## 1. Introduction

Cavity quantum electrodynamics (cavity QED) describes few atoms coupling to quantized electromagnetic fields inside an optical cavity (Mabuchi & Doherty, 2002). The core of cavity QED is the strong coherent interaction between the single-mode electromagnetic field and the internal states of the atom. It is one of few experimentally realizable systems in which the intrinsic quantum mechanical coupling dominates losses that due to dissipation (Cirac et al., 1997). Furthermore, it represents an almost ideal and the simplest quantum system which allows quantitative studying of a dynamical open quantum system under continuous observation. Up to the present, three representative optical microcavities have been proposed for studying quantum optics and implementing quantum information (Vahala, 2004). The first one is the conventional Fabre-Perot (FP) type cavities consisting of two concave dielectric mirrors facing each other at a distance of the order of a few 100 μm, where single neutral atoms can be trapped through magneto-optical trap (MOT), optical dipole trap or magnetic trap for a long time (up to several seconds). The second is the microcavities supporting whispering gallery modes, including microspheres, microdisks, and microtoroids. The third type is the nanoscale cavities in photonic crystal (Foresi et al., 1997).

With FP-type microcavities, numerous theoretical schemes have been suggested for generating nonclassical states of cavity fields (Vogel et al., 1993, Parkins et al., 1993, Law et al., 1996) and entangled states of many atoms (Cabrillo et al., 1999), and realizing two-qubit logic gates (Pellizzari et al., 1995, Pachos and Walther, 2002) and universal gates for Fock-state qubits (Santos, 2005), which lead to experimental realization of the Einstein-Podolsky-Rosen (EPR) state of two atoms, Greenberger-Horne-Zeilinger (GHZ) states of three parties (two atoms plus one cavity mode), Schrödinger cat state (Brune et al., 1996), and single-photon state (Brattke et al., 2001) of a cavity field. However, FP-type microcavities have their inherent problems. For example, it is extremely difficult to realize a scalable quantum computation in experiment by integrating many microcavities, though theoretical protocols may be simple and elegant. Recently, whispering gallery microcavities have been studied for cavity QED toward quantum information processing (Xiao et al., 2006) due to their ultrahigh quality factors (Q, which is proportional to the confinement time in units of the optical period) and high physical scalability. Strong-coupling regime has been demonstrated when cold caesium atoms fall through the external evanescent field of a whispering gallery mode. Nevertheless, the cold atoms are ideal stationery qubits (quantum bits), but not suited for good flying qubits. Thus, a solid-state cavity QED (involved single quantum dot (QD), for example) system with whispering gallery microcavities seeks further advancements.

As a new resonant configuration, nanocavities in photonic crystal with high quality factors (Q) and ultrasmall mode volumes (V) are attracting increasing attention in the context of optical cavity QED (Faraon et al., 2008, Fushman et al., 2008, Hennessy et al., 2007, Badolato et al., 2006, Reithmaier et al., 2004, Yoshie et al., 2004). Combined with low loss and strong localization, they present a unique platform for highly integrated nanophotonic circuits on a silicon chip, which can also be regarded as quantum hardware for nanocavity-QED-based quantum computing. Toward this goal, strong interactions between a QD and a single photonic crystal cavity have been observed experimentally (Hennessy et al., 2007, Badolato et al., 2006, Reithmaier et al., 2004, Yoshie et al., 2004). Moreover, single photons from a QD coupled to a source cavity can be remarkably transferred to a target cavity via an integrated waveguide in an InAs/GaAs solid-state system (Englund et al., 2007a), which opens the door to construct the basic building blocks for future chip-based quantum information processing systems. Weak coupling nanocrystal ensemble measurements are reported in TiO2-SiO2 and AlGaAs cavity systems (below 1 m wavelengths) recently (Guo et al., 2006, Fushman et al., 2005) and also independently in silicon nanocavities with lead chalcogenide nanocrystals (a special kind of QDs) at near 1.55 m fibre communication wavelengths recently (Bose et al., 2007).

