Abstract

The controlled coupling of photon emitters with tailored nanophotonic structures offers an exciting platform for studying fundamental quantum electrodynamics (QED) and developing on-chip quantum information processing at telecom-compatible wavelengths. Here we introduce a three-dimensional polariton waveguide structure, capable of achieving strong coupling with a single quantum emitter. This polariton waveguide consists of a nanowire-based photonic crystal (PC) waveguide with a quantum dot (QD) embedded in each unit cell. Using realistic designs and parameters, we derive and calculate the fundamental electromagnetic properties of these polariton waveguides, with an emphasis on the local optical density of states (LDOS) and the photon Green function. We demonstrate dramatic increases, and rich fundamental control, of the LDOS due to strong light–matter interactions in each unit cell through periodic QD interactions and further show that these results are quite robust to structural disorder. As an example application, we consider the coupling of an external target QD with a finite-sized polariton waveguide, and show that the single QD strong coupling regime is easily accessible, even for modest dipole strengths. Our polaronic structures are fundamentally interesting and allow for the exploration of new regimes of waveguide QED. While our calculations are exemplified for a PC nanowire system, the general results apply to a wide range of PC waveguides including planar PC slabs and “alligator” PCs, as well as circuit-QED systems.

© 2016 Optical Society of America

1. INTRODUCTION

The design of nanoscale solid-state devices to control light–matter interactions is highly desirable, both for applications in quantum information science [1] and to explore novel regimes of quantum electrodynamics (QED) [2,3]. By exploiting periodicity to tailor the local optical density of states (LDOS), photonic crystals (PCs) with embedded quantum dots (QDs) have had much success in this regard [47]. However, these PC slab structures have been partly hindered by fabrication issues including surface roughness [810] and limited control of the QD properties [11]. Arrays of nanowires (NWs) grown through a molecular beam epitaxy (MBE) technique [1214] have been proposed as an alternative PC platform to help mitigate these issues [15,16], with the potential to contain identical or very similar QDs with controllable properties inside each NW of a given radius [1719]. This allows for the conception of NW PC waveguides where each waveguide channel NW contains an identical QD embedded in its center. Recent work with “alligator” PC waveguides has also demonstrated the ability to trap chains of atoms and explore many-body physics in such systems [20,21].

Coupled dipole chains in free space have interesting collective properties and can act as subwavelength waveguides [22]; implementing such chains in PC waveguides could result in new physical behavior or improved performance. Metamaterials, composed of multiple elements that are engineered to have unique and useful properties, have been used to produce exotic waveguides with a dramatically different LDOS than simple dielectric structures, e.g., manifesting in large spontaneous emission rate enhancements [23,24] for dipole emitters, even with material losses. Plasmonic polariton waveguide structures [25] and metal nanoparticle chains coupled to traditional waveguides [26] have demonstrated chain-mediated coupling between plasmonic and photonic excitations leading to an anti-crossing in the band structure. This anti-crossing as a consequence of normal mode splitting is well understood to occur in simple one-dimensional waveguides containing resonator chains [27,28] or quantum wells [29,30]. These devices have used normal mode splitting to slow and trap light [30]; however, one-dimensional structures are incapable of LDOS enhancements leading to quantum optical effects at the single-quantum level, which also require tight localization of light within a small volume, a feat entirely inaccessible in a planar geometry or conventional waveguide.

