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Single-photon Transport in a Waveguide-cavity-emitter System

We investigate the single-photon transport properties in a hybrid waveguide quantum electrodynamics system, in which a one-dimensional waveguide is simultaneously coupled to a cavity and a driven Λ-type three-level atom. The cavity and the atom are also coupled to each other. We show that when the waveguide is coupled to the cavity and the atom at the same point, double electromagnetically induced transparency (EIT) can be observed from the system. The physical mechanism of the double EIT effect has been given by setting up the eigenstate structure for the whole system. When the waveguide is coupled to the cavity and the atom at different points, we demonstrate that the controllable nonreciprocal scattering effect can be achieved by modulating the non-zero cavity-emitter separation and the phase associated with the cavity-emitter coupling strength. These results have potential applications in realization of photonic coherent control.


Introduction
Single photons are considered as one of ideal carriers for quantum information, therefore, how to obtain the ability to control and manipulate single photon transport in quantum information processing has been an important issue. In last decades, waveguide QED system [1], which studies the interactions between emitters and one-dimensional (1D) waveguide, has been regarded as an excellent platform for realizing the controllable single-photon transport

Model
The system we considered is illustrated in Fig. 1, where a 1D waveguide is coupled to a cavity and a Λ-type three-level atom simultaneously with corresponding coupling strengths g and f. The cavity and the atom are also coupled to each other through the coupling strength J = |J|e iϕ . The position of the cavity and the emitter are located at x = 0 and x = x 0 , respectively. The atom is characterized by the excited state |e〉, the metastable state |f〉 and the ground state |g〉. As reported in Ref. [50], for our current proposal, the most relevant implementations is the circuit QED system, in which an LC circuits and a transmon qubit can be coupled to each other, serving as the cavity-emitter system. A superconducting transmission line, which supports many bosonic modes with a linear dispersion relation, can act as a waveguide.
The Hamiltonian of the whole system can be written as H = H w + H ae + V. Here, H w describes the propagation of the photon in the waveguide, which can be expressed as where υ g is the group velocity of the photons. c † R (x) (c R (x)) and c † L (x) (c L (x)) are the bosonic creation (annihilation) operators of the right-and left-going photons at position x, respectively.
The second term H ae denotes the Hamiltonian of the cavity-emitter system, Fig. 1 Schematic configuration of the coupling system. A cavity and a Λ-type three-level atom are coupled to a 1D waveguide at the points x = 0 and x = x 0 through the coupling strengths g and f, respectively. They are also directly coupled to each other through the coupling strength J = |J|e iϕ where a and a † are the photon creation and annihilation operators of the single mode cavity with frequency ω a . σ ef = (σ fe ) † is the transition operator between states |e〉 and |f〉. ω e and ω f are the frequencies of the states |e〉 and |f〉, respectively. The classical control field with strength Ω and frequency ω d is used to induce the transition between |e〉 and |f〉. In the rotating frame, H ae can be written as the time independent form The third term V in Hamiltonian represents the waveguide-cavity and the waveguideatom interactions. The detailed expressions are described as follows: In the single-excitation subspace, the eigenstate of the system can be written as where |ø,m a ,n〉 (m = 0,1;n = e,g,f) represents that the waveguide is in the vacuum state while the cavity and atom are in the states |m a 〉 and |n〉, respectively. ϕ R (x) and ϕ L (x) are the single-photon wave function of the right-going and left-going modes in the waveguide. μ a , μ e and μ f are the excitation amplitudes of the cavity field, the three-level atom in the excited state |e〉 and metastable state |f〉, respectively. Solving the stationary Schrödinger equation H|E k 〉 = E|E k 〉, the series of coupled equations can be obtained as follows: We now consider the transport behavior when a single photon with wave vector k is incident from the left side of the waveguide. In this case, the wave functions ϕ R (x) and ϕ L (x) can be written as Here, the photon propagates from the regime of x < 0, which will experience reflection and transmission when it arrives at the connecting point x = 0 and x = x 0 , respectively. We use r (B) and A (t) to represent the reflection and transmission coefficients of the photon at x = 0 (x = x 0 ). (x) shown in (13) is a step function. Combing Eqs. (6-10) with Eqs. (11)(12)(13), we obtain the transmission amplitude t as (14), if we set Ω = 0 and ω e = ω a , the three-level atom is equivalent to a two-level emitter, then we can recover the results given in Ref. [50].

