# Controlled On-Chip Single-Photon Transfer Using Photonic Crystal Coupled-Cavity Waveguides

Authors: Hubert Pascal Seigneur; Matthew Weed; Michael Niklaus Leuenberger; Winston Vaughan Schoenfeld
Journal: Advances in OptoElectronics
 (2011)

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## Abstract

To the end of realizing a quantum network on-chip, single photons must be guided consistently to their proper destination both on demand and without alteration to the information they carry. Coupled cavity waveguides are anticipated to play a significant role in this regard for two important reasons. First, these structures can easily be included within fully quantum-mechanical models using the phenomenological description of the tight-binding Hamiltonian, which is simply written down in the basis of creation and annihilation operators that move photons from one quasimode to another. This allows for a deeper understanding of the underlying physics and the identification and characterization of features that are truly critical to the behavior of the quantum network using only a few parameters. Second, their unique dispersive properties together with the careful engineering of the dynamic coupling between nearest neighbor cavities provide the necessary control for high-efficiency single-photon on-chip transfer. In this publication, we report transfer efficiencies in the upwards of 93% with respect to a fully quantum-mechanical approach and unprecedented 77% in terms of transferring the energy density contained in a classical quasibound mode from one cavity to another.


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## Body

## 1. Introduction

In order to obtain an efficient quantum computing architecture, the general consensus is that various implementations of the qubit should be combined. This calls on the one hand for stationary qubits that are good for storage, such as atoms to be used at quantum network nodes, and on the other hand for flying qubits that have desirable properties for travel, such as photons to be used as quantum interconnects. Moreover, the storage qubits can map their quantum state onto the traveling qubits and vice versa by means of coherent interfaces [1–3]. With the intention of realizing an efficient quantum computing architecture, this composite qubit approach to a quantum technology has been proposed for ion trap qubits [4] and also for neutral atoms [5]. We, in addition, have proposed a similar approach in connection with semiconductor-based artificial atoms or quantum dots [6]. Regardless of choice, these various implementations of the composite qubit architecture are only possible if single photons are able to be guided from one node to another with both high efficiency and fidelity.Recently, the on-chip generation and transfer of microwave single photons have been demonstrated in connection with superconducting qubits via transmission line cavities [7–10]. In addition, the generation and transfer of single photons on photonic crystal chips using a 25  μm long defect waveguide structure has been studied by England et al. showing 12% transfer efficiency with quantum dots inside the nodes and 49% transfer efficiency without [11]. In this publication, we investigate the use of a photonic crystal coupled-cavity waveguide (CCW) for the generation and transfer of single photons on-chip. Such approach to an on-chip quantum network present several advantages. For example, since the photons we use have a wavelength of around  1 μm (IR) as opposed to 1 cm  or more (microwave), the surface area of a quantum network in any optical system would be a great deal smaller and thus much more suitable for on-chip integration. In addition, the transfer of microwave photons was shown to take few hundred nanoseconds, whereas the transfer of optical photons inside the photonic crystal chip is anticipated to take no more than few tens of picoseconds.Furthermore, CCWs offer a truly unique and sophisticated control over the transport of single photons [12]. Because modes of CCWs resemble those of the high-Q cavity modes and possess the same field symmetries, these devices can be used to make bends with no reflection. In addition, they can dramatically slow down optical waves, and because of their versatile dispersion properties (both positive and negative dispersion are achievable), they allow for a great deal of control over a single-photon pulse propagation. Moreover, since each mode is strongly localized, the guided mode is composed of a linear combination of these individual bound modes. This renders the propagating mode easy to model quantum-mechanically [13], therefore allowing for their guiding behavior to be optimized with respect to maximized fidelity of quantum operations inside the quantum network. And, once their guiding behavior has been characterized with a few parameters, namely, the coupling coefficient κ and resonant frequencyωc, their physical structure can be constructed to emulate their intended behavior.The paper is organized as follows. The modeling of the on-chip single-photon transfer dynamics is described in Section2. Included in that section are the single-photon pulse propagation characteristics, the engineering of the source/target cavity to waveguide dynamic coupling as well as resulting transfer efficiencies. A physical implementation of a CCW based on a photonic crystal structure for the purpose of transferring single photon on-chip is presented in Section 3.  An analysis and discussion of the performance of CCWs is provided in Section 4, followed by a final summary and conclusions in Section 5.


## 2. Modeling On-Chip Single-Photon TransferDynamics

The device under investigation is depicted in Figure1. It consists of two cavities, cavity 1 and cavityN, linked to one another by means of a CCW, consisting of cavity 2 through cavityN-1. In this specific example,N is 12. The coefficient Γ describes the weak coupling between the cavities forming the waveguide.Figure 1
Two high-Q cavities connected via a coupled-cavity waveguide.A general Hamiltonian for this system is derived from the tight-binding Hamiltonian and it is shown in (1). âj†and âj are the creation and the annihilation for the field in the jth cavity, ωc is the resonant cavities frequency, VC1-W(t)is the time-dependent interaction between the first cavity and the waveguide, and VW-CN(t) is the time-dependent interaction between the waveguide and the last cavity(1)Ĥ=∑jℏωcâj†âj+∑2N-2Γ(âj†âj+1+âj+1†âj)+VC1-W(t)+VW-CN(t).
Our key concern in this publication is the engineering of the interaction between cavities and the waveguide structure VC1-W(t) andVW-CN(t). First, we consider the waveguide by itself and design its characteristics. Then, both the cavities and the waveguide are considered, and the system is engineered such that the photon is unloaded into the waveguide and transferred to its destination with minimum loss.


### 2.1. Waveguide

First, it is useful to consider a CCW by itself. The Hamiltonian that can describe such a waveguide is exactly the tight-binding Hamiltonian in(2)ĤTB=∑jℏωcâj†âj+∑jΓ(âj†âj+1+âj+1†âj),
In the Wannier representation, essentially a real-space picture of localized orbitals, this Hamiltonian matrix is written as(3)ĤTB=〈1n|〈2n|〈3n|〈4n|〈5n|〈6n|〈7n|〈8n|〈9n|〈10n|〈11n|〈12n|[EcΓ000000000ΓΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓΓ000000000ΓEc]|1n〉|2n〉|3n〉|4n〉|5n〉|6n〉|7n〉|8n〉|9n〉|10n〉|11n〉|12n〉,where state |1n〉 corresponds to the Wannier function localized in the first cavity forming the CCW, and state |2n〉 corresponds to the Wannier function localized in the second cavity forming the waveguide, and so on. The diagonal matrix element for each site isEc, this is the energy of the resonant mode for each cavity. Periodic boundary conditions are expressed at the waveguide ends as â1†=â12† andâ1=â12.