In this Chapter, we theoretically study the coherent interaction between single nanocrystals and nanocavities in photonic crystal. This Chapter is organized as follows. In section 2, our attention is focused on a single QD embedded in a single nanocavity. First, we introduce, derive, and demonstrate the explicit conditions toward realization of a spin-photon phase gate, and propose these interactions as a generalized quantum interface for quantum information processing. Second, we examine single-spin-induced reflections as direct evidence of intrinsic bare and dressed modes in our coupled nanocrystal-cavity system. In section 3, however, our attention is switched on the *N* coupled cavity-QD subsystems. We examine the spectral character and optical delay brought about by the coupled cavities interacting with single QDs, in an optical analogue to electromagnetically induced transparency (EIT) (Fleischhauer et al., 2005). Furthermore, we then examine the usability of this coupled cavity-QD system for QD-QD quantum phase gate operation and our numerical examples suggest that a two-qubit system with high fidelity and low photon loss.

## 2. Nanocrystals in silicon photonic crystal standing-wave cavities

In this section, we examine the single-photon pulse (or weak coherent light pulse) interactions of a single semiconductor nanocrystal in a system comprised of standing-wave high-Q/V silicon photonic crystal nanocavities (Xiao et al., 2007a). In contrast to earlier travelling-wave whispering gallery cavity studies (Xiao et al., 2006), we show here that a QED system based on coupled standing-wave nanocavities can realize a spin-photon phase gate even under the bad-cavity limit and provide a generalized quantum interface for quantum information processing. In addition, we demonstrate numerically a solid-state universal two-qubit phase gate operation with a single qubit rotation. This theoretical study is focused within the parameters of near 1.55 m wavelength operation for direct integration with the fiber network, and in the silicon materials platform to work with the vast and powerful silicon processing infrastructure for large-array chip-based scalability.

### 2.1. Theoretical model

We begin by considering a combined system consisting of coupled point-defect high-*Q/V* photonic crystal cavities, a line-defect photonic crystal waveguide, and an isolated single semiconductor nanocrystal. We offer some brief remarks on this system before building our theoretical model. When a photon pulse is coupled into the cavity mode via a waveguide (Fig. 1(a)), photons can couple out of the cavity along both forward and backward propagating directions of the waveguide because the cavity supports standing-wave modes. While each cavity can each have a Faraday isolator to block the backward propagating photon, such implementation may not be easily scalable to a large-array of cavities. To obtain only forward transmission, here we examine theoretically a defect cavity system with accidental degeneracy (Fan et al., 1998, Xu et al., 2000, Min et al., 2004) as a generalized study of *cavity-dipole-cavity* systems, and which also provides close to 100% forward-only drop efficiency. This framework is also immediately applicable to non-reciprocal magneto-optic cavities which have larger fabrication tolerances. Both systems support two degenerate even |*e* and odd |*o* cavity modes (*h*-polarized, dominant in-plane *E*-field) that have opposite parity due to the mirror symmetry, as shown in Fig. 1(a). The waveguides can support both *v*-polarizations (dominant in-plane *H*-field) and *h*-polarizations for polarization diversity (Barwicz et al., 2007).

Fig. 1(b) shows the energy levels and electron-exciton transitions of our cavity-dipole-cavity system. In order to produce nondegenerate transitions from the electron spin states, a magnetic field is applied along the waveguide direction (Atatüre et al., 2006). | and | play the rule of a stationary qubit, which have shown much longer coherence time than an exciton (dipole or charge). The transition

Now we construct our model by studying the interaction between the nanocrystal and the cavity modes. The Heisenberg equations of motion for the internal cavity fields and the nanocrystal are (Duan et al., 2003, Duan & Kimble et al., 2004, Sørensen & Mølmer, 2003)

where the interaction Hamiltonian

is in a rotating frame at the input field frequency
*e* and |*o* modes in the standing-wave cavities in order for forward-only propagation of the qubit. The cavity dissipation mechanism is accounted for by

When the two degenerate modes have the same decay rate, i.e.,

where the effective single-photon coupling rates are

The nanocrystal-cavity system is excited by a weak monochromatic field (e.g., single-photon pulse), so that we solve the above motion equations for the below explicit analytical expressions

andNote that orthogonality of the |*e* and |*o* basis modes (as shown in Fig. 1a) forces the nanocrystal to choose only either
*e* and |*o* are uniquely zero), but no other possibilities. Photon qubit input from only the left waveguide forces only one of the cavity states (|*e* + i|*o*) to exist (Fan et al., 1998), and we assume this cavity environment from the existing photon qubit enhances the
*true* one-way transmission through the cavity-dipole-cavity system.