Single-photon Transmission
In this section, we first study the single-photon transmission properties under the situation of x 0 = 0, which means that the waveguide are coupled to the cavity and the atom at the same point. Then, the result shown in (14) turns out to be We plot the transmission rate T = |t| 2 as a function of the detuning Δ in Fig. 2. The parameters we chosen in this paper are based on the recent experimental and theoretical progress [50][51][52][58][59][60][61]. As shown in Fig. 2(a), when the waveguide is only coupled to the cavity, one can find a perfect reflection valley (T = 0) located at Δ = 0, which is induced by the destructive interference between the incoming photon and the re-emitted photon by the cavity. Here we have set ω e = ω a , thus, Δ can also represent the detuning between the waveguide and the cavity. In contrast, when the waveguide is only coupled to the Λ − type three-level atom, as shown in Fig. 2(b), the interference between the two transition paths �g⟩ → �e⟩ and �g⟩ → �e⟩ → �f ⟩ → �e⟩ leads to the occurrence of a wide transparency window (T = 1) located at Δ = 0. This phenomenon is similar with the traditional EIT [54]. Here, the waveguide serves as a bath for the atom. In Fig. 2(c-e), we consider a more complicate case in which the waveguide is simultaneously coupled to the cavity and the atom. It's shown that the transmission spectra are characterized by double EIT. Moreover, by tuning the additional coupling strength f between the waveguide and the atom, the position and the width of the two transmission peaks can be effectively adjusted; while, the location of the three absorption dips keep unchanged.
The three absorption valleys and the double EIT can be explained by using the energy level diagram of the waveguide-cavity-emitter system shown in Fig. 3. As shown in Fig. 3(a), under the two-photon resonance condition Δ = E − ω e = E − δ, the control field applied to the three-level atom can induce the transition between the states �f ⟩ ↔ �e⟩ , leading to two dressed states �±⟩ = 1 √ 2 (�e⟩ ± �f ⟩) with energies shifted by ±Ω.
Hence, the direct cavity-emitter coupling can induce two transitions between |1,g〉 and |0,+〉 or between |1,g〉 and |0,−〉. The photon in waveguide can be absorbed by the The corresponding dressed energy structure of the system cavity-emitter system, and will induce the transition between |0,g〉 and |1,g〉 or between |0,g〉 and |0,±〉 with the coupling strength g or f.
In this cavity-emitter system, the Hamiltonian H ae can be rewritten in the new basis {|1,g〉,|0,±〉}, which yields three eigenstates [57] where ℵ 0,± are the normalization factors. The corresponding eigenvalues are ± = e ± √ J 2 + Ω 2 and ω 0 = ω e , respectively. As shown in Fig. 3(b), in the dressed picture, these eigenstates compose an anharmonic ladder structure. The incident photon will be completely absorbed by the cavity-emitter system, whenever the incident photon is resonant with the three different transition pathways �0, g⟩ ↔ � 0 ⟩ , �0, g⟩ ↔ � + ⟩ or �0, g⟩ ↔ � − ⟩ . The absorbed photon will be re-emitted through the interaction between the waveguide and the cavity-emitter system. The directly transmitted and the re-emitted rightmoving photons are disappeared via the destructive-interference process [50]. Therefore, the perfect reflections (T = 0) located at Δ = ± √ J 2 + Ω 2 and Δ = 0 can be observed from the transmission spectra.
Based on above analysis, we know that the eigenvalues of the three dressed states can affect the transmission behavior, which implies that the cavity-atom coupling strength J and the Rabi frequency are also important factors affecting the single-photon transmission properties. Figure 4(a) shows the transmission rate against the detuning Δ with different J. It's shown that with increasing of J, the symmetry of the transmission spectra can be broken. Moreover, the locations of the perfect transmission and reflection are shift with J. Since similar results can be obtained by tuning the Rabi frequency Ω, thus, we wouldn't discuss the effect of Ω on the transmission spectra in here. On the other hand, since the transitions shown in Fig. 3(a) can form a cyclic energy-level structure, the phase ϕ cannot be completely eliminated in the dressed picture and it affects the scattering behavior naturally. As shown in Fig. 4(b), by changing the phase from ϕ = 0 to ϕ = π, the transmission peaks move towards to the positive direction along the Δ axis.
(16) It's worthy to indicate that the three eigenvalues are obtained under the condition ω e = ω a = δ or Δ = Δ a = Δ f . In general cases, the expression of the eigenvalues depends not only on the cavity-atom coupling strength J and the Rabi frequency of the driving field Ω, but also on the two detuings between the atom and the cavity or the driving field. The effect of the detuning between the cavity and the atom on the transmission rate is explored in Fig. 5. It's clearly shown that for ω e < ω a (ω e > ω a ), the position of the three absorption dips and the two transmission peaks move towards to the negative (positive) direction along the Δ axis. More interestingly, if we tune the cavity-atom detuning from resonance (ω e = ω a ) to 0.4υ g (ω e = 0.6ω a ), the two absorption dips located at Δ ∼ 0.5 υ g and Δ = 0 can convert to two transmission peaks (see the blue solid and the red dot-dashed curves). Thus, the singlephoton switch from absorption to transmission can be realized by controlling the cavityatom detuning. Similar conclusion can be obtained by changing ω d , therefore, we wouldn't discuss the non-resonant driving case, i.e., δ≠ω e .

