level, the frequencies of the input nd output photons re limited in very smll scle. Yet, the tunble highly efficient frequency conversion in which both the frequencies of the input nd output photons cn be tuned in lrge scle hs not been explored. Here we propose scheme to chieve the tunble efficient single-photon frequency conversion. We show tht both of the frequencies of the input nd output photons cn be tuned in lrge scle by djusting the system prmeters in the efficient single-photon frequency conversion. When the frequency of the output photon is tuned higher thn the input photon, the up conversion is chieved, while the down conversion cn be chieved in the opposite sitution. We demonstrte this control with five-level emitter coupled to Sgnc Interferometer. Compred to the Sgnc interferometer coupled to three-level emitter,3, we show tht the efficient frequency conversion cn be chieved in either resonnce or off-resonnce cse. In the resonnce cse, the condition tht the different tomic trnsition-wveguide loop coupling strengths re equl is not necessrily essentil to obtin high conversion efficiency. This is more relizble under prcticl conditions. In prticulr, the efficient conversion cn lso be chieved for off-resonnce cse which permits tunble photon frequencies. The structure of the system under considertion is shown in Fig.. The Sgnc interferometer consists of 555 coupler nd wveguide loop. It cretes superposition of two counter-propgting photon sttes when single photon is injected into the setup. To void the output photon returning to the light source, supplementry route which is not illustrted here is necessry, s shown in,3. The emitter which will be mentioned s n tom below cn be rel tom or mnul tom-like object. The two tomic long-live sttes re denoted by jbæ nd jcæ, nd the excited sttes jæ, jdæ, nd jf æ. The tomic level frequencies re represented by v i (i 5, b, c, d, f). The tomic trnsitions jæ«jbænd jdæ«jcære coupled to the photons in the wveguide loop with strengths g nd g, respectively. The coupling strengths re ssumed to be independent of the wveguide wve number which is SCIENTIFIC REPORTS 3 : 3555 DOI:.38/srep3555

2 b g Ω Ω f Figure A Sgnc interferometer coupled to five-level emitter. Two externl clssicl fields re employed to drive the tomic trnsitions. equivlent to the Weisskopf-Wigner pproximtion. We employ two externl clssicl fields with frequencies (Rbi frequencies) v L ðv Þ nd v L ðv Þto drive the tomic trnsitions jæ«jfænd jdæ«jfæ, respectively. The five-level tomic configurtions hve been been studied extensively, for exmple 8,9. In this report, we first derive the trnsport property of the five-level tom coupled to the wveguide loop. We then find the system output stte of the tom coupled to the whole Sgnc interferometer by the scttering mtrix to study the controllble single-photon frequency conversion. The scttering mtrix of the Sgnc interferometer is S 5 S c S l S c, with S c ~ representing the bem splitter, nd { S l ~ representing tht the photon goes out from nother port different from the previous input port fter round in the wveguide loop. The wveguide loop cn be treted s one dimensionl wveguide which hs been studied extensively, for exmple, see 3 5. The time-independent Hmiltonin of the tom coupled to wveguide reds, H~ X ð ð v i sii {i dx { e ðþl x x e ðþ{i x dx { o ðþl x x o ðþ x i hp z ffiffi ð g dxdðþ x e ðþs x b ðþ p z ffiffi ð g dxdðþ x e ðþs x dc zv s f zv s df zh:c:, with v f ~v f zv L, v d ~v dzðv L {v L Þ, v c ~v czðv L {v L Þ nd s ij 5 jiæ Æjj denoting the tomic rising, lowering nd energy level popultion opertors. Here we hve tken h~, nd the photonic group velocity v g 5. It cn be seen tht the externl fields shift the tomic levels. The expressions of the even nd odd opertors re { e ðþ~ x h i { R ðþz{ x L ð{xþ nd { o ðþ~ x h i { R ðþ{{ x L ð{xþ, with the opertor { R ðþnd x { LðÞcreting x clockwise nd counterclockwise moving photon in the wveguide 3,33, respectively. Note tht the effective tomic frequency, v cb ~v c{v b zðv L {v L Þ,is relted to the externl field frequencies. We ssume tht, initilly, the tom is in the stte jbæ, nd photon with the wve number k is injected into the wveguide loop. After scttering, the tom is in the stte jbæ or jcæ, with the corresponding wve number of the output photon