### 2.2. Spin-photon phase gate

To examine more of the underlying physics, we consider first the case of exact resonance (
*global* phase change

This two-qubit phase gate combined with simple single-bit rotation is, in fact, universal for quantum computing. More importantly, this interacting system can be regarded as a quantum interface for quantum state sending, transferring, receiving, swapping, and processing.

To efficiently evaluate the quality of the gate operation, the gate fidelity is numerically calculated, as shown in Fig. 2. Considering specifically a lead chalcogenide (e.g. lead sulphide) nanocrystal and silicon photonic nanocavity system for experimental realization, we choose the spontaneous decay as _{s} ~ 2 MHz and all non-radiative dephasing _{p} ~ 1 GHz at cooled temperatures. Photonic crystal cavities have an ultrasmall mode volume
^{6} experimentally and ~10^{7} theoretically (Asano et al., 2006; Kuramochi et al., 2006) has been achieved in photonic crystal cavities.

With these parameters, as shown in Fig. 2a, the gate fidelity of the cavity-dipole-cavity system can reach 0.98 or more, even when photon loss is taken into account, and even when the vacuum Rabi frequency g_{e} is lower than the cavity decay rate (bad-cavity limit). The gate fidelity increases initially as the cavity approaches more into the over-coupling regime due to less photon loss and eventually decreases as the nanocrystal-cavity system moves away from the strong coupling regime. Secondly, we note that with non-zero detuning (/_{o}=2; Case III and VI), the gate fidelity slightly decreases but is still adequate. With increasing nanocrystal dissipation rate (Fig. 2b), the fidelity decreases as expected and the system moves away from strong coupling (less nanocrystal interactions with the cavity). The physical essence behind such high fidelities is the true one-way transmission where the nanocrystal couples to |+ mode, with only forward propagation with no backward scattering of the qubit. In addition, accidental degeneracy mismatch may degrade the gate performance. To validate the feasibility of the present scheme, we perform a direct calculation of gate fidelity for different frequency and lifetime of the opposite-parity cavity modes. Even with degeneracy mismatch (_{e} - _{l} = _{el}
_{ol} = _{o} - _{l}; in Case IV and VII) with some backward scattering of the qubit, the gate fidelity is shown to remain high. Moreover, with different lifetimes of the cavity modes (Case VII), the fidelity remains high as long as the

### 2.3. Single-spin-induced reflections

Furthermore, we show that the above cavity-dipole-cavity interaction mechanism can result in interesting transmissions and reflections based on the presence or absence of dipole interaction, and with different detunings. We examine the case of

## 3. Coupled electrodynamics in photonic crystal cavities

Over the past few years, theoretical and experimental interests are mainly focused on a single cavity interacting with atoms, and tremendous successes have been made ranging from strongly trapping single atoms and deterministic generation of single-photon states, to observation of atom-photon quantum entanglement and implementation of quantum communication protocols. For more applications, current interest also lies in the coherent interaction among distant cavities. The coherent interaction of cavity arrays has been studied as an optical analogue to EIT in both theory (Smith et al., 2004, Xiao et al., 2007b) and experiment (Xu et al., 2006, Totsuka et al., 2007). Coupled cavities can be utilized for coherent optical information storage because they provide almost lossless guiding and coupling of light pulses at slow group velocities. When dopants such as atoms or QDs interact with these cavities, the spatially separated cavities have been proposed for implementing quantum logic and constructing quantum networks. Recent studies also show a strong photon-blockade regime and photonic Mott insulator state (Hartmann et al., 2006, Hartmann & Plenio, 2007), where the two-dimensional hybrid system undergoes a characteristic Mott insulator to superfluid quantum phase transition at zero temperature (Greentree et al., 2006, Angelakis et al., 2007). Recently, it has shown that coupled cavities can also model an anisotropic Heisenberg spin-1/2 lattice in an external magnetic field (Hartmann et al., 2007). The character of a coupled cavity configuration has also been studied using the photon Green function (Hughes, 2007).

