Single photon transport along a one-dimensional waveguide with a side manipulated cavity QED system

Cong-Hua Yan; Lian-Fu Wei

doi:10.1364/OE.23.010374

1. Introduction

It is well-known that the interaction between a two-level atom (TLA) and a resonant electromagnetic wave is particularly fundamental to nanophotonics and optical quantum information processing [1]. Generally, perfect coupling (with 100% extinction of resonant waves) is difficult to achieve in the three-dimensional free space [2–5], as the spatial mode mismatches between the incident direction and the scattered waves along various directions. In principle, this limitation can be overcome by efficiently coupling the TLA to various one-dimensional quantum waveguides, such as superconducting transmission lines [6–10], photonic crystal waveguides [11], nanoscale surface plasmons confined on the conducting nanowires [12–14], etc.. For example, when a resonant single photon is incident upon the TLA, the spontaneously emitted photon inevitably interferes with the incident wave. As the forward and backward directions are the only two directions in the one-dimensional waveguide, this interference results in the resonant photons being completely reflected [15]. This implies that the TLA coupled to the waveguide can be served as a completely reflected mirror for the resonant photon, even when the TLA weakly couples to the waveguide [15]. Therefore, how to manipulate resonant photons along the one-dimensional optical waveguide is significant for nanophotonic devices, such as quantum amplifiers [7], single-photon transistors [12], and quantum switches [16].

Until now, a number of schemes have been proposed to control the resonant photons transporting along various waveguide structures, such as manipulating the transition frequency of the TLA in the coupled-resonator waveguide [16], introducing additional classical control fields to drive the auxiliary atomic transitions [17–19], using the asymmetric couplings between the incident photon and the TLA [20], adopting two waveguides to adjust the transporting properties [21,22], and coupling a three-level atom to a Sagnac interferometer [23,24], etc.. Certainly, the experimental demonstrations of these proposals still meet various challenges, e.g., the accurate manipulation of the dipole interactions between the TLA and the photon in nanoscale [2–5], the induced multiple-photon processes disturbing the measurements [25].

Interestingly, Carmele et al. [26] proposed an effective approach to stabilize the intrinsic quantum cavity electrodynamics by coupling an external mirror mode to feedback control the cavity mode. They showed that, with the feedback control, the Rabi oscillation between the occupation numbers of the atomic levels can be significantly strengthened. As a consequence, this oscillation can be observed even when the cavity-atom system works in the weak coupling regime. This approach can drive the TLA-cavity interaction into the strong coupling regime without changing the energy spectra of the TLA and the cavity or introducing multiple-photon processes.

In the present work, an external mirror mode coupling to the cavity mode is adopted to realize all-optical control of single photon transport along the one-dimensional optical waveguide. Our work is based on the expansion of the investigations on the Rabi-oscillation (in the time-domain) [26] to the Rabi-splitting (the counterpart of the Rabi-oscillation in the frequency domain). The results clearly show that the transmission probability of the resonant photon along the waveguide can be controlled from 0 to 1, via adjusting the cavity-mirror coupling strengths. Also, the relevant Fano resonance can be modulated by adjusting the coupling strengths between the cavity and the external mirror. Given that the amplitude of the controlled Rabi oscillations in the time domain (induced by feedback controls from the mirror) is only 5% [26], the Rabi splittings and Fano resonances in transmission spectra caused by the external mirror, shown here, are significantly enhanced.

This paper is organized as follows. Our generic model is presented in Sec. 2, wherein an effective real-space Hamiltonian of the system is introduced and the exact transmission spectrum of one-photon solution is given. In Sec. 3 we numerically analyze how the external mirror can be utilized to modulate the photon transport in the frequency domain with and without the dissipation of the system, respectively. Finally, in Sec. 4, we conclude our work and suggest an experimental demonstration of our proposal with the current photonic crystal waveguide technique.

2. Model and solutions

We consider the single photons traveling along the waveguide structure shown in Fig. 1, where a microcavity with a TLA is on the side. A mirror, over the cavity with a distance D and defining a boundary condition to the external mode, results in the desirable self-feedback of the cavity photons (i.e., the cavity photons leak from the cavity and then return to the cavity by the reflection of the mirror). The distance D is chosen close enough to ensure a pure single mode and no free dispersion occurs. Therefore, the system corresponds to a cavity with a TLA inside coupling to an empty cavity [26].