k nd k9, respectively. The former corresponds to the elstic scttering nd the ltter to the inelstic scttering. For the inelstic scttering, the frequency of the output photon depends on the externl field frequencies. Therefore, it is essentil to mke sure tht the input photon is merely inelsticlly scttered for vrious vlues of d g c the externl field frequencies to chieve the tunble frequency conversion. Results Single-photon frequency conversion properties. For n input photon split by the 555 coupler, the superposition of the clockwise nd counterclockwise moving sttes cn be prepred in the wveguide loop. In certin cses, the interference resulting from the superposition hs constructive effect on the inelstic scttering nd destructive effect on the elstic scttering. Once the reltive phse between the photonic clockwise nd counterclockwise moving sttes is zero, the scttered stte cn be obtined s jyi~t jb, k izt jc, k i: ðþ with t ~ h l, t ~ i ffiffiffiffiffiffiffiffi C C V V, l h~d ðd {D ÞðD {D zd Þ{V D zc C ðd {D Þ{V ð D {D zd Þ zi½c ðd {D ÞðD {D zd Þ {C ðd {D ÞD {V C zv C l~d ðd {D ÞðD {D zd Þ{V D {C C ðd {D Þ{V ð D {D zd Þ {i½c ðd {D ÞðD {D zd Þ zc ðd {D ÞD {V C {V C where D 5 v k, D ~v f {v L, D ~v df {v L, nd C m ~ g m ðm~,þ representing the tomic decy rte into the v g wveguide loop due to the coupling. When t 5, the inelstic scttering process converts the input single photon into n output photon of the wve number k9 with unity conversion efficiency. The control of the frequency of the output photon for high conversion efficiency is our prime concern. The frequency of the output photon fter the inelstic scttering is obtined s v ~v{ ½v cb zðv L {v L ÞŠ which cn be controlled by tuning the frequencies of the externl lsers. This cn be understood by the energy conservtion. When v cb zðv L {v L Þw, the down conversion cn be chieved fter the inelstic scttering, nd when v cb zðv L {v L Þv, the up conversion cn be chieved. Obviously, if the resonnce condition is stisfied i.e., D 5 D 5 5, we cn obtin the unity conversion efficiency when D C ~ V C V. The coupling strength g is usully different from the other strength g becuse they depend on the tomic dipole. Hence, the controllble Rbi frequencies enble us to obtin unity conversion efficiency in the resonnce cse. Fig. shows the conversion properties jt j nd jt j ginst the frequency of the input single photon when the externl lsers drive the tomic trnsitions resonntly when C ~ V C V. For smll Rbi frequencies, the spectr re shped like the Lorentzin line. The spectr split with the incresing Rbi frequencies. When C 5 C 5 C, nd V ~V ~V, we cn find D D t ~ {V zc D 3 {V D {C D {i CD. Obviously, when C {CV V $, the unity conversion efficiency cn be chieved only when the input photon intercts with the tom resonntly. However, when C V,, the unity conversion efficiency cn lso be obtined even when the input photon is off-resonnt to the tomic trnsition s shown in Fig. (d). ð3þ SCIENTIFIC REPORTS 3 : 3555 DOI:.38/srep3555

3 () (b) Δ 8 6 /Γ 8 4 Δ /Γ 4 8 (c) (d) Δ /Γ Δ /Γ 4 Figure Frequency efficiency conversion properties ginst the input-photon frequency. The red dshed lines re t nd the blue solid lines re t. We hve tken C =C ~V V nd D p 5 D 5 in ll of the plots. The respective prmeters re ()C 5 C, V ~5 ffiffi p C, (b)c 5 C, V ~ ffiffi p C, (c)c 5 C, V ~:5 ffiffi C, (d)c 5 C, V 5 C. Tunble single-photon frequency conversion. In the cse discussed bove, the externl clssicl frequencies re fixed nd then cn not be tuned to stisfy the resonnce condition. In order to chieve the tunble frequency of the converted output photon, the unity conversion efficiency in the off-resonnce cse is required. In the detuned cse, the condition t 5 requires V ~ C ðd {D Þ D zc, ðd {D zd ÞC zd C V ~ C ðd {D Þ ðd {D zd Þ zc : ðd {D zd ÞC zd C ð4þ C ðd {D Þ D Therefore, the conditions zc w nd ðd {D zd ÞC zd C C ðd {D Þ ðd {D zd Þ zc w re essentil to obtin ðd {D zd ÞC zd C unity conversion efficiency. Although these conditions cn not be stisfied for ny rbitrry vlue of the frequencies of the externl fields, they cn be fulfilled in lrge rnge of the frequency vlues. This fesible rnge is enough for the djusting of the convertedphoton frequency in wide scle. To explin this, we plot the Rbi frequencies V nd V ginst the frequencies of the externl fields when t 5 in Fig. 3. In Fig. 3() nd 3(b), we show the required Rbi frequencies when we djust both the externl frequencies together. 5 (c) Ω /Γ Ω /Γ 5 (d) Ω /Γ Ω /Γ Δ /Γ Δ /Γ Figure 3 The vlues of Rbi frequencies ginst the frequencies of externl driving lsers when the photon conversion efficiency is unity. () nd (b) re V nd V ginst the two lser frequencies, respectively. The prmeters re D 5 3C, C 5 C. We tke D 53C in (c), nd D 5 5C in (d). SCIENTIFIC REPORTS 3 : 3555 DOI:.38/srep3555 3