### 2.2. Dispersion, Group Velocity, and GVD

It is assumed that the wavelength of the single photonλph of 1.182 μm, which is a realistic wavelength for a GaAs/InGaAs-based QD emitter. This wavelength corresponds to the cavity resonant frequencyωc=1.594·1015 rad/s, whereEc=ℏωc. Next, assuming a quality factor Q=1000 for each cavity of the CCW, the coupling rate between neighboring cavities is thus calculated to beΓ=1.594·1012 rad/s, where Γ is defined asωc/Q. This definition of Γ assumes that coupling to the nearest neighbor cavity is the only loss channel, that is, out-of-plan losses are ignored.The dispersion relation of this CCW is solved from the diagonalization of the Hamiltonian matrix. Although working in a real space representation, the problem is fully equivalent to the reciprocal space-based Bloch representation. In fact, forN sites, Nk points exist in reciprocal space and are defined as k=2πK/Na where K is a quantum number such that K=-N/2+1,…,N/2 and a the lattice constant of the periodic waveguide. As a result, the same energy eigenvalues are obtained. They are plotted in red for the normalized positive k points of the first Brillouin zone in Figure 2.Figure 2
Dispersion of a 12 coupled-cavities waveguide.The dispersion obtained numerically matches very well the theoretical dispersion for CCWs in (4) in the linear dispersion approximation. Under this approximation, we assume weak coupling, which means that photons may only leak into the nearest neighbor cavity [14–18]. The tight-binding model yields the optical carrier frequencies(4)ωk=ωc[1+κcos⁡(ka)],
where ωc is the resonant frequency of a single cavity, κ is the coupling coefficient between cavities, k is the Bloch wave number, and a is the lattice constant. For this waveguide, the coupling coefficient was found to beκ=-0.002.An expression for the group velocityvg can be derived from (4) as follows:(5)vg=∇kωk=-ωcκasin⁡(ka).
The group velocity vg normalized over c (the speed of light in vacuum) is plotted in Figure 3 throughout the normalized coupled cavity waveguide band. It is obtained by taking the derivative of obtained energy eigenvalues with respect tok. At the edges of the waveguide band, that which corresponds to when k is 0 or 1, the group velocity tends toward zero.Figure 3
Group Velocity.This result has two important consequences. First, the group velocity dispersion is ill defined whenk is 0 or 1 according to the standard definition in (6). Second, the group velocity dispersion may be either positive or negative depending on which band edge k is closest to(6)GVD=-(2πcλ2)ddω(1vg(k)).
The group velocity dispersion (GVD) for the 12 cavities forming the waveguide is plotted in Figure 4 within the CCW band. For practical applications, Mookherjea derived more appropriate definitions of GVD that satisfy a small fractional change of the GVD coefficient over the range of frequencies of interest [19].Figure 4
Group Velocity Dispersion.


### 2.3. Single-Photon Propagation

A single-photon pulse is shown propagating for the waveguide composed of 12 cavities in Figure5. The time dependence for the coupled-cavity waveguide is obtained numerically using the Louiville or Von Neumann Equation (similar to the approach discussed in [20]) in conjunction with the tight-binding Hamiltonian to solve for the time evolution of the density matrix whose states correspond to the Wannier functions localized in the cavities forming the waveguide. It is assumed that there are neither any out-of-plane losses or material absorption causing the CCW mode to decay nor any scattering resulting in a sudden change of the phase.Figure 5
Traveling Photon.The single photon pulse can be seen traveling from cavity 1 to Cavity 12 in about 9 picoseconds before being reflected. An oscillatory structure at the trailing edge of the pulse can be noticed. We believe this feature has to do with the single-photon pulse defined initially at a precise point in space and time. The consequence is a wideband single-photon pulse, and therefore higher-order terms can no longer be neglected resulting in envelope distortion. It is also helpful to compare that propagation time with the decoherence time of atoms or QDs. The time it takes to transfer a photon from a source cavity to a target cavity is a relatively fast process compared to the decoherence processes associated with electron spin qubits in self-assembled QD (our storage qubit of interest). In a recent publication of ours [20], we showed how the coherent exchange between a photon qubit and an electron spin qubit in a QD is affected by decoherence resulting from hyperfine interactions. Although these decoherence processes prevented the interacting qubits from reaching maximum entanglement, there were at least partially entangled. It is important to note that these interactions were taking place over a longer period of time, about 1 to 2 orders of magnitude longer than the time needed for a photon to propagate down 12 cavities. In addition, a novel scheme based on ultrafast optical spin echo shows that decoherence times on the order of microseconds are achievable [21]. Therefore, it is anticipated that the propagation time of the photon will not be much of an issue over small to medium range distances.Figure5 also depicts the single-photon pulse propagation in the linear region of the CCW; therefore, the single-photon frequency is tuned to the middle of the waveguide band. (This corresponds to a normalized frequency equal to 1 in Figure 2.) As a result, ignoring the oscillatory structure at the trailing edge of the pulse, a pulse propagates mostly unchanged in shape. On the other hand, when the single-photon pulse propagation is determined by the flat region of the CCW dispersion corresponding to one of the edges of the transmission band, there exists a considerable slowing of the group velocity. Figure 6 depicts the travel time from cavity 1 to cavity 12 as a function of coupling coefficientκ.Figure 6
Time Delay.


### 2.4. Engineering Cavity-Waveguide Couplings

Dynamically coupling the end cavities to the waveguide can be achieved in two different ways. On one hand, this can be done by setting all the resonant modes to be the same for all cavities including the end cavities while varying the coupling coefficients between the end cavities and the waveguide in time in order to load and unload the photon from and onto the waveguide. This is depicted in Figure7, and its corresponding Hamiltonian is described in(7)Ĥ=∑jℏωcâj†âj+∑2N-2Γ(âj†âj+1+âj+1†âj)+Γ1(t)(â1†â2+â2†â1)+Γ2(t)(âN-1†âN+âN†âN-1).
On the other hand, dynamic coupling can also be completed by setting coupling coefficients to be the same for all cavities including the end cavities while varying the resonant modes of cavity 2 and cavity N-1 in time in order to load and unload the photon from and onto the waveguide. This is depicted in Figure 8, and its corresponding Hamiltonian is described in (8)Ĥ=∑j≠2,N-1ℏωcâj†âj+ℏω2(t)â2†â2+ℏωN-1(t)âN-1†âN-1+∑1N-1Γ(âj†âj+1+âj+1†âj).
In practice, both of these approaches can be realized by means of a spatial modulation of the refractive index within the quantum network. In order for these loading and unloading operations to be feasible in a functioning quantum network though, they must be performed at speeds much greater than usual decoherence possesses therefore requiring the assistance of ultrafast optical pulses. Methods for the dynamical tuning of refractive index are based on nonlinear effects, carrier injection by linear absorption of an optical pump (free-carrier plasma dispersion effect), carrier injection using a PIN diode, or thermal tuning via optical heating. We are interested in carrier injection by linear absorption of an optical pump which many groups have shown to be a viable method. For example, Tanaka et al. demonstrated a change in cavity Q from 12,000 to 3,000 in 4 ps [22]. Lipson’s group at Cornell University used a 100 fs pump pulse to generate a 18 ps index change pulse [23] and also demonstrated how a 1.5 ps pump pulse corresponding to a 25 ps index change pulse could change the Q factor of a cavity from 60000 to 17000 [24]. Tanabe et al. generated a 14 ps pump pulse resulting in a change in photon lifetime from 320 ps to 70 ps inside a high Q cavity [25]. Carrier injection using PIN diode is an interesting alternative and allows for both the injection and extraction of carrier simultaneously. The resulting shape of the free carrier index change looks a lot like a square pulse. Gardes et al. showed a 7 ps rise and fall time in the index change was possible [26].Figure 7
Unloading/loading by means of time-varying coupling coefficients.Figure 8
Unloading/loading by means of time-varying resonant frequencies