Fig. 1 Schematic configuration for controlling photonic transport by cavity-mirror coupling. A microcavity interacting with a TLA is coupled to a single-mode waveguide, and a perfect reflection mirror placed in a distance of D is utilized to produce the external mode beside the cavity mode. The TLA in the cavity is assumed initially at its ground state |g〉.

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The system can be described by the following Hamiltonian in the real space (with h̄ = 1) [20, 26–29]:

H = H_{p} + H_{a} + H_{c} + H_{m} + H_{ac} + H_{pc} + H_{cm} .

Here,

H_{p} = \int d x [C_{R}^{†} (x) (- i V_{g} \frac{\partial}{\partial x}) C_{R} (x) + C_{L}^{†} (x) i V_{g} \frac{\partial}{\partial x} C_{L} (x)],

is the free photon Hamiltonian with

C_{R (L)}^{†} (x)

and C_R(L)(x) being the bosonic creation and annihilation operators of a right-moving (left-moving) photon at position x, respectively. V_g is the group velocity of the photon.

H_{a} = Ω_{g} a_{g}^{†} a_{g} + (Ω_{e} - i \frac{1}{τ_{a}}) a_{e}^{†} a_{e},

is the Hamiltonian of the TLA with the eigenfrequency Ω Ω_e − Ω_g, and

a_{g (e)}^{†}

and a_g(e) are the creation and annihilation operators of its ground (exited) state, respectively. The number operator of the state |i〉 of the TLA reads

a_{i}^{†} a_{i} = n_{i}

(i = g, e), and 1/τ_a is the dissipation of the TLA. Note that adding an imaginary term in the relevant Hamiltonian is an effective approach to describe the dissipation of the system [28], wherein the effective Hamiltonian becomes non-Hermitian. This approximation can only be done for a single excitation. The Hamiltonian of the single mode cavity (with the frequency ω_c and the dissipation 1/τ_c) is

H_{c} = (ω_{c} - i \frac{1}{τ_{c}}) c^{†} c,

where c^† and c are the bosonic creation and annihilation operators of the cavity mode, respectively. The external mirror mode can be represented as

H_{m} = (ω_{m} - i \frac{1}{τ_{m}}) m^{†} m,

with 1/τ_m being the dissipation and m^†(m) being the bosonic creation (annihilation) operator of the external mode.

The interaction between the TLA and the cavity mode, under rotating wave approximation (RWA), is

H_{ac} = g [c a_{e}^{†} a_{g} + c^{†} a_{g}^{†} a_{e}],

where g means the coupling strength. The interaction term describing the scattering process between the waveguide and the cavity is given by

H_{pc} = \int d x δ (x) V [C_{R}^{†} (x) c + c^{†} C_{R} (x) + C_{L}^{†} (x) c + c^{†} C_{L} (x)],

where V describes the coupling strength between the cavity mode and the traveling photons, and the Dirac delta function δ(x) indicates that the cavity is located near the location x = 0 of the waveguide. Finally, the interaction between the cavity and the external mode reads

H_{cm} = V_{m} [c^{†} m + m^{†} c],

with V_m being the coupling strength. Without loss of generality, the coupling strength between the internal cavity mode c and the auxiliary empty cavity mode m is assumed to be a real constant [29].

In this work, we concentrate on the transport of single photons with a constant frequency along the waveguide. It is assumed that the TLA is originally prepared at the ground state |g〉. After being scattered by the cavity, the incident single photon may be absorbed by the TLA, may excite the cavity mode or the external mode, and also may be scattered into the left or the right direction along the waveguide. Therefore, the general eigenstate of the system Hamiltonian H should take the following form

\begin{array}{l} | Ψ 〉 & = & \int d x [ϕ_{R} (x) C_{R}^{†} (x) | 0 〉 + ϕ_{L} (x) C_{L}^{†} (x) | 0 〉] \\ + e_{c} c^{†} | 0 〉 + e_{m} m^{†} | 0 〉 + e_{a} a_{e}^{†} a_{g} | 0 〉 . \end{array}

Here, ϕ_R(x) and ϕ_L(x) are the probability amplitudes of right- and left-traveling photon. e_c and e_m are the excitation amplitudes of the cavity- and the external mode, respectively. e_a is the excitation amplitude of the TLA in the state |e〉. Certainly, |0〉 is the vacuum with zero photon among the waveguide, the cavity, and the external mode, and the TLA is at the ground state |g〉.

