Highly Efficient Two-Channel Drop Filter of Single Photons with One Drop Waveguide

A novel double-channel drop filter of single photons with only a drop wave-guide is proposed. In the structure, two microresonators with a qubit are used to connect the drop waveguide and the bus waveguide with a reflector. By tuning the system parameters, multiple wavelengths with high drop efficiencies in two output channels of the drop waveguide are obtained. As the reflector is independent on its exact location, and one drop waveguide with two drop channels saves the integrated space, this may suggest a potential filtering device which functions in photonic integrated circuits and dense wavelength division multiplexing communication.


Introduction
Optical drop filters are the essential elements in photonic integrated circuits and dense wavelength division multiplexing (WDM) communication systems, since they have been applied as multiplexers, selector, modulators, and so on.Over the past two decades, a variety of investigation on optical drop filtering has been carried out in various systems, including fiber Bragg gratings [1]- [6], photonic crystal structures [7]- [17], plasmonic systems [18] [19], and microresonators [20]- [25], etc. Especially, whispering-gallery mode (WGM) resonators as filter components have attracted much attention owing to their high quality factors and compact sizes [26].With such a filter coupled with a single-mode resonator, the desired wavelength with high efficiency is selected to drop in one output channel of the drop waveguide, while another output port is free, as the single-mode direction does not match the transmission direction in the free port.
As an extension of single channel filters, multi-channel filtering has drawn wide concern, since multiple signals can be dropped to meet the demands of multiplexing.Usually, numerous filters based on single coupled resonators were designed to implement single resonance wavelength outputs within each drop waveguide channel.It should be interesting to perform two-channel filtering by only one drop waveguide, when only drop function is considered in two ports.
Using the free port increases the drop channel capacity with different wavelengths, and reduces the complexity of the structure, which saves the integrated space of the coupled filtering device.
Inspired by these considerations, we extend the single coupled-resonator configuration [20] and propose a two-channel filter with a drop waveguide, which are side-coupled to two resonators with a qubit.A reflector requiring no exact location is utilized to reflect the photons to investigate how to perform two-channel filtering of single photons with high drop efficiency.By employing a real-space Hamiltonian, the photon scattering amplitudes in two output ports of the drop waveguide are obtained theoretically.Our results show that multiple peak wavelengths in the two-channel structure with high drop efficiencies can be achieved, by adjusting the waveguide-resonator and qubit-resonator coupling strengths.
Hence, the proposed compact filtering systems provide potential applications in photonic integrated circuits and dense WDM networks.

Theoretical Model
As shown in Figure 1, a two-channel filtering system is formed by a bus waveguide and a drop waveguide, and these two waveguides are connected by two The second resonator is also coupled to a two-level system (qubit).A perfect mirror is used to reflect the photons in the bus waveguide to transmit along the right port in the drop waveguide.The incident photon from the left port of the bus waveguide is reflected, or dropped to the two ports of the drop waveguide by two coupled resonators.† n c ( 1, 2 n = throughout the paper), with the resonance frequency cn ω .The second resonator is also coupled to a two two-level system (qubit) characterized by the ground state g and excited state e , with the coupling strength ξ .
When a single photon is input from the left side of the bus waveguide, it is coupled to the resonator-1 at 0 x = , and dropped to the left port of the waveguide-b.To increase the drop channel number, a perfect reflecting mirror located at 2 x d = is utilized to redirect the photons to couple another resonator-2, owing to direction matching of the resonator and guided-wave modes, and then transfer the photons to the right port of the drop waveguide.As seen in Figure 1, the right port of the drop waveguide is also employed to save the device area, and a two-channel drop filter with only one drop waveguide is constructed effectively.
The real-space Hamiltonian [27] for the filtering system can be expressed as Here, we have taken the ground-state level of the qubit as the energy reference.
( )  represents the generation of a right-travelling (left- travelling) photon at x in the waveguide-m, and g υ is the group velocity of the photon propagation.( ) ( )  indicates that the resonator-waveguide interaction occurs at ( ) and g e σ − = denote the raising and lowering operators of the qubit with a transition frequency e ω , re- spectively.mn V describes the coupling strength between the resonator-n and the waveguide-m, and cn γ and e γ stand for the energy loss rates for the reso- nators and qubits, respectively.
Assume that a single photon is incident from the left port of the bus waveguide with the energy . The scattering eigenstate of the Hamiltonian ( 1) is written as  The corresponding wave functions take the forms where, 1 t and 2 t represent the transmission amplitudes for two ports in the drop waveguide, and a t and a r represent the transmission and reflection am- plitudes in the bus waveguide, respectively.m r is the reflection amplitude from the reflector.( ) .
Here, ( ) detuning between the resonator-1 and the coupled qubit.

