Tunable single-photon frequency conversion in a Sagnac interferometer

Quantum information carriers like photons might be manipulated, stored and transmitted in different quantum systems. It is important to integrate those systems efficiently. The capability of converting photons from one wavelength to another wavelength is a key requirement for combining the photons in telecommunications band for quantum transmission and the photons in near-visible band for quantum storage. Here, we investigate the tunable single-photon frequency conversion in the five-level emitter-Sagnac interferometer system. We show that the efficient single-photon conversion can be achieved in this scheme, at the same time, the frequencies of the input and output photons can be tuned in a large scale by controlling the frequencies and Rabi frequencies of the external driving fields. The realization of this scheme may lead to the efficient combination of quantum storage system with the quantum communication system.

equivalent to the Weisskopf-Wigner approximation. We employ two external classical fields with frequencies (Rabi frequencies) v L1 V 1 ð Þ and v L2 V 2 ð Þ to drive the atomic transitions jaae « jf ae and jdae « jf ae, respectively. The five-level atomic configurations have been been studied extensively, for example 28,29 . In this report, we first derive the transport property of the five-level atom coupled to the waveguide loop. We then find the system output state of the atom coupled to the whole Sagnac interferometer by the scattering matrix to study the controllable single-photon frequency conversion. The scattering matrix of the Sagnac interferometer is S 5 S c S l S c , with representing the beam splitter, and S l~0 1 1 0 representing that the photon goes out from another port different from the previous input port after a round in the waveguide loop. The waveguide loop can be treated as a one dimensional waveguide which has been studied extensively, for example, see  .
The time-independent Hamiltonian of the atom coupled to a waveguide reads, ð dxd x ð Þa e x ð Þs dc zV 1 s af zV 2 s df zh:c: with ð Þ and s ij 5 jiae AEjj denoting the atomic raising, lowering and energy level population operators. Here we have taken h~1, and the photonic group velocity v g 5 1. It can be seen that the external fields shift the atomic levels. The expressions of the even and odd operators are with the operator a { R x ð Þ and a { L x ð Þ creating a clockwise and counterclockwise moving photon in the waveguide 32,33 , respectively. Note that the effective atomic frequency, v 0 cb~v c {v b z v L1 {v L2 ð Þ , is related to the external field frequencies. We assume that, initially, the atom is in the state jbae, and a photon with the wave number k is injected into the waveguide loop. After scattering, the atom is in the state jbae or jcae, with the corresponding wave number of the output photon k and k9, respectively. The former corresponds to the elastic scattering and the latter to the inelastic scattering. For the inelastic scattering, the frequency of the output photon depends on the external field frequencies. Therefore, it is essential to make sure that the input photon is merely inelastically scattered for various values of the external field frequencies to achieve the tunable frequency conversion.