4 () (b) Δ /Γ. (c) 5 5 Δ 5 /Γ Δ /Γ 5 (d) Δ /Γ 5 Figure 4 The photon conversion properties influenced by dissiption. (), (b) nd (c) show the probbilities t, t, nd the totl probbility t jt t j, which correspond to the red dshed lines, blue solid lines, nd blck dshed dotted lines, respectively. (d) shows the probbility jt j. The blue zjt j solid line, red dshed line, nd blck dshed dotted line denote the sitution s shown in (), (b) nd (c), respectively. For ll the plots, the dissiption rte is tken 5. nd the coupling strengths C 5 C. The respective prmeters re () V ~ 5 C, V ~ 5 C p, D 5 D 5, (b) V ~5 ffiffi C, V ~5C, ffiffiffi ffiffiffiffiffi 9 D 5 5C, D 5 D 5, (c) V ~ 3 C 4, V ~ C, D 5 D 54C. 3 Fig. 3(c) nd 3(d) show the Rbi frequency requirement when we djust one of the externl frequency while the other frequency is fixed. Fig. 3 shows tht for the lrge scle of the externl lser frequencies, the essentil conditions bove cn be stisfied nd the pproprite vlues of the Rbi frequencies cn be found. Therefore, we cn control the frequency of the converted output photon by controlling the frequencies of the externl lser nd tune the Rbi frequencies to obtin unity conversion efficiency. Although the injected photon is not resonnt with the tom, the suitble prmeters of the externl lsers cn ensure the conversion complete. The frequency conversion process cn be understood s photon trpping process. After the inelstic scttering, the injected photon is trpped nd the tom is in the stte jcæ, with nother photon b creted. Besides, the trpped photon cn be retrieved by injecting the photon b. It mens tht, the photon is trpped for complete conversion. The retrievl processing corresponds to the complete conversion b R. The retrievl efficiency cn be computed when the tomic initil stte is jcæ by similr clcultion done bove. Obviously, the retrievl efficiency cn be unity under suitble condition. Dissiption cse. The intrinsic dissiption is hrmful to chieve the unity conversion efficiency. This dissiption cn be incorported by introducing the nonhermitin Hmiltonin H non ~{i P j~,f,d j ji j hjin j the quntum jump picture, with j being the decy rte to other modes except the mode of the wveguide loop from the level jjæ for rel tom nd being the decy rte plus dephse rte for mnul tom-like object. As shown bove, complete conversion cn be chieved in the resonnce nd off-resonnce cses under the idel condition. Fig. 4(), 4(b) nd 4(c) plot the conversion properties in both the cses fter considering the dissiption. The strong coupling nd lrge detuning cn tolerte the dissiption better. Fig. 4(d) plots the probbility F~ jt j jt j zjt j. The high conversion efficiencies cn be obtined in the cse s shown in Fig. 4(b) nd Fig. 4(c). The probbility F cn be nerly unity which mens tht the input photon is dissipted nd converted, nd little elstic scttering exists. We note nother restricting condition tht the Rbi frequencies cn not be too smll in order to tolerte the dissiption. Fortuntely, fter considering this condition, the tunble frequency scle of the output photon is little ffected, which cn be understood from Fig. 3. We hve study the cse tht the input light is monochromtic. For input pulse with finite bndwidth, the conversion efficiency decreses, which cn be seen in Fig. 4(c). To chieve the efficient single-photon frequency conversion, the nrrow bndwidth of the input pulse is necessry. Discussion We propose tunble single-photon frequency conversion scheme with high efficiency. The inelstic scttering shifts the frequency of the input photon. Especilly, in the off-resonnce cse, the frequency shift cn be tuned by djusting the externl clssicl fields. Thus, the output frequency is tunble. The dissiption will in generl diminish the conversion efficiency from the unity nd the output photon