Inspired by the metamaterial concept and advances in NW growth techniques, the inclusion of a QD chain in a PC waveguide can similarly reshape and greatly enhance the system LDOS and result in drastically different properties and behavior, but without the large metallic losses typically associated with plasmonic metamaterials. In this article, we introduce and explore the optical properties of a nanophotonics system consisting of a PC waveguide with an embedded QD in each unit cell, which we describe as a “polariton waveguide” because its excitations are mixed light–matter states due to the collective coupling of the QDs with the waveguide Bloch mode. Figure 1(a) shows a schematic of our proposed structure implemented in a NW PC geometry, although the fundamental polariton waveguide concept of a nanophotonic waveguide coupled to a dipole chain and resultant physics is not at all restricted to this specific type of structure. To elucidate the underlying physics of these systems, we derive the photon Green function (GF) of both infinite and finite-sized systems polariton waveguides, and explore their coupling with a single external “target” QD, which is found to be in the strong coupling regime. We stress that the polariton anti-crossing in this system is fundamentally different from classical normal mode splitting in simple one-dimensional structures: in a three-dimensional geometry with periodic small mode volumes, such as our NW PC system, the slow-light enabled large LDOS can be further enhanced through normal mode splitting with an embedded dipole chain. The resultant achievement of true strong coupling between a single quantum emitter and a realistic waveguide mode, to the best of our knowledge first predicted here, allows for the creation of devices relying on quantum cavity physics, such as photon blockades and single photon switches [3,7], with reliable input/output coupling on chip. Moreover, although we analyze a specific NW PC for realistic fabrication purposes, our polariton waveguide architecture can be generalized to platforms such as alligator or conventional slab PCs and coupled-cavity waveguides [1,31,32], and also be adapted to other systems such as circuit QED [2].

 figure: Fig. 1.

Fig. 1. (a) Proposed polariton waveguide, with yellow QDs embedded inside the PC waveguide channel, and a red external QD coupling to the structure. Red arrows denote the waveguide direction (ex) and NWs are cut out so embedded QDs can be seen. (b) Slow-light region of the band structure on top, and Im{G(rn,rn)} in units of ρh (see text) on bottom; the infinite polariton waveguide in solid blue is compared with the original PC waveguide in dashed orange.

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2. PROPOSED STRUCTURE

Our proposed structure exploits the elevated NW PC waveguide design of Ref. [15], where a PC is formed from a square lattice array of GaAs NWs extended from an AlO substrate [33]. The waveguide channel is introduced by reducing the radius of a row of NWs from rb=0.180a to rd=0.140a, which we define to be along ex. Importantly, this structure is based on current fabrication techniques, and properties are determined through full three-dimensional calculations including radiative coupling effects. This PC waveguide, with lattice constant a=0.5526  μm, contains a single vertically polarized (ez) below-light-line waveguide band with a mode edge near the telecom wavelength of 1.550 μm (ranging from 755 to 795 meV). Light is confined to the upper GaAs portion of the NWs (height 2.27a), while the 2a AlO layer separates them from the substrate and an array width of 7a is sufficient to prevent in-plane losses. As this waveguide band approaches the mode edge, it flattens due to symmetry and the group velocity goes to zero, causing the LDOS to diverge as seen in Fig. 1(b), an effect that is inevitably spoiled by disorder-induced losses in real systems [8]. To form the polariton waveguide of Fig. 1(a), we embed a vertically polarized QD in the center of the GaAs layer of each waveguide NW at rn,0=r0+naex, where n is an integer. We take the embedded QDs to have a Lorentzian polarizability α=α(ω)ez=2ω0|d|2ez/(ε0(ω02ω2iΓ0ω)), with a realistic dipole moment |d|=30  D (0.626e-nm) and polarization decay rate Γ0=1  μeV (including both nonradiative decay and coupling into nonwaveguide modes); these are similar to experimental parameters for InAs QDs at 4 K [34]. The QD exciton line is taken to be ω0=794.5  meV, which is in the moderately slow-light regime of the waveguide band corresponding to a group velocity vg=c/30.4 and Fz=40.1 (relative LDOS enhancement, Fz=Im{G}/Im{Gh}). Since PC waveguide frequencies simply scale with the inverse lattice constant, these systems can be designed to couple with QDs with an arbitrary exciton resonance. Vertically polarized QDs can be formed in NWs by controlling growth conditions to produce elongated QDs [19,35], and their exciton line can be controlled through QD composition and dimension [17,35]. The coupling of the embedded QDs to this nontrivial photonic environment is treated through the photonic GF approach described below.