### 3.1. Model of coupled N cavity–QD subsystems

Using transmission theory, we study coherent interactions in a cavity array that includes N cavity–QD subsystems (Xiao et al., 2008), with indirect coupling between adjacent cavities through a waveguide (Fig. 4). First, we investigate a subsystem in which a single cavity interacts with an isolated QD. Here for simplicity we suppose that only a single resonance mode (h-polarized) is present in the cavity, although two-mode cavity–QD interactions have been considered in the previous section. The cavity–QD–waveguide subsystem has mirror-plane symmetry, so that the mode is even with respect to the mirror plane. We can easily obtain the Heisenberg equations of motion

where

In the weak excitation limit excited by a weak monochromatic field or a single photon pulse with frequency

Here the transmission matrix is

where

where

When studying only the spectral character of the coupled cavity-QD interaction (Section 3.2), we note that this is analogous to classical microwave circuit design, where the transmission and reflection characteristics from Eq. (14) can also be examined with coupled-mode theory with dipole terms inserted. Examining the spectral character first (Section 3.2) helps to understand the coupled cavity-QD controlled quantum phase gate operation and performance (Sections 3.3 and 3.4).

### 3.2. Spectral character of coupled cavity-QD arrays

To examine the physical essence, we need to first examine the spectral character of the coupled cavity-QD system. The reflection and transmission coefficients are defined as

Fig. 5a describes the transmission spectra of two coupled empty cavities (without QD) with different detuning

In the presence of QDs, Fig. 6a (top) shows the spectral characteristics in which a single QD resonantly interacts with the first cavity. When both cavities are resonant, there exist two obvious sharp peaks located symmetrically around

Fig. 6a (bottom) illustrates the case where both cavities resonantly interact with a single QD each. Similar to the above analysis, we can explain the number and locations of sharp peaks with respect to different

Phase shift and photon storage.– To further examine this coupled cavity-QD system, Fig. 6c shows the transmission phase shift for various detunings of the input photon central frequency, where the cavity and QD transition are resonant for both subsystems. The phase shift has a steep change as we expected intuitively, which corresponds to a strong reduction of the group velocity of the photon. As shown in Fig. 6d, the delay time (

### 3.3. Quantum phase gate operation

In the section above, we have shown the novel transport character of the coupled cavity-QD system. Now we study the possibility of quantum phase gate operation of the QDs based on this transport character. The schematic to realize this multi-QD coupled cavity-cavity system is illustrated in Fig. 7. The QDs are represented by two ground states |g and |r, where the state |r is largely detuned with the respective cavity mode. The two ground states can be prepared via QD spin-states such as demonstrated remarkably in experiment in Ref. (Atatuer et al., 2006) with near-unity fidelity. The input weak photon pulse is assumed h-polarized, with an input pulse duration D (e.g. 1 ns) larger than the loaded cavity lifetime for the steady-state approximation. To remove the distinguishability of the two output photon spatial modes in the waveguide (transmitted and reflected), a reflecting element is inserted in the end of waveguide (such as a heterostructure interface), as shown in Fig. 7. This ensures that the photon always exits in the left-propagating mode |L (from a right-propagating input mode|R) without any entanglement with the QD states. Alternatively, a Sagnac interferometer scheme such as introduced in Reference (Gao et al., 2008) can also be implemented to remove the spatial mode distinguishability and QD-photon entanglement. In this single input single output mode scheme, |h and |v represent the two polarization states of the input photon. We emphasize that in the below calculations we have considered the complete characteristics of the full system (including the end reflecting element and the resulting "standing wave" due to the long photon pulse width) where we examined the final left-propagating output mode |L from a right-propagating input mode |R (Fig. 7). The reflection interference is included where we force