Formally, the spatial dependent photonic amplitudes can be expressed as

\begin{array}{l} ϕ_{R} (x) & = & e^{i k x} [θ (- x) + t θ (x)], \\ ϕ_{L} (x) & = & e^{- i k x} r θ (- x), \end{array}

with k =ω/V_g, and θ(x) being the step function. T = |t|² and R = |r|² represent the probabilities that an input photon is transmitted and reflected, respectively.

The eigenvalue equation

H | Ψ 〉 = ω | Ψ 〉,

directly yields

- i V_{g} \frac{\partial}{\partial x} ϕ_{R} (x) + Ω_{g} ϕ_{R} (x) + V δ (x) e_{c} = ω ϕ_{R} (x),

i V_{g} \frac{\partial}{\partial x} ϕ_{L} (x) + Ω_{g} ϕ_{L} (x) + V_{δ} (x) e_{c} = ω ϕ_{L} (x),

Ω_{g} e_{c} + (ω_{c} - i \frac{1}{τ_{c}}) e_{g} + g e_{a} + V \int d x δ (x) ϕ_{R} (x) + V d x δ (x) ϕ_{L} (x) + V_{m} e_{m} = ω e_{c},

(Ω_{e} - i \frac{1}{τ_{a}}) e_{a} + g e_{c} = ω e_{a},

Ω_{g} e_{m} + (ω_{m} - i \frac{1}{τ_{m}}) e_{m} + V_{m} e_{c} = ω e_{m} .

Integrating the above five-element equations, we get

t = [(ω - ω_{c} + i \frac{1}{τ_{c}}) (ω - Ω + i \frac{1}{τ_{a}}) (ω - ω_{m} + i \frac{1}{τ_{m}}) - g^{2} (ω - ω_{m} + i \frac{1}{τ_{m}}) - V_{m}^{2} (ω - Ω + i \frac{1}{τ_{a}})] / X

,

r = - i \frac{V^{2}}{V_{g}} (ω - Ω + i \frac{1}{τ_{a}}) (ω - ω_{m} + i \frac{1}{τ_{m}}) / X

,

e_{a} = g V (ω - ω_{m} + i \frac{1}{τ_{m}}) / X

,

e_{m} = V_{m} V (ω - Ω + i \frac{1}{τ_{a}}) / X

, and

e_{c} = V (ω - Ω + i \frac{1}{τ_{a}}) (ω - ω_{m} + i \frac{1}{τ_{m}}) / X

, with

X = (ω - ω_{c} + i \frac{1}{τ_{c}} + i \frac{V^{2}}{V_{g}}) (ω - Ω + i \frac{1}{τ_{a}}) (ω - ω_{m} + i \frac{1}{τ_{m}}) - g^{2} (ω - ω_{m} + i \frac{1}{τ_{m}}) - V_{m}^{2} (ω - Ω + i \frac{1}{τ_{a}})

. The physical analysis based on these equations will be given numerically in the following sections.

3. Single photons scattered by a singe-mode cavity coupling to a completely reflection mirror

3.1. Without the dissipations

Without loss of generality, we typically take the decay rate of the cavity mode to the waveguide mode as V²/V_g = Γ = 0.09Ω throughout our calculation. Certainly, the generic conclusions are independent of this choice [27].

Under the usual weak TLA-cavity coupling limit, i.e., g ≪ Γ, Fig. 2 shows how the transmitted probability T = |t|² depends on the rate of frequency ω of the incident photon to the eigenfrequency Ω. The TLA is resonant with the cavity and mirror modes and all the potential dissipations are neglected. It is clearly seen that, when the coupling V_m between the cavity mode and the mirror mode is absent, the Rabi splitting does not appear. This means that the incident resonant photon is completely reflected by the cavity aside [30]. Phenomenally, this can be explained as: due to the negligible TLA-cavity interaction (when g ≪ Γ), the energy structure of the cavity mode can not be modified by the TLA. Consequently, there is not any Rabi splitting of the transmitted photon along the waveguide.