Throughout this work, we connect to the photonic GF G(r,r;ω), which describes the system electric-field response at r to a point source at r (fully including all light-scattering events):

[××ω2c2ε(r)]G(r,r;ω)=ω2c21δ(rr),
where ε(r) is the complex material permittivity and 1 the unit dyad, since G is a second-rank tensor. The imaginary and real components of the GF at equal space points are directly proportional to the system LDOS (and dipole emitter spontaneous emission rate) and Lamb shift, respectively [1,16]. These GFs are projected along the relevant ez component for vertically polarized QDs, G=ez·G·ez, and given in units of the imaginary part of free-space GF Im{Gh(r,r;ω)}=ω3/(6πc3)ρh(ω) [36] to connect directly to LDOS enhancement. When we consider the addition of an external target QD to the waveguide, this will be embedded, e.g., on top of the waveguide NW at position rn, where n indexes the unit cell, as seen in Fig. 1(a). This position is chosen both for fabrication purposes and because this is where the antinode of the waveguide mode resides.

3. POLARITON WAVEGUIDES FOR AN INFINITE PC

We first consider an infinite embedded QD array to help explain these structures’ underlying physics, exploiting Bloch’s theorem and treating the QDs as a perturbation to each PC unit cell Δε(r)=δ(rr0)α(ωk)ez, which shifts the waveguide resonance ωk for a given kkx to ωk. We obtain from perturbation theory ωk2=ωk2(1VcΔε(r)|ez·uk(r)|2dr) [3739], where Vc is the unit-cell volume and uk(r)=uk(r+naex) is the waveguide unit-cell function. The unit-cell function is normalized through Vcε(r)|uk(r)|2dr=1, and corresponds to waveguide mode fk(r)=aLuk(r)eikx, where L is the waveguide length. We find that the waveguide band is split by the QD–Bloch-mode interaction, resulting in a pair of complex eigenfrequencies ωk,±=ωk,±+iΓk,±/2 at each k, with

ωk,±=ω0+ωk2±12(ω0ωk)2+4gk2,
and
Γk,±=Γ0ωk,±2ωk2(ωk,±2ωk2)+(ωk,±2ω02).
The QD–unit-cell coupling parameter gk=ω02ε0d·uk(r0) gives the strength of the light–matter interaction leading to anti-crossing and is the continuum form of g, the single-mode quantum optical coupling rate. This modified band structure is compared with that of the original PC in Fig. 1(b), where the strong mode splitting (gk0=120  μeVΓ0) flattens the dispersion as ωk,±ω0 and losses remain low: Γk,±Γ0ωk,±. The inclusion of the QD array has caused the waveguide band to split, corresponding to a mixed light–matter excitation: the polariton waveguide. The above approach was also applied to the waveguide Bloch modes: away from ωk,±ω0 where perturbation theory naturally breaks down we find that only the waveguide Bloch-mode contribution is significant and uk,±=uk. Throughout this work we only discuss results in the frequency regime where these perturbative results hold.

We proceed to write the GF of this infinite polariton waveguide as a sum over these waveguide Bloch modes, GP(r,r;ω)=k,±ω±(k)2fk,±(r)fk,±*(r)ω±(k)2ω2. Converting this sum to an integral in the complex plane [39], we arrive at an analytic expression similar in form to that for a regular PC waveguide [cf. Eq. (5)]:

GP(r,r;ω)=iaω2v˜g[Θ(xx)ukω(r)ukω*(r)e(ikωκω)(xx)+Θ(xx)ukω*(r)ukω(r)e(ikωκω)(xx)],
but with v˜g(ω)=vg(ω)i2dΓ±(k)dk|kω, where vg(ω)dΓ±dk is the polariton waveguide group velocity and κω=Γ±(kω)2vg(ω). We emphasize that both vg(ω) and kω are found from the polariton waveguide ω±k relationship, which differs greatly from that of the original PC near ω0 due to the polariton splitting. This was derived assuming κωkω, which remains valid until vgc/2300, where |ωω0|<20  μeV. The effect of the QD array, demonstrated in Fig. 1(b), is thus to produce large and tunable enhancements to the PC LDOS near resonance by flattening the band structure; as one moves away from ω0, the original PC waveguide GF is quickly recovered. At the above minimum detuning, these polariton waveguides yield Fz=3100—a dramatic rate enhancement and indicative that these structures should enter the strong coupling regime of QED, even at the single-quantum level.