is mostly the inelsticlly scttered photon. Hving considered the dissiption nd the nrrow bndwidth of the input pulse, the high efficiency cn lso be chieved. The reliztion of this scheme my combine the quntum informtion processing system with the longdistnce quntum communiction system. Methods The one-excittion stte of the wveguide-tom system cn be written s ð jyi~ dxbðþ x { e ðdxc ðþz x ðþ x { e ðþscb x zas b zfs fb zds db jb,i, ð5þ where B(x), C(x), A, F, nd D re mplitude probbilities, nd jb, æ represents tht the tom is in the stte jbæ nd the photon number in the wveguide is zero. Under the nstz B(x) 5 [h(x) t h(x)]e ikx nd C(x) 5 t h(x)e ik9x, we cn find the solution of the time-independent Shrödinger eqution HjYæ 5 EjYæ. The sttionry stte evolves with time s jy(t)æ 5 e iet jyæ. After clcultion 43, the trnsport properties t nd t re obtined s in Eq. (3). SCIENTIFIC REPORTS 3 : 3555 DOI:.38/srep3555 4

5 Going bck to the clockwise nd counterclockwise picture from the even nd odd picture, the scttering mtrix of the emitter coupled to the wveguide loop cn be 3,33 derived from t nd t nd then the whole system scttering mtrix cn be clculted. As long s ny one of the Rbi frequencies {V, V } is zero, the frequency conversion efficiency is zero due to the fct tht the tomic trnsition jdæ «jcæ decouples from the photon in the wveguide nd hence the inelstic scttering vnishes. In detil, when V 5, we cn find t ~ D zic nd t 5, which is the D {ic sme s two-level system coupled to the wveguide And when V 5, we cn find t ~ D ðd {D Þ{V zi ð D {D ÞC D ðd {D Þ{V {i D nd t 5, corresponding to L threelevel tom coupled to the wveguide 44. This lso revels tht the frequency conversion ð {D ÞC cn be switched off by shutting off the externl clssicl field, which is equivlent to the control of the reltive phse shift between the clockwise nd counterclockwise moving photon. When the reltive phse is p, n odd-mode qusi prticle is prepred in the wveguide loop nd the destructive interference mkes the frequency conversion efficiency zero. The reltionship jt j jt j 5 cn be esily checked. The mximl frequency conversion efficiency is when photon moves only clockwise or only counterclockwise towrds the tom in the wveguide loop. In this cse, the output stte hs i~ p ffiffi jb, k mximlly entngled stte. the form of jy p e iw jc, k i with w being rel number, which is iz ffiffi. Kumr, P. Quntum frequency conversion. Opt. Lett. 5, 476 (99).. Hung, J. & Kumr, P. Observtion of quntum frequency conversion. Phys. Rev. Lett. 68, 53 (99). 3. Gibbs, H. M. Opticl Bistbility: Controlling Light with Light. (Acdemic, Orlndo, 985). 4. Bouwmeester, D., Ekert, A. & Zeilinger, A. The Physics of Quntum Informtion. (Springer, Berlin, ). 5. McCutcheon, M. W., Chng, D. E., Zhng, Y., Lukin, M. D. & Loncr, M. 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Few-photon trnsport in lowdimensionl systems: interction-induced rdition trpping. Phys. Rev. Lett. 4, 36 (). 48. Zheng, H., Guthier, D. J. & Brnger, H. U. Cvity-free photon blockde induced by mny-body bound sttes. Phys. Rev. Lett. 7, 36 (). 49. Zheng, H. & Brnger, H. U. Persistent quntum bets nd long-distnce entnglement from wveguide-medited interctions. Phys. Rev. Lett., 36 (3). 5. Hung, J. F., Shi, T., Sun, C. P. & Nori, F. Controlling single-photon trnsport in wveguides with finite cross-section. Phys. Rev. A 88, 3836 (3). Acknowledgments This work is supported by 973 progrm (CB994), NSFC (7548), nd grnts from Chinese Acdemy of Sciences. Author contributions W.-B.Y. nd H.F. proposed the model. W.-B.Y. clcultes the system scttering properties while J.F.H. nd H.F. provide the technology dvices. W.-B.Y., J.-F.H. nd H.F. nlyzed the results. W.-B.Y. nd H.F. wrote the pper. Additionl informtion Correspondence Correspondence nd requests for mterils should be ddressed to H.F. or W.-B. Y. 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6 Competing finncil interests: The uthors declre no competing finncil interests. How to cite this rticle: Yn, W.-B., Hung, J.-F. & Fn, H. Tunble single-photon frequency conversion in Sgnc interferometer. Sci. Rep. 3, 3555; DOI:.38/srep3555 (3). This work is licensed under Cretive Commons Attribution 3. Unported license. To view copy of this license, visit SCIENTIFIC REPORTS 3 : 3555 DOI:.38/srep3555 6