### 2.5. Transfer Efficiencies

On-chip on-demand single-photon transfer stipulates that the photon is unloaded onto a waveguide from a cavity 1 at a timet1 allowing it to travel down the waveguide before being loaded into a cavity 2 at a timet2. Considering the two approaches mentioned in the previous section, a critical question to be answered is what transfer efficiencies can be obtained.First, we consider the case of time-varying coupling coefficients. The unloading of the photon onto the waveguide is achieved by dynamically switching the magnitude of the coupling coefficient between cavity 1 and cavity 2 starting att1. Perhaps, this can be realized using an approach similar to Noda’s [22]. Then, the loading of the photon from the waveguide into cavity N is achieved by once again dynamically switching the magnitude of coupling coefficient between cavity N-1 and cavity N starting att2.Design parameters of interest for both switching functions,Γ1(t)andΓ2(t), are their shape or time dependence, the range of coupling strength over which they are varied, the window in time over which they should be varied. There are also few design constraints. The temporal width of Γ1(t) or FWHM should be larger than the natural cavity decay into the CCW so as to allow the photon to escape yet smaller than round trip time. Similarly, properties of Γ2(t) will greatly depend on factors such as the CCW length, the group velocity, or the group velocity dispersion. Figure 9 shows the switching functions Γ1(t) and Γ2(t) qualitatively.Figure 9
Switching functionsΓ1(t) and Γ2(t).It was found that the optimum shape of switching functionsΓ1(t) and Γ2(t) is a Gaussian profile. In a CCW with constant coupling between its cavities, when a photon is allowed to propagate freely, though highly localized initially (effectively represented by a delta function in space), it eventually exhibits a distribution in space that happens to be well approximated by a Gaussian. It may be that the distribution could be also approximated by other functions, for example, a squared hyperbolic secant. The authors believe it is worth further investigating the mechanism behind the broadening associated with the probability of finding the photon in space. In any case, this is certainly related to the fact that the coupled cavities have a finite Q which introduces an uncertainty in the time over which the photons actually hops from one cavity to the next. That being said, the context of Figure 9 is slightly different as it relates not just to a waveguide with constant coupling coefficient but to time-varying coupling coefficients. In other words, the coupling constant between two quasi-bound modes that are degenerate in frequency is varied. And the cavity Q depends on the coupling Γ(t) whereQ=ω/Γ(t). Providing that the switching function Γ(t) has a Gaussian shape, the Q(t) will also have a Gaussian shape, thus allowing for minimum reflections at the target cavity boundaries since the incoming photon also has a Gaussian-like probability distribution in time.As far as the range over which the coupling strength should be varied forΓ1(t), it is assumed that there is not any coupling, initially, between cavity 1 and cavity 2, yieldingΓ1(0)=0. However, the maximum of the Gaussian shaped switching function is designed to beΓmax=1.594·1012rad/s, which corresponds to the regular coupling rate between neighboring cavities of the CCW. The minimum FWHM for Γ1(t) that allows the entire photon to leak out of the cavity was found to be3ps. The characteristics of Γ2(t) are engineered so as to maximize the transfer efficiency.Figure10 shows transfer efficiencies for various FWHM for Γ2(t) and a large range oft2. These transfer efficiencies are calculated assuming Γ1(t) with a FWHM equal to 3 ps, a peak coupling rate equal to1.594·1012 rad/s, and t1 equal to 1.5 ps. A maximum transfer efficiency of 93% is obtained for Γ2(t) with a FWHM equal to 2.75 ps and a starting time t2 equal to 7.25 ps. The transfer efficiency is defined as the ratio of the probability of finding a photon in cavity N following its capture (once Γ2(t) goes back to zero) over the probability of finding a photon in Cavity 1 before its release (whenΓ1(0)=0), which is always unity.Figure 10
Transfer efficiency with time-varying coupling coefficients.Next, the case of time-varying resonant frequencies is considered. The unloading of the photon onto the waveguide is achieved by dynamically switching the resonant frequency of cavity 2 fromωc+Δ to ωc starting att1. Then, the loading of the photon from the waveguide into cavity N is achieved by dynamically switching the resonant frequency of cavity N-1 from ωc+Δ to ωc starting at t2 and subsequently from ωc back toωc+Δ.Design parameters of interest for both switching functions,S1(t)andS2(t), are their shape or time dependence, the range of frequencies over which the cavity resonant frequencies are varied, the time window over which resonant frequencies should be varied. As far as design constraints, it is desirable that the amount of detuning Δ for the resonant frequency of the “barrier” cavities to be larger than the waveguide bandwidth to avoid any significant coupling between the waveguide and the end cavities. Also, the duration ofS1(t), which is the time during which the resonant frequency of cavity 2 is switched from ωc+Δ toωc, should be large enough for the photon to escape, yet smaller than the round trip time. Similarly, S2(t)needs to be large enough for the photon to be captured. Figure 11 shows S1(t) and S2(t) qualitatively.Figure 11
Switching functionsS1(t) and S2(t).It was found that the optimum shape forS1(t) and S2(t) is a square profile. In the case of the time-varying frequency scheme, the coupling is mostly dictated by ω (Γ≪ω), so until the frequency of the “barrier” cavity matches the frequencies of adjacent cavities, the probability of tunneling through is insignificant. That explains why a square profile is more appropriate. Earlier, in the case of the time-varying coupling coefficient scheme, the difference was that all theω’s were the same; therefore, the probability of tunneling through was only depending onΓ(t). In fact, much higher transfer efficiencies were achieved for the time-varying frequency scheme with a square shape switching function with the prescribed duration (about 75%) compared to a Gaussian shape switching function with a wide range of duration (no more than 10%). The authors believe that the transfer efficiency could further be improved if the switching function was switched on and off adiabatically, resulting in a rounded square shape. In addition, the switching function duration is now chosen to correspond to approximately twice the photon lifetime in the “barrier” cavity since the photon has to both enter and exit the “barrier” cavity before it can reach the target cavity.As far as the range over which the resonant frequency of the “barrier” cavities should be varied, we assume that each cavity in the CCW has aQ of 1000 resulting in a coupling coefficient of Γ=1.594·1012 rad/s  at the wavelength of interest and a bandwidth of BW=ω±κ=1±0.002 in normalized units of frequency. In our case, this corresponds toBW=1.59·1015±3.18·1012 rad/s. Consequently, we designed the detuning parameter to be Δ=3.24·1012 rad/s  such that|κ|≤Δ≪ω. Therefore, by switching “barrier” cavity frequencies from ωc toωc+Δ, we are able to prevent coupling between the end cavities and the CCW.Figure12 shows transfer efficiencies for various FWHM for S2(t) and a large range oft2. These transfer efficiencies are calculated assuming S1(t) with a FWHM equal to 4.875 ps and t1 equal to 1.5 ps. A maximum transfer efficiency of 75% is obtained for S2(t) with a FWHM equal to 3.985 ps and a starting time t2 equal to 5.7 ps.Figure 12
Transfer efficiency with time-varying resonant frequencies.


