TwoPhoton Scattering by a CavityCoupled TwoLevel Emitter in a OneDimensional Waveguide
Abstract
We show that twophoton transport can be modulated by a twolevel emitter coupled to a cavity in a onedimensional waveguide. In the ordinary case, the transmitted light has a wider frequency spectrum than the situation without the cavity because it is reflected and scattered many times. But when the two photons are resonant with the cavity resonance reflection frequency, the frequency spectrum of the transmitted light becomes narrower than that without the cavity. This means that properly tuning the cavity resonance frequency can improve the photonphoton interaction. In addition, we show that the twophoton intensity correlation functions are nearly opposite to each other at the two sides of the emitter transition frequency rather than be the same, which is exactly the Fano resonance line shape for two photons. Such an effect is important for lowering the power threshold in optical bistable devices and for sensing applications. When the emitter transition frequency equals to the cavity resonance frequency for a highQ cavity, our results agree with the recent experiments and theories.
keywords:
onedimensional waveguide, atomcavity system, twophoton scattering, Fano resonance line shapecompress
1 Introduction
Quantum internet is important for quantum computation, communication and metrology KimbleNat453 (). The main methods to realize the quantum networks are the cavityQEDbased protocol CiracPRL78 () and the protocol provided by Duan, Lukin, Cirac, and Zoller (DLCZ) DuanNat414 (); ChoiNat452 (). The former one uses the interaction between light and type threelevel atoms in a highQ cavity to store quantum states FleischOC179 (). The latter one involves the measurementinduced entanglement CabrilloPRA59 (), replaces single atoms by atomic ensembles so that it reduces the restriction of the cavity’s quality and utilize the builtin entanglementpurification to create the longdistance quantum communication efficiently.
Thus it is of great significance to consider the interaction between few photons and a cavitycoupled emitter. Here we focus on a unique onedimensional (1D) waveguide for photons because it is helpful to realize the strong photonphoton interaction FanPRL98 (); FanPRA76 (). Recently, people have already done some researches about the interaction between light and an atomcavity system in a 1D waveguide both in experiment WallraffNat431 (); BirnbaumNat436 () and in theory FanAPL80 (); FanJOSAA20 (); FanOL30 (); FanPRL95 (); BlaisPRA69 (); FanPRA79 (); ShiarXiv1009 (). These theoretical approaches include either scattering matrix FanAPL80 (); FanJOSAA20 (); FanOL30 (); FanPRL95 () or quantum field theory BlaisPRA69 (); FanPRA79 (); ShiarXiv1009 (). The two methods get the similar results for single photons FanPRL95 (); BlaisPRA69 ().
In this paper we study the interaction of two photons and a cavitycoupled twolevel emitter in a 1D waveguide. By analyzing twophoton wave packets, generalizing the scattering matrix HausWFO () to dispose frequencyspectrum transformation, we get the twophoton correlated momenta distribution, correlated transmitted coefficient, Fano resonance line shape and other important properties. Comparing with most existing works, our results are more general because our approach is not restricted to a highQ cavity. Because our model includes the decay inherently, our results exactly agree with the experimental results in Ref. BirnbaumNat436 (). In Sec. 2 we show the model and analyze twophoton overlaps. In Sec. 3 we generalize the scattering matrix HausWFO () to deal with the frequencyspectrum transformation. We show the results in Sec. 4 and make a summary in Sec. 5.
2 The Model and TwoPhoton Wave Packets
There are several methods to realize the atomcavity system in a 1D waveguide WallraffNat431 (); BirnbaumNat436 (); FanAPL80 (). Fig. 1(a) shows the schematic experimental setups FanPRA76 (). The incident beam is monochrome plane wave. After the two photons are scattered by the whole system, we separate them by a beam splitter (BS) and use two singlephoton counters to detect them. We register the probability of concurrent detections. Similarly with the Hanbury BrownTwiss effect ScullyQO (), the results can show correlation of the two photons. Fig. 1(b) shows the parameters of the twolevel emitter.
Fig. 2(a) shows wave packets of the two photons. After the incidence of the two photons on the reference plane, they can be scattered to different sides of the plane. Incident photons can also come from different sides of the system. So we cannot divide these wave packets into leftcoming or rightcoming parts. However, we find that if each one of the two photons runs out of the cavity, it cannot form the situation that two photons are at together again. So we take this situation as “loss” at first [see the “uncoupled” wave packets in Fig. 2(a)]. After calculation of wave packet 1⃝, we add wave packets 2⃝ and 3⃝ to make the complete resolution of this problem.
3 Scattering Matrix and FrequencySpectrum Transformation
For twophoton case, the emitted light has a frequencyspectrum distribution FanPRL98 (); FanPRA76 (). So we need to generalize the scattering matrix HausWFO () to dispose this transformation. In fact, we need to solve several integral equations to get the spectrums of the outgoing wave packets [wave packets 1⃝, 2⃝ and 3⃝ in Fig. 2(a)] from the incident wave packets [the wave packets denoted by uncircled numbers “1” and “0” in Fig. 2(a)], of which the amplitudes are known. Here we use scattering matrix to express and solve these equations. We define the scattering matrix as follow
(1) 
Here, et al. are frequency spectrums of the wave packets [Fig. 2(b)]. The operator () represents that two photons come together and transmit (reflect) together. describes that two photons come from different sides of the reference plane and are scattered to the same side, of which the reverse process is denoted by . denotes that two photons come from and to different sides of the system. According to FanPRA76 () we have
(2) 
(3) 
(4) 
(5) 
where denotes the frequency spectrum of the incident photons. and are the transmission and reflection amplitude of the emitter for single photons respectively FanPRA76 (). is the background fluorescence
(6) 
Here, is the coupling strength between the photons and the emitter.
From Eq. (2) we know that operator has two parts: the one is multiplied by and the other is multiplied by . The calculation of operators can be considered by each part respectively. In Tab. 1 we have the operatormultiplying formula (see A for details)
Operators  