To facilitate the discussion but without loss of generality, we describe the all resonance case (i.e.,

Case I: The two QDs are initially prepared in

Case II: The QDs initially occupy in

Therefore, with the exit of the photon of the single input single output system, the state of the two QDs after the interaction is now described by

We provide a few more notes on this designed coupled cavity-cavity multi-QD system. First, the temporal distinguishability is small for the single cavity-QD system, where in Fig. 8c we plot the shape function of the output photon pulse for cases when the QD is coupled (|g), decoupled (|r), or without the cavity, through numerical simulation of the dynamical evolution of the system. The pulse shape function overlaps very well. Secondly, the calculated temporal distinguishability in the coherently coupled cavity-cavity multi-QD system is also small compared to the pulse duration

### 3.4. Gate fidelity and photon loss

To exemplify the coupled cavity system, isolated single semiconductor QDs in high-
^{4} and 10^{5} are achievable experimentally, with intrinsic
^{6} reported recently (Noda et al., 2007, Tanabe et al., 2007).

To characterize the present gate operation, Figs. 9a and 9b present the two-qubit phase gate fidelity

## 4. Summary

In this Chapter, with the nanocavities in photonic crystal, we theoretically introduce, derive, and demonstrate the robust implementation of a single spin-photon phase gate in a cavity-dipole-cavity system. The conditions of accidental degeneracy are examined to enforce complete transfer, either in the forward transmission or in reflection, of the qubit. In addition, we observe that a photon pulse is strikingly reflected by a cavity interacting with a single spin, even under the bad-cavity limit. This combined nanocrystal-cavity system, examined in a silicon materials platform with lead chalcogenide nanocrystals in the near infrared, can serve as a QD spin-photon two-qubit quantum phase gate and, indeed, as a general quantum interface for large-array chip-based quantum information processing. To further utilize the high-Q and small-V nanocavities of photonic crystal, we also investigate the operation and performance of a scalable cavity–QD array on a photonic crystal chip towards controlled QD-QD quantum gates. The coupling among single-QD emitters and quantized cavity modes in a coherent array results in unique transmission spectra, with an optical analogue of EIT-like resonances providing potential photon manipulation. In the quantum phase gate operation, we note that the gate fidelity can reach 0.99 or more and the photon loss can be below 0.04 in a realistic semiconductor system, provided the non-ideal detunings are kept within the cavity decay rates. Our study provides a potential for a chip-scale quantum gate towards a potential quantum computing network with the platform of silicon photonic crystal.

For future experimental quantum information processing in photonic crystal, we note that it is possible to realize the initial idea in a single nanocavity-QD coupled system, as experimentally demonstrated in the context of strong cavity-QD coupling (Hennessy et al., 2007, Badolato et al., 2006, Reithmaier et al., 2004, Yoshie et al., 2004, Faraon et al., 2008, Fushman et al., 2008) and the quantum state transfer between a single QD and a target cavity (Englund et al., 2007a). This opens the door to construct the basic building blocks for future chip-based quantum information processing systems. However, it is still a challenge to implement quantum information with a nanocavity array in photonic crystal. The challenge includes several main technique difficulties. First, it is necessary to precisely place a single two-level nanocrystal (or other QDs) with respect to the corresponding nanocavity mode in photonic crystal, for the largest Rabi frequency, and also to position across an array. Second, both the cavity resonances and QD transitions should spectrally overlap within approximately the cavity or exciton linewidths, although our theoretical model is still robust with small QD-QD, cavity-QD, and cavity-cavity detunings. The former challenge depends on careful nanofabrication techniques, while the latter condition can be relaxed through high-precision tunability of the cavity resonances or QD transitions. Moreover, we note that the ultrahigh-Q and low-V regime is desired to operate well into the strong coupling regime, suppressed chip-scale photon losses or improved collection to improve quantum state transfer, as well as control of dephasing especially at high temperatures in order for chip-level scalability in solid-state quantum information nanosciences.