## 2.1. Waveguide

First, it is useful to consider a CCW by itself. The Hamiltonian that can describe such a waveguide is exactly the tight-binding Hamiltonian in(2)ĤTB=∑jℏωcâj†âj+∑jΓ(âj†âj+1+âj+1†âj),
In the Wannier representation, essentially a real-space picture of localized orbitals, this Hamiltonian matrix is written as(3)ĤTB=〈1n|〈2n|〈3n|〈4n|〈5n|〈6n|〈7n|〈8n|〈9n|〈10n|〈11n|〈12n|[EcΓ000000000ΓΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓ0000000000ΓEcΓΓ000000000ΓEc]|1n〉|2n〉|3n〉|4n〉|5n〉|6n〉|7n〉|8n〉|9n〉|10n〉|11n〉|12n〉,where state |1n〉 corresponds to the Wannier function localized in the first cavity forming the CCW, and state |2n〉 corresponds to the Wannier function localized in the second cavity forming the waveguide, and so on. The diagonal matrix element for each site isEc, this is the energy of the resonant mode for each cavity. Periodic boundary conditions are expressed at the waveguide ends as â1†=â12† andâ1=â12.


## 2.2. Dispersion, Group Velocity, and GVD

It is assumed that the wavelength of the single photonλph of 1.182 μm, which is a realistic wavelength for a GaAs/InGaAs-based QD emitter. This wavelength corresponds to the cavity resonant frequencyωc=1.594·1015 rad/s, whereEc=ℏωc. Next, assuming a quality factor Q=1000 for each cavity of the CCW, the coupling rate between neighboring cavities is thus calculated to beΓ=1.594·1012 rad/s, where Γ is defined asωc/Q. This definition of Γ assumes that coupling to the nearest neighbor cavity is the only loss channel, that is, out-of-plan losses are ignored.The dispersion relation of this CCW is solved from the diagonalization of the Hamiltonian matrix. Although working in a real space representation, the problem is fully equivalent to the reciprocal space-based Bloch representation. In fact, forN sites, Nk points exist in reciprocal space and are defined as k=2πK/Na where K is a quantum number such that K=-N/2+1,…,N/2 and a the lattice constant of the periodic waveguide. As a result, the same energy eigenvalues are obtained. They are plotted in red for the normalized positive k points of the first Brillouin zone in Figure 2.Figure 2
Dispersion of a 12 coupled-cavities waveguide.The dispersion obtained numerically matches very well the theoretical dispersion for CCWs in (4) in the linear dispersion approximation. Under this approximation, we assume weak coupling, which means that photons may only leak into the nearest neighbor cavity [14–18]. The tight-binding model yields the optical carrier frequencies(4)ωk=ωc[1+κcos⁡(ka)],
where ωc is the resonant frequency of a single cavity, κ is the coupling coefficient between cavities, k is the Bloch wave number, and a is the lattice constant. For this waveguide, the coupling coefficient was found to beκ=-0.002.An expression for the group velocityvg can be derived from (4) as follows:(5)vg=∇kωk=-ωcκasin⁡(ka).
The group velocity vg normalized over c (the speed of light in vacuum) is plotted in Figure 3 throughout the normalized coupled cavity waveguide band. It is obtained by taking the derivative of obtained energy eigenvalues with respect tok. At the edges of the waveguide band, that which corresponds to when k is 0 or 1, the group velocity tends toward zero.Figure 3
Group Velocity.This result has two important consequences. First, the group velocity dispersion is ill defined whenk is 0 or 1 according to the standard definition in (6). Second, the group velocity dispersion may be either positive or negative depending on which band edge k is closest to(6)GVD=-(2πcλ2)ddω(1vg(k)).
The group velocity dispersion (GVD) for the 12 cavities forming the waveguide is plotted in Figure 4 within the CCW band. For practical applications, Mookherjea derived more appropriate definitions of GVD that satisfy a small fractional change of the GVD coefficient over the range of frequencies of interest [19].Figure 4
Group Velocity Dispersion.


## 2.3. Single-Photon Propagation

A single-photon pulse is shown propagating for the waveguide composed of 12 cavities in Figure5. The time dependence for the coupled-cavity waveguide is obtained numerically using the Louiville or Von Neumann Equation (similar to the approach discussed in [20]) in conjunction with the tight-binding Hamiltonian to solve for the time evolution of the density matrix whose states correspond to the Wannier functions localized in the cavities forming the waveguide. It is assumed that there are neither any out-of-plane losses or material absorption causing the CCW mode to decay nor any scattering resulting in a sudden change of the phase.Figure 5
Traveling Photon.The single photon pulse can be seen traveling from cavity 1 to Cavity 12 in about 9 picoseconds before being reflected. An oscillatory structure at the trailing edge of the pulse can be noticed. We believe this feature has to do with the single-photon pulse defined initially at a precise point in space and time. The consequence is a wideband single-photon pulse, and therefore higher-order terms can no longer be neglected resulting in envelope distortion. It is also helpful to compare that propagation time with the decoherence time of atoms or QDs. The time it takes to transfer a photon from a source cavity to a target cavity is a relatively fast process compared to the decoherence processes associated with electron spin qubits in self-assembled QD (our storage qubit of interest). In a recent publication of ours [20], we showed how the coherent exchange between a photon qubit and an electron spin qubit in a QD is affected by decoherence resulting from hyperfine interactions. Although these decoherence processes prevented the interacting qubits from reaching maximum entanglement, there were at least partially entangled. It is important to note that these interactions were taking place over a longer period of time, about 1 to 2 orders of magnitude longer than the time needed for a photon to propagate down 12 cavities. In addition, a novel scheme based on ultrafast optical spin echo shows that decoherence times on the order of microseconds are achievable [21]. Therefore, it is anticipated that the propagation time of the photon will not be much of an issue over small to medium range distances.Figure5 also depicts the single-photon pulse propagation in the linear region of the CCW; therefore, the single-photon frequency is tuned to the middle of the waveguide band. (This corresponds to a normalized frequency equal to 1 in Figure 2.) As a result, ignoring the oscillatory structure at the trailing edge of the pulse, a pulse propagates mostly unchanged in shape. On the other hand, when the single-photon pulse propagation is determined by the flat region of the CCW dispersion corresponding to one of the edges of the transmission band, there exists a considerable slowing of the group velocity. Figure 6 depicts the travel time from cavity 1 to cavity 12 as a function of coupling coefficientκ.Figure 6
Time Delay.