4. FINITE-SIZED POLARITON PC WAVEGUIDES

We next consider polariton waveguides with a finite number of embedded QDs, more representative of real structures. We can no longer exploit Bloch’s theorem, but instead include QDs iteratively in the system GF via a Dyson equation approach that yields an exact solution to this system. The background Green function G(0) is that of the underlying PC waveguide, given by [1,40]

Gw(r,r;ω)=iaω2vg[Θ(xx)ukω(r)ukω*(r)eikω(xx)+Θ(xx)ukω*(r)ukω(r)eikω(xx)].
Starting with unit cell n=1, we introduce a QD at rn,0, calculate the resultant Green function G(n) including scattering from this QD, and then use this as the background Green function to introduce a QD in the subsequent unit cell. From the Dyson equation, QD n can be included self-consistently in the system GF through [41] G(n)(r,r;ω)=G(n1)(r,r;ω)+G(n1)(r,rn,0;ω)·α·G(n)(rn,0,r;ω). This was done numerically using the same system as for the infinite case (i.e., identical PC waveguide and QDs) for QD chains of increasing length, N=1101.

Figure 2 shows the G(r,r;ω) for increasing N, highlighting the N=101 result, corresponding to a chain sufficiently long to produce substantial LDOS enhancements and begin to recover the infinite chain result, while still demonstrating important finite-size effects. We emphasize the position on top of the central NW of the QD array in Fig. 2(a) (r=r51 for the N=101 structure, where a target QD will be embedded later) because constructive interference from repeated QD scattering maximizes the polariton effects and resultant LDOS enhancement at this point, as seen in Fig. 2(b). The addition of a single QD introduces a dip in G at ω0; however, with increasing N the buildup of off-resonant enhancement from QD scattering leads to the formation of a strong resonance in Im{G}, which exceeds G(0)(rn,rn;ω0)=49.5ρh(ω0) for N>15. This resonance is red-shifted from ω0, with a peak Fz=338.14 and FWHM Γ1=3.35  μeV at ω1=ω021.49  μeV for the 101 QD case. As more QDs are added, this polariton peak grows, narrows, and blue-shifts toward ω0 while additional weaker red-shifted resonances begin to appear. Above ω0, resonances also form, which grow and become more numerous with increasing N, indicating that they arise from QD chain Fabry–Perot (FP) modes. The higher two FP modes for the N=101 structure are at ωFP=ω1+25.12  μeV and ωFP=ω1+29.02  μeV. Away from ω0, G(N) converges to the PC waveguide GF, as was seen for the infinite case. The large number of resonances in these waveguides also produce a richly varying Re{G} that remains substantial over a broad frequency region, particularly for larger N. These findings were verified to hold for a range of QD decay rates Γ0=010  μeV, with the ω0 dip extending deeper and resonances initially appearing at lower N and closer to ω0 as Γ0 is reduced. Figure 2(b) further highlights the profound influence of the QD array on the PC LDOS: the strong resonances and depletion around ω0 are seen throughout the polariton portion of the waveguide. Even outside of the structure, scattering from the QD array leads to spatially varying constructive or destructive interference and a corresponding substantial increase or decrease in the LDOS.

 figure: Fig. 2.

Fig. 2. (a) Imaginary and real components of G(N)(rt,rt;ω) on right and left, respectively. G(101)(r51,r51;ω) in solid blue is compared with G(51)(r26,r26;ω) in solid orange, G(21)(r11,r11;ω) in dashed red, and N=0 results in dash-dotted gray. (b) Im{G(101)(rn,rn;ω)} near ω0, plotted on a logarithmic scale due to the narrowness of the primary resonance. Δx=n51 and the gray dashed lines denote the terminus of the QD array at Δx=±50. All values are in units of ρh(ω).