Single-Photon Filtering in the Coupled Waveguide System
For the two-channel structure, the filtering properties of single photons are characterized by these transmission ( ) and reflection . As a review, we first consider the case that there is no reflector at the right end of the bus waveguide.When a single photon enters from the left port of the bus waveguide, it will propagate from left to right along the bus waveguide, or transfers to the left side of the drop waveguide with the help of the coupled single-mode resonator-1, while it cannot be redirected to the right port of the drop waveguide, due to the mode-direction matching condition from the resonator and guided-wave modes.
Figure 2(a) shows the photon transmission versus the frequency detuning ω ∆ in two waveguides.As seen, the photon-flow conservation  excluded.In addition, single drop peak for T 1 with maximum drop efficiency of unity is exhibited at resonance 0 ω ∆ = when the coupling ratio This implies that the traveling photon along the bus waveguide is totally blocked by the first resonator, and then routed to the drop waveguide, as demonstrated in Ref. [20].To further investigate the dependence of the filtering efficiency on the coupling ratio P, we plot their relationship in Figure 2  arising from the strongest interference of two drop signals.
When increasing the inter-resonator detuning to separate their resonance frequencies, quantum interference and its induced crosstalk of two channel signals are lowered, and thus the drop efficiency of the additional channel is improved.Therefore, it reveals that the signal crosstalk in the two channels can be suppressed by adjusting the detuning between two resonators.Notice that these transmissions are independent of the accurate position of the reflecting mirror in the reflection feedback, which can be found in Equation ( 4), since the mode-direction matching condition is satisfied by the horizontal reflection direction and rotation mode direction of the resonator.In contrast to the previous schemes requiring the precise phase shifts in connection with the locations [7] [8] [9], the kind of filters have great advantages.
When the qubit is attached to the second resonator, high drop efficiencies can be achieved in another way.As described in Figure 3(a), T 2 is small when 12 0 ω ∆ = and there is not the coupled qubit ( 0 G = ).However, as G continues to increase, two peak values of T 2 gradually increase and reach to unities, as displayed in  remains unchanged at the resonance point 0 ω ∆ =, while the drop peak of T 2 expanded in a V-shaped area with the frequency shifts related to G, and goes slowly up to unity, as shown in Figure 4(b) and Figure 4(c).The resonator-qubit coupling strength can be controlled by their separation or the polarization of the atomic exciton.For instance, the coupling is tuned by using an external field to orient the direction of the dipole moment of the atomic exciton [27] [28].Consequently, controlling the resonator-qubit coupling can also serve as a novel path to perform highly-efficient filtering, because the widen frequency interval generated by the Rabi splitting can suppress the interference from these filtering signals.
To gain a deeper insight into the effect of the resonator-waveguide couplings on the drop efficiencies of two drop ports, Figure 5 plots the drop spectra for different coupling ratios.For simplification, resonator-waveguide couplings, when the qubit is coupled and there is not inter-resonator detuning, as depicted in Figure 5(b).In the case, a white straight line of 1 P = also passes through these peak regions.Note that the dissipations are not taken into account in the above discussions.
In fact, the coupled system inevitably experiences dissipations.In contrast with the ideal case with no dissipations, these drop efficiencies of T 1 and T 2 in the filtering spectra decrease with the increasing dissipations, and two drop frequencies of T 2 experience more leakages with the Rabi-splitting effects, as displayed in Figure 6.For a high-quality microresonator with very low losses, however, high drop efficiency is still available, such as the appropriate result around 0.8 for T 2 when 0.05 γ =

Conclusion
In summary, we have proposed a dual-channel quantum filter of single photons with only one drop waveguide, which saves the integrated space of the coupled device.Our results show that double or triple drop peaks with high drop efficiencies can be achieved in the transmission spectra, by adjusting the frequency detuning between two resonators or the resonator-qubit coupling.In addition, the reflection feedback in the system does not require exact location of the reflector, which has advantages over the schemes dependent on the phase shifts related to the locations.As the proposed structure is able to generalize easily to a multiple two-channel one by coupling more drop waveguides, it may have potential in photonic integrated circuits and dense WDM optical communication systems.

Figure 1 .
Figure 1.(Color online) Schematic diagram for a drop filter of single photons constructed by two waveguides coupled to two single-mode whispering gallery resonators.The second resonator is also coupled to a two-level system (qubit).A perfect mirror is used to reflect the photons in the bus waveguide to transmit along the right port in the drop waveguide.The incident photon from the left port of the bus waveguide is reflected, or dropped to the two ports of the drop waveguide by two coupled resonators.
single-photon wave functions along the right and left directions in the waveguide-m.
of waveguides and resonators and the ground state of the qubit.n η and ζ are the excitation amplitudes of the resonator modes and ex- cited state of the qubit, respectively.
θ is the Heaviside step function, with ( ) bus waveguide, one can obtain the expressions of these scattering amplitudes resonator modes and the atomic transition frequency may be different, thus we introduce these detunings, where the yellow straight line) for all frequencies is satisfied when the dissipations are

Figure 3 (
b) and Figure 3(c) show that the drop peak values of T 2 increase with a frequency shift equal to the inter-resonator detuning when varying the inter-resonator detuning.Figure 3(d) further displays that the drop peaks with maximum values of ones are exhibited in a narrow and straight crimson region with a threshold value around

Figure 4 .
Moreover, two separated drop peaks of T 2 , located around the resonance point 0 ω ∆ = with ±G (assuming 1 e ω ω = ), are distributed symmetrically due to the Rabi splitting effect.Similar to the above case of tuning inter-resonator detuning, the drop peak of T 1 with the maximum value of 1

12 ω∆
ratio for two resonator-waveguide couplings.It is shown in Figure 5(a) that two drop peaks can synchronously reach the maximum values under the condition 1 P = for an inter-resonator detuning drop efficiencies of unities can be realized in two frequency locations with a separation of , since the symmetrical coupling paths through two waveguides results in the complete transfer of these signals, as mentioned previously.Similarly, three drop peaks in two channels emerge for four equal