Fig. 2 Transporting properties of photons influenced by cavity-mirror coupling (with $ω_{c} = ω_{m} = Ω$ , $\frac{1}{τ_{c}} = \frac{1}{τ_{m}} = \frac{1}{τ_{a}} = 0$ , and Γ = V²/V_g = 0.09Ω). The cavity-TLA coupling is in the weak regime (g = 0.001Ω ≪ Γ). It is shown that the transmission probabilities of the resonant photon (T(Ω)) can be controlled from 0 to 1 by enhancing the coupling strength V_m. Especially, Rabi splitting is found, regardless of the cavity-TLA in the weak coupling regime. The inset shows that, for the strong coupling case (g = 0.3Ω and V_m = 0.4Ω), the perfect transmission (T = 1) is not located at ω = Ω but stabilizes in a proper frequency range. The Rabi-splitting dips are clearly found at $Ω \pm \sqrt{g^{2} + V_{m}^{2}}$ .

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Now, if the external mode supplied by the perfectly reflected mirror drives the cavity, i.e., V_m ≠ 0, then the Rabi splittings can be observed in the transmission spectra of the photons, even when the TLA-cavity interaction works in the weak coupling regime. The stronger coupling between the cavity mode and the mirror mode corresponds to the larger Rabi splittings. This implies that the energy structure of the cavity could be influenced by the interaction with the mirror mode, and thus two fundamental modes in the cavity are induced, each corresponding to one resonant point (resulting in the appearing of two perfect reflection frequencies). This is the physical origination of the Rabi splitting observed in the transmitted spectra of photons transporting along the above waveguide structure.

For a more generic case, i.e., when both the mirror-cavity and TLA-cavity interactions take the roles, the Rabi splittings of the incident photons transporting along the waveguide always appear. Furthermore, the inset of Fig. 2 shows that, for the strong couplings, e.g., V_m = 0.4Ω ≫ Γ and g = 0.3Ω ≫ Γ, incident photons with frequencies $| ω - Ω | ≪ \sqrt{g^{2} + V_{m}^{2}}$ can perfectly transmit.

In fact, the above discussions can be analytically verified by delivering the distance of the Rabi splittings. From formulas below Eq. (16) one can easily calculate the locations of the two dips for the Rabi splittings:

ω_{1} = Ω - \sqrt{g^{2} + V_{m}^{2}}, ω_{2} = Ω + \sqrt{g^{2} + V_{m}^{2}},

with g and V_m being the coupling strengths between the cavity-TLA and cavity-mirror modes, respectively. Finally, the numerical results shown in Fig. 2 naturally result in an argument, i.e., by adjusting the cavity-mirror coupling strength (e.g., via moving the distance between them), the transmission probabilities of the resonant photons can be changed from 0 to 1. This means that the device proposed here can serve as a potential single-photon switch in a full-optical way.

The above discussions are based on the assumption that the cavity mode is resonant with the mirror mode. Next, we investigate the two modes with certain detuning, i.e., ω_m ≠ Ω. It is seen from Figs. 3(a)–3(d) that, when the band gap of the TLA is in resonance with the single cavity mode, i.e., Ω = ω_c, the transmission amplitude of the resonant photon is always 1, regardless of the cavity-mirror detuning. Simultaneously, an asymmetric Fano peak reveals at the point ω = ω_m [31]. The asymmetric Fano line in the transmission spectrum is specified by two phase shifts [32]; they correspond to the photon directly passing (along the waveguide) and the indirectly resonance-assisted passing (along the pathway of the cavity coupled to the external mode). Furthermore, one can see from Fig. 3(e) that, when the TLA is resonant with the external mode rather than the cavity mode (i.e., Ω = ω_m ≠ ω_c), the Rabi splitting disappears. This indicates that the internal cavity mode and the external mode impact on the transporting properties of the incident photon in different ways. Especially, when the three kinds of frequencies are all detuned (i.e., ω_m ≠ Ω ≠ ω_c), two Fano peaks appear at ω = ω_m and ω = Ω, respectively. Figures 3(e) and 3(f) also show that, an extinction of transmitted photon with frequency ω = ω_c appears (T(ω_c) = 0), which is strikingly contrast to the cases in Figs. 3(a)–3(d) wherein T(ω_c) = 1. This means that, in the far-detuned case, the TLA is essentially decoupled from the cavity, and thus the transmission properties of the incident photons with the frequency ω = ω_c are mainly determined by the cavity mode.

