Two-Photon Scattering by a Cavity-Coupled Two-Level Emitter in a One-Dimensional Waveguide
We show that two-photon transport can be modulated by a two-level emitter coupled to a cavity in a one-dimensional 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 photon-photon interaction. In addition, we show that the two-photon 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 high-Q cavity, our results agree with the recent experiments and theories.
keywords:one-dimensional waveguide, atom-cavity system, two-photon scattering, Fano resonance line shape
Quantum internet is important for quantum computation, communication and metrology KimbleNat453 (). The main methods to realize the quantum networks are the cavity-QED-based 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 three-level atoms in a high-Q cavity to store quantum states FleischOC179 (). The latter one involves the measurement-induced entanglement CabrilloPRA59 (), replaces single atoms by atomic ensembles so that it reduces the restriction of the cavity’s quality and utilize the built-in entanglement-purification to create the long-distance quantum communication efficiently.
Thus it is of great significance to consider the interaction between few photons and a cavity-coupled emitter. Here we focus on a unique one-dimensional (1D) waveguide for photons because it is helpful to realize the strong photon-photon interaction FanPRL98 (); FanPRA76 (). Recently, people have already done some researches about the interaction between light and an atom-cavity 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 cavity-coupled two-level emitter in a 1D waveguide. By analyzing two-photon wave packets, generalizing the scattering matrix HausWFO () to dispose frequency-spectrum transformation, we get the two-photon 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 high-Q 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 two-photon overlaps. In Sec. 3 we generalize the scattering matrix HausWFO () to deal with the frequency-spectrum transformation. We show the results in Sec. 4 and make a summary in Sec. 5.
2 The Model and Two-Photon Wave Packets
There are several methods to realize the atom-cavity 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 single-photon counters to detect them. We register the probability of concurrent detections. Similarly with the Hanbury Brown-Twiss effect ScullyQO (), the results can show correlation of the two photons. Fig. 1(b) shows the parameters of the two-level 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 left-coming or right-coming 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 Frequency-Spectrum Transformation
For two-photon case, the emitted light has a frequency-spectrum 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
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
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
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 operator-multiplying formula (see A for details)
For calculation, we need the inverse operator, which satisfies
Here we make the substitution
where is the reflectivity of the cavity, is the length of the cavity.
The scattering matrix of the cavity and the free space are
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
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 two-photon wave function in part of the out states, in which both photons are transmitted.
4.1 Situation for Low-Q Cavity
Here we consider a low-Q 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 two-photon 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 photon-photon interaction by properly tuning the resonant frequency of the cavity, which is important for realization of controlled optical nonlinearity.
Then we study the non-resonant 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 ()
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 super-Poissonian statistics () when (the upside figures of Fig. 4) and anti-bunching behaviors and sub-Poissonian 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 High-Q Cavity
In a high-Q 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 center-of-mass 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 two-level emitter.
In conclusion, we study the two-photon correlated scattering by a two-level cavity-coupled emitter in a 1D waveguide. Our results agree with the recent works BirnbaumNat436 (); ShiarXiv1009 (). Our research is important for realization of strong photon-photon interaction at the case of few photons and for quantum network. In addition, we propose a general approach to two-photon, cavity-coupled and multi-frequency scattering problem in the 1D waveguide. The same approach can deal with the three-level emitter, which is useful to make the photon-transistor ChangNatphy3 (); WittNJP12 ().
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 Operator-Multiplying Formula
Here we derive the Operator-multiplying formula in Tab. 1. If operators and are defined as follow
then the operator can be derived as follow
Here we factorize the background fluorescence and as follow
For example, if [Eq. (2)], then
In Eq. (20)
Appendix B Derivations of Operator-inversion Formula
We use the operator-multiplying formula to derive the operator-inversion formula in Tab. 2. We order that , so we have
Appendix C Derivations of the Reduced Scattering Matrix of the Emitter
By using the following relation (see Fig. 2(a))
Here we make the substitution
where is the reflectivity of the cavity.
- (1) H. J. Kimble, Nature 453 (2008) 1023.
- (2) J. I. Cirac, P. Zoller, H. J. Kimble, H. Mabuchi, Phys. Rev. Lett. 78 (1997) 3221.
- (3) L.-M. Duan, M. D. Lukin, J. I. Cirac, P. Zoller, Nature 414 (2001) 413.
- (4) K. S. Choi, H. Deng, J. Laurat, H. J. Kimble, Nature 452 (2008) 67.
- (5) M. Fleischhauer, S. F. Yelin, M. D. Lukin, Opt. Commun. 179 (2000) 395.
- (6) C. Cabrillo, J. I. Cirac, P. García-Fernández, P. Zoller, Phys. Rev. A 59 (1999) 1025.
- (7) J.-T. Shen, S. Fan, Phys. Rev. Lett. 98 (2007) 153003.
- (8) J.-T. Shen, S. Fan, Phys. Rev. A 76 (2007) 062709.
- (9) A. Wallraff et al., Nature 431 (2004) 162.
- (10) K.M. Birnbaum, A. Boca, R. Miller, A.D. Boozer, T.E. Northup, H.J. Kimble, Nature 436 (2005) 87.
- (11) S. Fan, Appl. Phys. Lett. 80 (2001) 908.
- (12) S. Fan, W. Suh, J. D. Joannopoulos, J. Opt. Soc. Am. A 20 (2003) 569.
- (13) J.-T. Shen, S. Fan, Opt. Lett. 30 (2005) 2001.
- (14) J.-T. Shen, S. Fan, Phys. Rev. Lett. 95 (2005) 213001.
- (15) A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, R. J. Schoelkopf, Phys. Rev. A 69 (2004) 062320.
- (16) J.-T. Shen, S. Fan, Phys. Rev. A 79 (2009) 023837.
- (17) T. Shi, S. Fan, C. P. Sun, preprint at http://arxiv.org/abs/1009.2828.
- (18) H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, New Jersey, 1984.
- (19) M. O. Scully, M. S. Zubairy, Quantum Optics, Cambrige Univ. Press, Cambrige, 1997.
- (20) D. E. Chang, A. S. Sørensen, E. A. Demler, M. D. Lukin, Nat. Phys. 3 (2007) 807.
- (21) D. Witthaut, A. S. Søensen, New J. Phys. 12 (2010) 043052.