+  
+ 
where
(7) 
For calculation, we need the inverse operator, which satisfies
(8) 
Then we derive Tab. 2, which shows the operatorinversion formula (see B for details)
Operators  

where
(9) 
Having the formulae in Tab. 1 and 2, we can put the reduced scattering matrix of the emitter as follow (see C for details)
(10) 
where
(11)  
Here we make the substitution
(12) 
where is the reflectivity of the cavity, is the length of the cavity.
Similarly with the approach for single photons in Ref. FanAPL80 (), we rewrite Eq. (10) as follow
(13) 
The scattering matrix of the cavity and the free space are
(14) 
and
(15) 
We combine the scattering matrix of the cavity, the free space and the emitter in sequence to get the scattering matrix of the whole system
(16) 
Next, the procedure of our approach is described as follow [below the wave packets are shown in Fig. 2(a)]

By using the formulae in Tab. 1 and Tab. 2, we derive the operators and . So we get the frequency spectrums of wave packets 1⃝ and 4⃝, which are and respectively FanAPL80 (). Here is the frequency spectrum of the incident two photons, which is a delta function and denoted by the uncircled number “1” in Fig. 2(a).

By using the scattering matrix of the cavity [Eq. (14)], we derive the frequency spectrums of wave packets 2⃝ and 5⃝ from 1⃝ and wave packet 6⃝ from 4⃝.

By using the scattering matrix of the emitter [Eq. (1)], we derive the frequency spectrum of wave packet 7⃝ from 5⃝ and 6⃝. When one photon of wave packet 7⃝ transmits the cavity, we get wave packet 3⃝ from 7⃝.

we calculate the two photon transmitted wave packets associated with 2⃝ and 3⃝. In this situation, the problem turns to single photon transport, so we can use the results in Ref. FanAPL80 () directly.