## 2.4. Engineering Cavity-Waveguide Couplings

Dynamically coupling the end cavities to the waveguide can be achieved in two different ways. On one hand, this can be done by setting all the resonant modes to be the same for all cavities including the end cavities while varying the coupling coefficients between the end cavities and the waveguide in time in order to load and unload the photon from and onto the waveguide. This is depicted in Figure7, and its corresponding Hamiltonian is described in(7)Ĥ=∑jℏωcâj†âj+∑2N-2Γ(âj†âj+1+âj+1†âj)+Γ1(t)(â1†â2+â2†â1)+Γ2(t)(âN-1†âN+âN†âN-1).
On the other hand, dynamic coupling can also be completed by setting coupling coefficients to be the same for all cavities including the end cavities while varying the resonant modes of cavity 2 and cavity N-1 in time in order to load and unload the photon from and onto the waveguide. This is depicted in Figure 8, and its corresponding Hamiltonian is described in (8)Ĥ=∑j≠2,N-1ℏωcâj†âj+ℏω2(t)â2†â2+ℏωN-1(t)âN-1†âN-1+∑1N-1Γ(âj†âj+1+âj+1†âj).
In practice, both of these approaches can be realized by means of a spatial modulation of the refractive index within the quantum network. In order for these loading and unloading operations to be feasible in a functioning quantum network though, they must be performed at speeds much greater than usual decoherence possesses therefore requiring the assistance of ultrafast optical pulses. Methods for the dynamical tuning of refractive index are based on nonlinear effects, carrier injection by linear absorption of an optical pump (free-carrier plasma dispersion effect), carrier injection using a PIN diode, or thermal tuning via optical heating. We are interested in carrier injection by linear absorption of an optical pump which many groups have shown to be a viable method. For example, Tanaka et al. demonstrated a change in cavity Q from 12,000 to 3,000 in 4 ps [22]. Lipson’s group at Cornell University used a 100 fs pump pulse to generate a 18 ps index change pulse [23] and also demonstrated how a 1.5 ps pump pulse corresponding to a 25 ps index change pulse could change the Q factor of a cavity from 60000 to 17000 [24]. Tanabe et al. generated a 14 ps pump pulse resulting in a change in photon lifetime from 320 ps to 70 ps inside a high Q cavity [25]. Carrier injection using PIN diode is an interesting alternative and allows for both the injection and extraction of carrier simultaneously. The resulting shape of the free carrier index change looks a lot like a square pulse. Gardes et al. showed a 7 ps rise and fall time in the index change was possible [26].Figure 7
Unloading/loading by means of time-varying coupling coefficients.Figure 8
Unloading/loading by means of time-varying resonant frequencies


## 2.5. Transfer Efficiencies

On-chip on-demand single-photon transfer stipulates that the photon is unloaded onto a waveguide from a cavity 1 at a timet1 allowing it to travel down the waveguide before being loaded into a cavity 2 at a timet2. Considering the two approaches mentioned in the previous section, a critical question to be answered is what transfer efficiencies can be obtained.First, we consider the case of time-varying coupling coefficients. The unloading of the photon onto the waveguide is achieved by dynamically switching the magnitude of the coupling coefficient between cavity 1 and cavity 2 starting att1. Perhaps, this can be realized using an approach similar to Noda’s [22]. Then, the loading of the photon from the waveguide into cavity N is achieved by once again dynamically switching the magnitude of coupling coefficient between cavity N-1 and cavity N starting att2.Design parameters of interest for both switching functions,Γ1(t)andΓ2(t), are their shape or time dependence, the range of coupling strength over which they are varied, the window in time over which they should be varied. There are also few design constraints. The temporal width of Γ1(t) or FWHM should be larger than the natural cavity decay into the CCW so as to allow the photon to escape yet smaller than round trip time. Similarly, properties of Γ2(t) will greatly depend on factors such as the CCW length, the group velocity, or the group velocity dispersion. Figure 9 shows the switching functions Γ1(t) and Γ2(t) qualitatively.Figure 9
Switching functionsΓ1(t) and Γ2(t).It was found that the optimum shape of switching functionsΓ1(t) and Γ2(t) is a Gaussian profile. In a CCW with constant coupling between its cavities, when a photon is allowed to propagate freely, though highly localized initially (effectively represented by a delta function in space), it eventually exhibits a distribution in space that happens to be well approximated by a Gaussian. It may be that the distribution could be also approximated by other functions, for example, a squared hyperbolic secant. The authors believe it is worth further investigating the mechanism behind the broadening associated with the probability of finding the photon in space. In any case, this is certainly related to the fact that the coupled cavities have a finite Q which introduces an uncertainty in the time over which the photons actually hops from one cavity to the next. That being said, the context of Figure 9 is slightly different as it relates not just to a waveguide with constant coupling coefficient but to time-varying coupling coefficients. In other words, the coupling constant between two quasi-bound modes that are degenerate in frequency is varied. And the cavity Q depends on the coupling Γ(t) whereQ=ω/Γ(t). Providing that the switching function Γ(t) has a Gaussian shape, the Q(t) will also have a Gaussian shape, thus allowing for minimum reflections at the target cavity boundaries since the incoming photon also has a Gaussian-like probability distribution in time.As far as the range over which the coupling strength should be varied forΓ1(t), it is assumed that there is not any coupling, initially, between cavity 1 and cavity 2, yieldingΓ1(0)=0. However, the maximum of the Gaussian shaped switching function is designed to beΓmax=1.594·1012rad/s, which corresponds to the regular coupling rate between neighboring cavities of the CCW. The minimum FWHM for Γ1(t) that allows the entire photon to leak out of the cavity was found to be3ps. The characteristics of Γ2(t) are engineered so as to maximize the transfer efficiency.Figure10 shows transfer efficiencies for various FWHM for Γ2(t) and a large range oft2. These transfer efficiencies are calculated assuming Γ1(t) with a FWHM equal to 3 ps, a peak coupling rate equal to1.594·1012 rad/s, and t1 equal to 1.5 ps. A maximum transfer efficiency of 93% is obtained for Γ2(t) with a FWHM equal to 2.75 ps and a starting time t2 equal to 7.25 ps. The transfer efficiency is defined as the ratio of the probability of finding a photon in cavity N following its capture (once Γ2(t) goes back to zero) over the probability of finding a photon in Cavity 1 before its release (whenΓ1(0)=0), which is always unity.Figure 10
Transfer efficiency with time-varying coupling coefficients.Next, the case of time-varying resonant frequencies is considered. The unloading of the photon onto the waveguide is achieved by dynamically switching the resonant frequency of cavity 2 fromωc+Δ to ωc starting att1. Then, the loading of the photon from the waveguide into cavity N is achieved by dynamically switching the resonant frequency of cavity N-1 from ωc+Δ to ωc starting at t2 and subsequently from ωc back toωc+Δ.Design parameters of interest for both switching functions,S1(t)andS2(t), are their shape or time dependence, the range of frequencies over which the cavity resonant frequencies are varied, the time window over which resonant frequencies should be varied. As far as design constraints, it is desirable that the amount of detuning Δ for the resonant frequency of the “barrier” cavities to be larger than the waveguide bandwidth to avoid any significant coupling between the waveguide and the end cavities. Also, the duration ofS1(t), which is the time during which the resonant frequency of cavity 2 is switched from ωc+Δ toωc, should be large enough for the photon to escape, yet smaller than the round trip time. Similarly, S2(t)needs to be large enough for the photon to be captured. Figure 11 shows S1(t) and S2(t) qualitatively.Figure 11
Switching functionsS1(t) and S2(t).It was found that the optimum shape forS1(t) and S2(t) is a square profile. In the case of the time-varying frequency scheme, the coupling is mostly dictated by ω (Γ≪ω), so until the frequency of the “barrier” cavity matches the frequencies of adjacent cavities, the probability of tunneling through is insignificant. That explains why a square profile is more appropriate. Earlier, in the case of the time-varying coupling coefficient scheme, the difference was that all theω’s were the same; therefore, the probability of tunneling through was only depending onΓ(t). In fact, much higher transfer efficiencies were achieved for the time-varying frequency scheme with a square shape switching function with the prescribed duration (about 75%) compared to a Gaussian shape switching function with a wide range of duration (no more than 10%). The authors believe that the transfer efficiency could further be improved if the switching function was switched on and off adiabatically, resulting in a rounded square shape. In addition, the switching function duration is now chosen to correspond to approximately twice the photon lifetime in the “barrier” cavity since the photon has to both enter and exit the “barrier” cavity before it can reach the target cavity.As far as the range over which the resonant frequency of the “barrier” cavities should be varied, we assume that each cavity in the CCW has aQ of 1000 resulting in a coupling coefficient of Γ=1.594·1012 rad/s  at the wavelength of interest and a bandwidth of BW=ω±κ=1±0.002 in normalized units of frequency. In our case, this corresponds toBW=1.59·1015±3.18·1012 rad/s. Consequently, we designed the detuning parameter to be Δ=3.24·1012 rad/s  such that|κ|≤Δ≪ω. Therefore, by switching “barrier” cavity frequencies from ωc toωc+Δ, we are able to prevent coupling between the end cavities and the CCW.Figure12 shows transfer efficiencies for various FWHM for S2(t) and a large range oft2. These transfer efficiencies are calculated assuming S1(t) with a FWHM equal to 4.875 ps and t1 equal to 1.5 ps. A maximum transfer efficiency of 75% is obtained for S2(t) with a FWHM equal to 3.985 ps and a starting time t2 equal to 5.7 ps.Figure 12
Transfer efficiency with time-varying resonant frequencies.