Results
Single-photon frequency conversion properties. For an input photon split by the 50550 coupler, the superposition of the clockwise and counterclockwise moving states can be prepared in the waveguide loop. In certain cases, the interference resulting from the superposition has a constructive effect on the inelastic scattering and a destructive effect on the elastic scattering. Once the relative phase between the photonic clockwise and counterclockwise moving states is zero, the scattered state can be obtained as Þ representing the atomic decay rate into the waveguide loop due to the coupling. When t 2 5 1, the inelastic scattering process converts the input single photon into an output photon of the wave number k9 with unity conversion efficiency. The control of the frequency of the output photon for a high conversion efficiency is our prime concern. The frequency of the output photon after the inelastic scattering is obtained as which can be controlled by tuning the frequencies of the external lasers. This can be understood by the energy conservation. When v cb z v L1 {v L2 ð Þ w0, the down conversion can be achieved after the inelastic scattering, and when v cb z v L 1 {v L 2 ð Þ v0, the up conversion can be achieved. Obviously, if the resonance condition is satisfied i.e., D a 5 D 1 5 D 2 5 0, we can obtain the unity conversion efficiency when The coupling strength g 1 is usually different from the other strength g 2 because they depend on the atomic dipole. Hence, the controllable Rabi frequencies enable us to obtain a unity conversion efficiency in the resonance case. Fig. 2 shows the conversion properties jt 1 j 2 and jt 2 j 2 against the frequency of the input single photon when the external lasers drive the atomic transitions resonantly when For small Rabi frequencies, the spectra are shaped like the Lorentzian line. The spectra split with the increasing Rabi frequencies. When C 1 5 C 2 5 C, and V 2 1~V 2 2~V 2 , we can find Obviously, when C 2 2 2V 2 $ 0, the unity conversion efficiency can be achieved only when the input photon interacts with the atom resonantly. However, when C 2 2 2V 2 , 0, the unity conversion efficiency can also be obtained even when the input photon is off-resonant to the atomic transition as shown in Fig. 2(d). Tunable single-photon frequency conversion. In the case discussed above, the external classical frequencies are fixed and then can not be tuned to satisfy the resonance condition. In order to achieve the tunable frequency of the converted output photon, the unity conversion efficiency in the off-resonance case is required. In the detuned case, the condition t 1 5 0 requires Therefore, the conditions w0 are essential to obtain a unity conversion efficiency. Although these conditions can not be satisfied for any arbitrary value of the frequencies of the external fields, they can be fulfilled in a large range of the frequency values. This feasible range is enough for the adjusting of the convertedphoton frequency in a wide scale. To explain this, we plot the Rabi frequencies V 1 and V 2 against the frequencies of the external fields when t 1 5 0 in Fig. 3. In Fig. 3(a) and 3(b), we show the required Rabi frequencies when we adjust both the external frequencies together. We have taken C 1 =C 2~V 2 1 V 2 2 and D 1 5 D 2 5 0 in all of the plots. The respective parameters are (a)C 1  are V 1 and V 2 against the two laser frequencies, respectively. The parameters are D a 5 3C 2 , C 1 5 2C 2 . We take D 1 5 23C 2 in (c), and D 1 5 5C 2 in (d).
www.nature.com/scientificreports Fig. 3(c) and 3(d) show the Rabi frequency requirement when we adjust one of the external frequency while the other frequency is fixed. Fig. 3 shows that for the large scale of the external laser frequencies, the essential conditions above can be satisfied and the appropriate values of the Rabi frequencies can be found. Therefore, we can control the frequency of the converted output photon by controlling the frequencies of the external laser and tune the Rabi frequencies to obtain a unity conversion efficiency. Although the injected photon is not resonant with the atom, the suitable parameters of the external lasers can ensure the conversion complete. The frequency conversion process can be understood as a photon trapping process. After the inelastic scattering, the injected photon a is trapped and the atom is in the state jcae, with another photon b created. Besides, the trapped photon can be retrieved by injecting the photon b. It means that, the photon is trapped for a complete conversion. The retrieval processing corresponds to the complete conversion b R a. The retrieval efficiency can be computed when the atomic initial state is jcae by a similar calculation done above. Obviously, the retrieval efficiency can be unity under a suitable condition.
Dissipation case. The intrinsic dissipation is harmful to achieve the unity conversion efficiency. This dissipation can be incorporated by introducing the nonhermitian Hamiltonian H non~{ i P j~a,f ,d j 2 j j i j h j in the quantum jump picture, with j being the decay rate to other modes except the mode of the waveguide loop from the level jjae for a real atom and being the decay rate plus dephase rate for a manual atom-like object. As shown above, a complete conversion can be achieved in the resonance and off-resonance cases under the ideal condition. Fig. 4(a), 4(b) and 4(c) plot the conversion properties in both the cases after considering the dissipation. The strong coupling and large detuning can tolerate the dissipation better. Fig. 4(d) plots the probability Ft 2 j j 2 t 1 j j 2 z t 2 j j 2 . The high conversion efficiencies can be obtained in the case as shown in Fig. 4(b) and Fig. 4(c). The probability F can be nearly unity which means that the input photon is dissipated and converted, and little elastic scattering exists. We note another restricting condition that the Rabi frequencies can not be too small in order to tolerate the dissipation. Fortunately, after considering this condition, the tunable frequency scale of the output photon is little affected, which can be understood from Fig. 3. We have study the case that the input light is monochromatic. For a input pulse with finite bandwidth, the conversion efficiency decreases, which can be seen in Fig. 4(c). To achieve the efficient single-photon frequency conversion, the narrow bandwidth of the input pulse is necessary.

Discussion
We propose a tunable single-photon frequency conversion scheme with high efficiency. The inelastic scattering shifts the frequency of the input photon. Especially, in the off-resonance case, the frequency shift can be tuned by adjusting the external classical fields. Thus, the output frequency is tunable. The dissipation will in general diminish the conversion efficiency from the unity and the output photon is mostly the inelastically scattered photon. Having considered the dissipation and the narrow bandwidth of the input pulse, the high efficiency can also be achieved. The realization of this scheme may combine the quantum information processing system with the longdistance quantum communication system.

Methods
The one-excitation state of the waveguide-atom system can be written as    Going back to the clockwise and counterclockwise picture from the even and odd picture, the scattering matrix of the emitter coupled to the waveguide loop can be derived from t 1 and t 2 32,33 and then the whole system scattering matrix can be calculated. As long as any one of the Rabi frequencies {V 1 , V 2 } is zero, the frequency conversion efficiency is zero due to the fact that the atomic transition jdae « jcae decouples from the photon in the waveguide and hence the inelastic scattering vanishes. In detail, when V 1 5 0, we can find t 1~D a ziC 1 D a {iC 1 and t 2 5 0, which is the same as a two-level system coupled to the waveguide [30][31][32][33] . And when V 2 5 0, we can and t 2 5 0, corresponding to a L threelevel atom coupled to the waveguide 44 . This also reveals that the frequency conversion can be switched off by shutting off the external classical field, which is equivalent to the control of the relative phase shift between the clockwise and counterclockwise moving photon. When the relative phase is p, an odd-mode quasi particle is prepared in the waveguide loop and the destructive interference makes the frequency conversion efficiency zero. The relationship jt 1 j 2 1 jt 2 j 2 5 1 can be easily checked. The maximal frequency conversion efficiency is 1 2 when a photon moves only clockwise or only counterclockwise towards the atom in the waveguide loop. In this case, the output state has the form of Y j i~1ffiffi ffi 2 p b,1 k j iz 1 ffiffi ffi 2 p e iw c,1 k 0 j iwith w being a real number, which is a maximally entangled state.