By adding the two photon transmitted wave packets associated with 2⃝ and 3⃝ to the wave packet 1⃝, we derive the frequency spectrum of the two photon transmitted wave packet finally. After fourier transformation, we get the twophoton wave function in part of the out states, in which both photons are transmitted.
4 Results
4.1 Situation for LowQ Cavity
Here we consider a lowQ cavity. We set the cavity reflectivity , cavity transmission amplitude , the coupling strength of the emitter .
Firstly, We consider the situation that two photons are near resonant with the emitter transition frequency [i.e. , see Fig. 1(b)]. We find that, in ordinary case [e.g. , see Fig. 3(b)], the correlated transmission probability density contracts, which indicates that the transmitted light has a wider frequency spectrum than the situation without the cavity FanPRL98 (); FanPRA76 (). This is because the photons reflected and scattered many times. But when the two photons are resonant with the cavity resonance reflection frequency [e.g. , see Fig. 3(c)], the twophoton wave packet extends. So the cavity makes the frequency spectrum of the transmitted light narrower than that without the cavity. As a consequence, we achieve a strong photonphoton interaction by properly tuning the resonant frequency of the cavity, which is important for realization of controlled optical nonlinearity.
Then we study the nonresonant case that two photons are not resonant with the emitter transition frequency (Fig. 4). Here we consider the intensity correlation function , which is defined as follow ScullyQO ()
(17) 
where . is the normalization constant and independent of .
We show that the intensity correlation functions are nearly opposite to each other at the two sides of the emitter transition frequency rather than be the same. The transmitted two photons exhibit bunching behaviors and superPoissonian statistics () when (the upside figures of Fig. 4) and antibunching behaviors and subPoissonian statistics () when (the underside figures of Fig. 4). We already know that, for single photons, Fano resonance exhibits a sharp asymmetric line shape with the transmission coefficients varying from 0 to 100% over a very narrow frequency range FanJOSAA20 (). So here we show the Fano resonance line shape for two photons. Because of its sharp asymmetric line shape, such an effect is important for lowering the power threshold in optical bistable devices and for sensing applications FanAPL80 ().
4.2 Situation for HighQ Cavity
In a highQ cavity, we can compare our results with the recent works done by Birnbaum et al. BirnbaumNat436 () and Shi et al. ShiarXiv1009 (). We set our parameters as follow to fit their conditions: the cavity’s reflectivity and transmission amplitude , the emitter transition frequency (which is also the cavity resonance frequency when is a real number), the coupling strength . We also consider the intensity correlation function (Fig. 5).
Fig. 5(a) represents the relation between and , which agrees with the results in Ref. BirnbaumNat436 (); ShiarXiv1009 (). When or , gets its minimum. Comparing with Ref. BirnbaumNat436 (); ShiarXiv1009 (), we also consider the situation when [Fig. 5(b) and Fig. 5(c)]. Because our model includes the decay of the cavity inherently, our results exactly match the Figure. 3 of Ref. BirnbaumNat436 (), except where has a modulation arising from centerofmass motion of the trapped atom. In Ref. ShiarXiv1009 (), however, the arises to unity after infinite time because it does not consider the decay of the cavity or that of the twolevel emitter.
5 Summary
In conclusion, we study the twophoton correlated scattering by a twolevel cavitycoupled emitter in a 1D waveguide. Our results agree with the recent works BirnbaumNat436 (); ShiarXiv1009 (). Our research is important for realization of strong photonphoton interaction at the case of few photons and for quantum network. In addition, we propose a general approach to twophoton, cavitycoupled and multifrequency scattering problem in the 1D waveguide. The same approach can deal with the threelevel emitter, which is useful to make the photontransistor ChangNatphy3 (); WittNJP12 ().
6 Acknowledgments
We acknowledge the financial support of Special Prophase Project on the National Basic Research Program of China (No.2011CB311807), the National Natural Science Foundation of China (No.10804092), the Natural Science Basic Research Plan in Shaanxi Province of China (No.2010JQ1004), the Fundamental Research Funds for the Central Universities (No.xjj20100099).
Appendix A Derivations of OperatorMultiplying Formula
Appendix B Derivations of Operatorinversion Formula
Appendix C Derivations of the Reduced Scattering Matrix of the Emitter
Bellow we derive Eq. (10) and Eq. (11). From the third equation of Eq. (1) we have
(27) 
By using the following relation (see Fig. 2(a))
(28) 
we have
(29) 
By putting Eq. (29) into the first and second equations of Eq. (1)
(30)  
we get
(31)  
Here we make the substitution
(32) 
where is the reflectivity of the cavity.
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