## 3. Physical Design

As a sort of proof of principal, a classical optical system was designed to compare and contrast the propagation properties of light in CCW systems. Using photonic crystal as a means to integrate such a coupled system allows great potential for high density integration of waveguides and high-Q cavities that are readily reproducible in fabrication. There has been much work showing from both theory and experiment, that photonic crystal CCW systems can exploit the unique dispersive properties discussed previously for applications of slow light pulse compression and transparency [27–30]. Many different systems have been shown to stop light including traditional defect waveguides with side-coupled integrated sequence of resonators. Initially, these systems seem identical to CCW structures; they both exhibit a cosine-like dispersion relation inside the photonic bandgap, which is eventually flattened adiabatically to stop the light. Differences become apparent when one consider the photonic wave function. In the CCW system under consideration, a highly localized photonic wave function is anticipated once the light is stopped, that is, within a single cavity. On the other hand, for defect waveguides with side-coupled integrated sequence of resonators, the photonic wave function would be spread over the defect waveguide and several cavities. For the reasons just mentioned, a CCW structure decrease device footprint. In addition, CCW structures lend themselves more readily to cavity QED applications. In this publication we focus on the maneuvering of light through a network of cavities only.The coupling of cavity modes to waveguide modes has been a topic of intense research effort for the realization of both quantum and classical integrated optical circuit [11, 31, 32]. It is therefore useful, in photonic crystal devices, to design similarities between isolated cavity modes and propagating CCW modes. This similarity makes for efficient coupling toward the effort of release and capture of optical energy as well as bends and splitters for waveguide routing [33, 34]. The bandwidth of a CCW band may also be adjusted by changing localization properties of the cavities, or the coupling strength (overlap integral) between the localized cavity modes. For instance, decreasing the intercavity distance leads to a wider bandwidth [35].With many such degrees of design freedom that impact the modes of both the isolated cavity and the CCW, use of photonic crystal cavities for analysis of a coupled cavity system is a fruitful choice. The remainder of this section will review the methods and findings of the design and characterization of a CCW system with isolated cavity release and capture switching ability.


### 3.1. Numerical Methods

Analysis of our classical system makes use of the standard numerical simulation tools employed in photonic crystal devices, Plane Wave Expansion (PWE), and Finite-Difference Time Domain (FDTD). The proposed embodiment of this CCW system in a photonic crystal featuring a triangular lattice of circular holes will be performed for TE (even) polarized light only. This choice of polarization allows both a more comfortable band gap in which to engineer defects as well as a more accurate approximation from a 3D slab to a 2D effective index method approximation for future work. All results presented herein may be taken as approximations to a 3D finite slab or simply as a proof of concept in an infinite 2D system. The potential barriers and loss mechanisms for moving to a fully 3D model will be discussed later.PWE provides an initial glimpse of the resonant behavior of these index-periodic metamaterials. The band in which the structure exhibits this periodicity-induced resonance, as shown in Figure13 represents a range of frequencies in which light is not allowed to propagate in the structure.Figure 13
TE band diagram of a 2D photonic crystal with circular scatterers in a triangular lattice. The structure simulated here is a hole type crystal of index 3.4.It is then, in this forbidden spectral range that engineered crystal defects lead to allowed optical modes that may be localized (cavities) or allowed to propagate (waveguides). PWE will also provide the means to find the dispersion relation for our coupled-cavity waveguide modes which is key to the calculation of GVD.Finally, FDTD is implemented to study the spectral response as well as the operational efficiency of these devices. This simulation tool allows the designers to observe how their device would operate under perfect conditions.


### 3.2. Structure

The envisioned system will consist of coupled cavities which exhibit guided modes as well as well-confined cavities that will exhibit highQ-factors in comparison. Lengths of CCW cavities will act as transmission lanes between the high-Q cavities and so the resonances of the two structures will have to align.  We choose, as a simple proof of concept, to implement L3s cavities for both the CCW unit cavity as well as the high Q cavity. L3 is the representation of a defect line cavity of three missing holes, while the s denotes that the holes on the ends of the cavity are reduced in size. This size reduction enables the designer to sculpt the cavity resonance but is not considered variable for the current study. The CCW will be a chain of these cavities as defects in a triangular array of air hole circles ofr=0.4a with 2 hole spacers (r=0.3a) between each cavity as shown in Figure 14. The variable a is the lattice spacing of holes in the crystal.Figure 14
Structure of the high-Q cavity intended as a classical embodiment to the j=1 and j=N cavities in the quantum model above.In order to create the optical isolation necessary between the high-Q cavity and the CCW chain, barriers of variable index material will be used. These regions represent targets for optically induced refractive index switching using off resonant excitation [22].