Nonreciprocal Behavior of the System
In this section, we focus on the case of x 0 ≠ 0, that is, the cavity and the emitter locate at the different points of the waveguide. Based on (14), we can calculate the location of the three complete reflections under the resonant condition ω e = ω a = δ, which gives Note that the term 0 = kx 0 in (17) functions like the accumulated phase as the travelling photon moves from one coupling point to the other. According to (17), Δ ± has been slightly modulated by x 0 . Consequently, the position of the transmission valleys naturally shift with 0 . More interestingly, the nonreciprocal photon transmission can be realized for x 0 ≠ 0, which means that the transmission behavior will change if the incident photon is  (14) to − x 0 , which gives [50] Figure 6(a) clearly shows that T L→R can be larger or smaller than T R→L , depending on incident photon with certain frequency. In addition, in order to quantitatively describe the nonreciprocality in the present system, we define the isolation depth I as I = |T L→R − T R→L | . The dependence of I on detuning is also illustrated in Fig. 6(a). One can find that, when Δ = 0.9υ g , the isolation depth I can surpass 0.9, which implies that the right-incident photon can be completely transmitted to the left, while the left-incident photon is nearly blocked. Further, this nonreciprocal scattering behavior is very sensitive to the position phase 0 and the phase ϕ. As shown in Fig. 6(b)-(d), the non-reciprocity can be modulated . , we can find that if 0 and ϕ equal to some certain values, the incident photon is from the left (right) side, it will be transmitted to the right (left) side with a relatively large probability, and will be nearly blocked when it's incident from the right (left) side. For example, when [ 0 = 0.35π + 2nπ,ϕ = 2nπ] ([0.65π + 2nπ,ϕ = (2n + 1)π]) or [ 0 = 0.35π + (2n + 1)π,ϕ = (2n + 1)π] ([ 0 = 0.65π + (2n + 1)π,ϕ = 2nπ]), T L→R → 1 ( T R→L → 1 ) and T R→L → 0 ( T L→R → 0 ). Hence, in those cases, the isolation depth I → 1 (see Fig. 6(d)).
We also investigate the effect of the cavity-emitter detuning with ω e ≠ω a on the nonreciprocal transmission in Fig. 7. It's shown that the nonreciprocity is still robust to the cavity-emitter detuning. Especially, for ω e = 0.7ω a , the maximum value of the isolation depth I is still possible to reach as high as 0.98.
Those results confirm that a controllable single-photon nonreciprocal transmission can be achieved in our system. The physical reason behind the single-photon nonreciprocal transmission is that when we rewrite the Hamiltonian in the momentum space and change i to − i, the expression of the Hamiltonian cannot keep the original form [50]. In other words, when the cavity and the emitter are located at different points of the waveguide, the time-reversal symmetry cannot maintain, which leads to the single-photon nonreciprocal transmission.

Conclusion
In conclusion, we investigated the single-photon scattering in a 1D waveguide which is simultaneously coupled to a single cavity and a driven Λ-type three-level atom. The atom acts as a quantum node and is directly coupled the cavity as well. We showed that when the waveguide is coupled to the cavity-atom system at the same point, the double EIT phenomenon can be observed from the single-photon transmission spectra. In order to give an explicit explanation of the double EIT, we set up the eigenstate structure for the system. Moreover, the effects of the waveguide-emitter coupling, the value and the phase ϕ of the cavity-emitter coupling strengths, and the detuning between the cavity and the atom on the Fig. 7 The isolation depth I as a function of Δ for different cavity-emitter detunings. Here, we set 0 = π/3 and ϕ = 0. Other parameters are the same as that in Fig. 6  a ω e =1.3ω a singe photon properties are discussed in more detail. When the waveguide is coupled to the cavity and the atom at different points, we have shown that the controllable nonreciprocal scattering can be realized by adjusting the the position phase 0 and the phase ϕ. These results can be applied to realize photonic coherent control in waveguide systems.