Fig. 3 Transmission spectra of detuned photons with g = 0.03Ω. (a) Without the external mode. (b)–(d) With the external mode driving the cavity. Here, Fano resonances are obviously found. (e) Cavity-TLA is detuned (ω_c ≠ Ω). It is clearly shown that the cavity-TLA detuning suppresses the Rabi-splitting but does not influence the Fano-resonance. (f) Both cavity-TLA and cavity-mirror are detuned (ω_c ≠ Ω, ω_m ≠ Ω). Two Fano resonances occur at ω = Ω and ω = ω_m, respectively.

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3.2. With the dissipations

Because the unavoidable intrinsic dissipations in the present cavity-waveguide system always result in the leakage of photons into the non-waveguide mode, it is necessary to investigate how the mode-mismatch and mode drifting affect the transporting properties of the photons. Generally, the dissipations decrease the transmission probabilities of the photons transporting along the optical waveguide [28]. However, the results of our investigations show that the dissipative processions (among the TLA, the cavity, and the external mirror) influence the single-photon transmission spectra very differently.

Firstly, for the tuning case, i.e., ω_m = Ω = ω_c, the upper panel of Fig. 4 shows that, when the incident photon is resonant with the cavity (i.e., ω = Ω), the amplitudes of the transmission spectra are always equal to 1, which are insensitive to various dissipations in the system. This presents a distinct contrast to the single-mode cavity (with a TLA inside) side coupled to the waveguide, as presented in [28], in which Shen et al. show that the resonant photon can not be perfectly transmitted (i.e., T < 1) due to the dissipation.

Fig. 4 Transmission spectra affected by the dissipations of the system. (a)–(c) All the modes are resonant (i.e., ω_c = ω_m = Ω). It is seen that resonant photons are perfectly transmitted and insensitive to the dissipations of the TLA, cavity-, and the external modes. Other parameters are chosen as g = 0.03Ω and V_m = 0.04Ω. (d)–(f) Cavity-TLA is resonant but the cavity-mirror is detuned, e.g., ω_c = Ω and ω_m = 0.9Ω. It can be seen that: i) the dissipation of the cavity mainly influences the minimum values of the transmission spectra; ii) the dissipation of the external mode plays an important role in Fano resonance; and iii) the dissipation of the TLA affects the formation of Rabi splittings. Corresponding parameters are set as: g = 0.03Ω and V_m = 0.01Ω.

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In order to confirm the dissipative effects, comparing with Fig. 3(d), we further investigate the dissipations for the detuned case (e.g., ω_m = 0.9Ω) with the cavity photon being resonant with the TLA (ω_c = Ω). According to Fig. 4(d), the dissipation of the cavity mainly influences the minimum values of the Rabi splitting in the transmitted spectrum and the Fano resonance dip. When the dissipative value of the external mode is large enough, as shown in Fig. 4(e), the Fano effect is totally suppressed. Figure 4(f) shows that the dissipation of the TLA destroys the Rabi splitting and hardly influences the Fano effect. These distinct phenomena illuminate that the dissipations affect the transmission properties and change the phase relation among the excitation amplitudes of the cavity mode, the TLA, and the external mode in different ways.

4. Discussion and conclusion

To demonstrate our theoretical results on controlling the photon transport, we now propose an experimental device based on the usual photonic crystal technique, as depicted in Fig. 5. This structure consists of three parts: a parallel line-defect photonic waveguide sustaining the photons transporting, a point-defect cavity with a quantum dot embedded, and a vertical line-defect waveguide formed in the vicinity of the cavity and terminated at one end by a perfect mirror (induced by a hetero-interface). This kind of hetero-interface has been successfully fabricated and widely used to dynamically control the Q factor of the photonic crystal cavity [33], modulate the emission of the quantum dot [34], create frequency gaps of waveguide [35], and strongly couple two distant photonic nanocavities [36], etc.

Fig. 5 A potential setup constructed by two pieces of photonic crystals with different lattice constants a₁ and a₂. A cavity with a two-level quantum dot inside (the small red ball) is coupled to both a parallel waveguide and a vertical waveguide. The vertical waveguide with width $\sqrt{3} a_{1}$ is terminated at one end by the other vertical waveguide (with width $\sqrt{3} a_{2}$ ) to form a hetero-interface. Utilizing the external mode provided by the complete- reflect mirror (placed in the distance of D over the cavity), the transporting properties of the photons along the parallel waveguide, such as Rabi-splitting and Fano-resonance, can be controlled by the cavity-mirror coupling demonstrated above.