### 3.3. Coupled Cavity Waveguide

Implementation of our PWE scheme with the above device shows that light will be nondispersive and therefore strongly confined along the M crystal direction but will exhibit dispersion and therefore propagate along the K crystal direction, as shown in Figure15.Figure 15
PWE calculation of our defect mode of interest over the full first Brillouin Zone. The cavity supercell is shown in the inset. There are only confined, nondispersive modes in the M crystal direction but along the K crystal direction, our intended direction of CCW propagation, dispersion is observed.By orthogonalizing thek-vectors calculated in the PWE scheme so as to consider only the K direction, we may increment the PWE’s eigenvalue calculations along only the direction of propagation to simulate the CCW structure’s dispersion relation, shown in Figure 16.Dispersion relation for the L3s2 CCW structure described above. (a) we see that many modes exist in the crystal’s photonic band gap, some with high confinement that result in no dispersion and others that allow coupled-cavity resonance for propagation that result in the sinusoidal k-relation from (4). The mode of interest is selected and zoomed in upon for (b).
(a)(b)The resulting spectral signature of this mode dispersion is shown in Figure17 and will result in as many peaks over the range of resonant frequencies as there are coupled cavities in the chain.(a) represents the spectra inside a single L3s cavity isolated from any other cavities, while (b) represents the spectra through a 7 cavity coupled L3s2 system. Results calculated from FDTD.
(a)(b)Finally, it is important for the spatial field mode profiles of our cavity and CCW to be similar in order to maximize coupling between the two structures. Using both PWE and FDTD simulation methods, we verify the spatial field dependence shown in Figure18 of our mode of choice.Figure 18
Simulated CCW modes in the L3s2 system. On the left, a PWE calculation of the allowed mode corresponding to the dispersive mode in Figure15 matches well with the FDTD simulation of the propagating mode on the right, both showing the out of plane field component (Hy).


### 3.4. Cavity

Relying on FDTD and the analysis of spectra, the design of our cavity is made through observation of the behavior of cavity resonance with changes to surrounding regions. First, an isolated L3s cavity is simulated to establish a basis for the types of modes this sort of cavity is likely to support. This sort of isolated defect is unable to be readily released into a CCW chain, and so a heterostructured cavity is implemented by including index-switchable regions to act as barriers to confine light away from the CCW portion of the device. Because the resonances of identical cavities shift in materials of different refractive index, shown in Figure19, it is possible to use this resonance mismatch as a confining mechanism.Figure 19
Cavity resonance shifting due to a change in material index of refraction.  Using a standard index of 3.4 (red) as our base, to comply with Si or GaAs substrates, switching the barrier regions of our heterostructured cavity to an index of 3.3 (yellow) offers negligible overlap between “barrier” cavity resonances and the waveguide bandwidth (grey). Further confinement would be possible with lower switched index but is less physically reasonable.By implementing switched index cavities as barriers for the generation of a high-Q cavity, the mode of interest for both barrier-open (left) and barrier-closed (right) three cavity segments are shown in Figure 20. While the heterostructured cavity is greatly diminished in Q-factor when compared to a single well isolated L3 cavity, it still represents a functional and well-confined optical mode that is effectively isolated from the neighboring CCW chain. With a Q-factor just under 2000, this proof of concept cavity represents an area for future research in optimization if strong coupling is desired.(a) spectra of 3 coupled cavities simulated through FDTD. (b) the first and third cavities have been index switched to show spectra that now represents an isolated heterostructured High-Q cavity.  The spatial mode profile of this High-Q mode is also shown in (c).
(a)(b)(c)Now, as we look toward coupling this cavity mode to the CCW chain, it should be noted that the cavity resonance of our isolated cavity is now centered to the CCW spectral feature. With these tools, the release and capture of an optical cavity mode is now outlined.


### 3.5. Implementing Coupling Mechanism

To perform release of the cavity mode into the CCW chain, one cavity barrier is index switched (opened) by optically induced carrier injection. Now matching the chain region, the cavity mode is allowed to couple to the CCW resonance and propagate from cavity to cavity down the chain. As the energy propagates down the chain, it becomes distributed amongst the coupled cavities and must be collected in a two step process by which the propagating mode is stopped by the outside secondary cavity barrier and allowed to accumulate before the inside secondary cavity barrier is activated, trapping the mode in the secondary heterostructured cavity.  The full three step release and capture process is diagrammed here in Figure21.Figure 21
HighQ cavity mode release and capture process. A shows the confined cavity with both barriers activated with an index of 3.3. B shows the opening of the inside cavity by index switching to n=3.4. With the second cavity’s outside barrier closed, C, the energy accumulates and is then trapped by closing the inner barrier of the second cavity in D.


## 3.1. Numerical Methods

Analysis of our classical system makes use of the standard numerical simulation tools employed in photonic crystal devices, Plane Wave Expansion (PWE), and Finite-Difference Time Domain (FDTD). The proposed embodiment of this CCW system in a photonic crystal featuring a triangular lattice of circular holes will be performed for TE (even) polarized light only. This choice of polarization allows both a more comfortable band gap in which to engineer defects as well as a more accurate approximation from a 3D slab to a 2D effective index method approximation for future work. All results presented herein may be taken as approximations to a 3D finite slab or simply as a proof of concept in an infinite 2D system. The potential barriers and loss mechanisms for moving to a fully 3D model will be discussed later.PWE provides an initial glimpse of the resonant behavior of these index-periodic metamaterials. The band in which the structure exhibits this periodicity-induced resonance, as shown in Figure13 represents a range of frequencies in which light is not allowed to propagate in the structure.Figure 13
TE band diagram of a 2D photonic crystal with circular scatterers in a triangular lattice. The structure simulated here is a hole type crystal of index 3.4.It is then, in this forbidden spectral range that engineered crystal defects lead to allowed optical modes that may be localized (cavities) or allowed to propagate (waveguides). PWE will also provide the means to find the dispersion relation for our coupled-cavity waveguide modes which is key to the calculation of GVD.Finally, FDTD is implemented to study the spectral response as well as the operational efficiency of these devices. This simulation tool allows the designers to observe how their device would operate under perfect conditions.


## 3.2. Structure

The envisioned system will consist of coupled cavities which exhibit guided modes as well as well-confined cavities that will exhibit highQ-factors in comparison. Lengths of CCW cavities will act as transmission lanes between the high-Q cavities and so the resonances of the two structures will have to align.  We choose, as a simple proof of concept, to implement L3s cavities for both the CCW unit cavity as well as the high Q cavity. L3 is the representation of a defect line cavity of three missing holes, while the s denotes that the holes on the ends of the cavity are reduced in size. This size reduction enables the designer to sculpt the cavity resonance but is not considered variable for the current study. The CCW will be a chain of these cavities as defects in a triangular array of air hole circles ofr=0.4a with 2 hole spacers (r=0.3a) between each cavity as shown in Figure 14. The variable a is the lattice spacing of holes in the crystal.Figure 14
Structure of the high-Q cavity intended as a classical embodiment to the j=1 and j=N cavities in the quantum model above.In order to create the optical isolation necessary between the high-Q cavity and the CCW chain, barriers of variable index material will be used. These regions represent targets for optically induced refractive index switching using off resonant excitation [22].


## 3.3. Coupled Cavity Waveguide

Implementation of our PWE scheme with the above device shows that light will be nondispersive and therefore strongly confined along the M crystal direction but will exhibit dispersion and therefore propagate along the K crystal direction, as shown in Figure15.Figure 15
PWE calculation of our defect mode of interest over the full first Brillouin Zone. The cavity supercell is shown in the inset. There are only confined, nondispersive modes in the M crystal direction but along the K crystal direction, our intended direction of CCW propagation, dispersion is observed.By orthogonalizing thek-vectors calculated in the PWE scheme so as to consider only the K direction, we may increment the PWE’s eigenvalue calculations along only the direction of propagation to simulate the CCW structure’s dispersion relation, shown in Figure 16.Dispersion relation for the L3s2 CCW structure described above. (a) we see that many modes exist in the crystal’s photonic band gap, some with high confinement that result in no dispersion and others that allow coupled-cavity resonance for propagation that result in the sinusoidal k-relation from (4). The mode of interest is selected and zoomed in upon for (b).
(a)(b)The resulting spectral signature of this mode dispersion is shown in Figure17 and will result in as many peaks over the range of resonant frequencies as there are coupled cavities in the chain.(a) represents the spectra inside a single L3s cavity isolated from any other cavities, while (b) represents the spectra through a 7 cavity coupled L3s2 system. Results calculated from FDTD.
(a)(b)Finally, it is important for the spatial field mode profiles of our cavity and CCW to be similar in order to maximize coupling between the two structures. Using both PWE and FDTD simulation methods, we verify the spatial field dependence shown in Figure18 of our mode of choice.Figure 18
Simulated CCW modes in the L3s2 system. On the left, a PWE calculation of the allowed mode corresponding to the dispersive mode in Figure15 matches well with the FDTD simulation of the propagating mode on the right, both showing the out of plane field component (Hy).


## 3.4. Cavity

Relying on FDTD and the analysis of spectra, the design of our cavity is made through observation of the behavior of cavity resonance with changes to surrounding regions. First, an isolated L3s cavity is simulated to establish a basis for the types of modes this sort of cavity is likely to support. This sort of isolated defect is unable to be readily released into a CCW chain, and so a heterostructured cavity is implemented by including index-switchable regions to act as barriers to confine light away from the CCW portion of the device. Because the resonances of identical cavities shift in materials of different refractive index, shown in Figure19, it is possible to use this resonance mismatch as a confining mechanism.Figure 19
Cavity resonance shifting due to a change in material index of refraction.  Using a standard index of 3.4 (red) as our base, to comply with Si or GaAs substrates, switching the barrier regions of our heterostructured cavity to an index of 3.3 (yellow) offers negligible overlap between “barrier” cavity resonances and the waveguide bandwidth (grey). Further confinement would be possible with lower switched index but is less physically reasonable.By implementing switched index cavities as barriers for the generation of a high-Q cavity, the mode of interest for both barrier-open (left) and barrier-closed (right) three cavity segments are shown in Figure 20. While the heterostructured cavity is greatly diminished in Q-factor when compared to a single well isolated L3 cavity, it still represents a functional and well-confined optical mode that is effectively isolated from the neighboring CCW chain. With a Q-factor just under 2000, this proof of concept cavity represents an area for future research in optimization if strong coupling is desired.(a) spectra of 3 coupled cavities simulated through FDTD. (b) the first and third cavities have been index switched to show spectra that now represents an isolated heterostructured High-Q cavity.  The spatial mode profile of this High-Q mode is also shown in (c).
(a)(b)(c)Now, as we look toward coupling this cavity mode to the CCW chain, it should be noted that the cavity resonance of our isolated cavity is now centered to the CCW spectral feature. With these tools, the release and capture of an optical cavity mode is now outlined.


## 3.5. Implementing Coupling Mechanism

To perform release of the cavity mode into the CCW chain, one cavity barrier is index switched (opened) by optically induced carrier injection. Now matching the chain region, the cavity mode is allowed to couple to the CCW resonance and propagate from cavity to cavity down the chain. As the energy propagates down the chain, it becomes distributed amongst the coupled cavities and must be collected in a two step process by which the propagating mode is stopped by the outside secondary cavity barrier and allowed to accumulate before the inside secondary cavity barrier is activated, trapping the mode in the secondary heterostructured cavity.  The full three step release and capture process is diagrammed here in Figure21.Figure 21
HighQ cavity mode release and capture process. A shows the confined cavity with both barriers activated with an index of 3.3. B shows the opening of the inside cavity by index switching to n=3.4. With the second cavity’s outside barrier closed, C, the energy accumulates and is then trapped by closing the inner barrier of the second cavity in D.


## 4. Results and Discussion

Through the switching protocol described above, transfer of energy from the optical mode in one high-Q cavity, through a chain of 5 coupled cavities, to a second high-Q cavity was achieved in a 2D FDTD scheme with 77% efficiency. Confinement at the first cavity, propagation between cavities, and capture in the second cavity is shown in Figure 22.Three step release of an optical mode in a high-Q cavity into a CCW and capture into a second high-Q cavity with 77% efficiency. Energy density (a) and field evolution (b) are shown to illustrate the process.
(a)(b)


## 5. Conclusion

We have demonstrated that in theory using both a quantum model and a classical model that single photons can be transferred efficiently on-chip from a high-Q cavity to another using coupled cavity waveguides if the dynamic coupling between nearest neighbor cavities is carefully engineered. From the quantum model, as far as single-photon trapping and releasing mechanisms are concerned, dynamically switching coupling coefficients between “end” cavities and the waveguide yields much higher single photon transfer efficiencies (93% versus 75%) than dynamically switching the resonant frequencies of the “barrier” cavities. However, so far, only the single-photon trapping and releasing mechanism based on dynamically switching the resonant frequencies of the “barrier” cavities was able to be implemented in a practical way within our classical model. The endeavor to realize this system in a realistic form of a 3D photonic crystal slab (PCS) presents two hurdles: increased numerical cost, and out of plane loss mechanisms. It is expected that the increased numerical cost may be greatly alleviated by using 2D approximation methods for TE (even) polarization.  However, due to the low group velocity of CCW modes and the zero group velocity of the high-Q state, careful engineering of the Fourier components of the field distributions in this system will be necessary to avoid coupling to out-of-plane radiative modes. Until then, though, the two models shown in this work yield strikingly similar single-photon transfer efficiencies (~75%), which reinforces the versatility of CCWs and the advantage of using such structures to implement photonic quantum networks.



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*Source: 893086-2010-12-16.xml*