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Technically, the transmitting band of the waveguide (with lattice constant a₂ and width $\sqrt{3} a_{2}$ ) should be designed to exclude the cavity mode. Therefore, the interface acts as a perfect mirror. Incident photons transporting along the parallel waveguide will resonate with the cavity mode, and the cavity photons leak out not only into the free space but also into the vertical waveguide. This makes the cavity mode couple with the external mode along the vertical wavegudie. Then, the hetero-interface mirror can modify the internal cavity mode and influence the transporting properties of photons along the parallel waveguide.

Actually, there must be two key steps in experiments to realize the above arguments: (1) The two-level quantum dot must be accurately placed and directly coupled to the single cavity mode; (2) The coupling strength between the cavity mode and the external mirror mode could be precisely controlled. Technically, the single quantum dot has been successfully placed in the cavity through precise positioning techniques [37], and the resonance of the quantum dot can be controlled by adjusting the temperature of the system from 20K to 40K [38]. Additionally, the velocity of light in the vertical photonic crystal waveguide could be electrically controlled [39,40], wherein the cavity-mirror coupling directly relates to the speed of light outside the cavity [26]. Then, the frequencies of the external mode (determining the Fano resonances) have already been precisely tuned by laser-assisted oxidization [41]. Typically, in [38], Majumdar et al. show that very high quality factors (10⁴) of coupled cavity modes can be achieved and the cavity-cavity coupling strength (110 GHz) can be larger than field decay rates (20 GHz). This indicates that the auxiliary empty cavity can drive effectively the cavity-TLA interaction into the strong coupling regime [29]. Therefore, adopting the current photonic crystal techniques, the desired quantum manipulation of single photons in nano optical waveguides could be implemented.

Additionally, the photonic transmission behaviors presented in this work look like the phenomena of electromagnetic induced transparency (EIT). The intrinsic differences are: the heart of EIT is a quantum interference effect, due to the coherence between a classical driving laser field and one of the two transitions of three-level atomic ensembles. Comparatively, in the present work, the transparency window is induced by the quantized photon-photon interaction. This analogous EIT is due to the energy structure modifications of the cavity by the cavity-mirror coupling. Here, the photon transport properties are controlled by the quantized mirror mode, which can be used to overcome the multiple-photon processes of the laser fields. Therefore, the present proposal could be used to design a single-photon switch in a full-optical way. Actually, the cavity-mirror coupling considered here refers to the energy transformation between the photons in different cavities. This is essentially different from that in optomechanical structures [32], wherein the photon-phonon interaction dominates the energy transformation. Also, although the full quantum mechanical method can also be used to investigate the coherent transport of the surface plasmons in nanowire [12], the plasmon modes themselves are not suitable as the carriers of information for long distances due to their strong dissipation in metals.

In summary, we investigated the single photon transporting along the one dimensional optical waveguide influenced by a side manipulated cavity QED system. By coupling the internal cavity mode to the external mode provided by the perfect reflection mirror, the Rabi-splitting and the Fano-resonance of transmission spectra can be controlled by adjusting the amplitude of cavity-mirror coupling, even when the TLA-cavity coupling is in the weak regime. Actually, the photonic transmission spectra along waveguides have been successfully detected by using homodyne detections [10], phase-sensitive vector network analyzers [1], and spectrum analyzers [8]. We hence conclude that the structure proposed here provides a realizable experimental single-photon switch by utilizing the controllable photonic transports along the one-dimensional waveguide.

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China (NSFC) under Grants No. 11304210, No. 91321104, No. 11174373, and No. U1330201, the National Fundamental Research Program of China through Grant No. 2010CB923104, and the Construction Plan for Scientific Research Innovation Teams of Universities in Sichuan Province with Grant No. 12TD008.

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Single photon transport along a one-dimensional waveguide with a side manipulated cavity QED system

Abstract

1. Introduction

2. Model and solutions

3. Single photons scattered by a singe-mode cavity coupling to a completely reflection mirror

3.1. Without the dissipations

3.2. With the dissipations

4. Discussion and conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (17)

Optics Express