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Figure 3 shows the propagator |G(101)(rn,r51;ω)|, which is proportional to the coupling strength between the target QD location and various points in the structure, and far greater coupling rates are found than for a bare ideal PC waveguide. For instance, |Im{G(101)(rn,r51;ω)}|>300ρh(ω) and |Re{G(101)(rn,r51;ω)}|>150ρh(ω) are found for Δn15. One can produce any arbitrary combination of Im{G} and Re{G} through careful choice of separation and frequency, as G(N)(rn,rn;ω) acquires a phase shift of eikωa|nn| as n and n are varied. The interaction between a pair of dipole emitters in a photonic medium at r and r is governed by both the real and imaginary portions of G(r,r;ω) [16]; these structures can thus be used to produce arbitrary interactions and possibly even form the platform of a many-body quantum simulator. This GF reshaping again persists even past the mode edge of the polariton waveguide: once one is outside of the QD chain |G(101)(rn,r51;ω)| has a peak of 101.8ρh(ω) at ω1 (ω021.6  μeV), more than double the bare PC waveguide result: |G(0)(rn,rn;ω1)|=49.5ρh(ω1). Notably, the above results fully include finite-size effects and radiative loss, and reproduce the physics of the infinite structure, namely strong light–matter interactions resulting in substantial enhancements in the system LDOS.

 figure: Fig. 3.

Fig. 3. |G(101)(rn,r51;ω)| in units of ρh(ω) for the N=101 polariton waveguide, where the termination of the QD array at Δx=±50 is again denoted with dashed gray lines.

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5. INFLUENCE OF STRUCTURAL DISORDER

An additional advantage of the above Dyson equation approach is that it can be adapted to systems with arbitrarily inhomogeneous QD properties, allowing for the study of structural disorder, which inevitably occurs in real systems. We have thus evaluated the influence of fluctuations in QD exciton frequency, as well as position and dipole moment, on the LDOS of the N=101 polariton waveguide, and a selection of representative results is plotted in Fig. 4. We first consider disorder in QD position: since the internal QDs are embedded at local maxima of the waveguide Bloch mode, any shift in position will reduce the field amplitude at the QD uk(rn,0). This field amplitude only appears in G in the form |dn·uk(rn,0)|2|gk|2, so fluctuations in QD position are equivalent to reductions in QD dipole strength or the site-specific coupling parameter. We calculate the LDOS for random reductions in |gk| for each QD with a normal distribution and standard deviation of up to σg=10%. This corresponds to a shift in position of up to 0.1a=55  nm, yielding a 3 D reduction in |d|, or a dipole moment rotation of up to 26° (although this is unlikely to be an issue in NW systems). Denoting the change in polariton peak LDOS (at ω1) ΔFz and the shift in this peak Δω1, we find that σg=5% yields a mean Δω1=1.5±0.1  μeV and small fluctuations in peak height ΔFz from -20 to 5, and increasing σg to 10% has little influence on ΔFz but increases Δω1 to 3.1±0.1  μeV. A pair of single instance LDOS results is shown on the left of Fig. 4, where it can be seen that these QD-coupling-strength fluctuations only shift the primary polariton peak slightly.

 figure: Fig. 4.

Fig. 4. Im{G(101)(rn,rn;ω)} in units of ρh(ω) under various instances of disorder compared with the disorder-free result in dashed–dotted gray. On left, sample results for coupling-strength reductions σg=5% and 10% are plotted in dashed red and solid blue. On right, the influence of fluctuations in ω0 is seen, with a σω0=5  μeV result in solid orange compared with a σω0=10  μeV result in solid blue. An instance of the same set of σω0=10  μeV frequencies and random σg=10% reductions is plotted in dashed red.

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We then considered fluctuations in QD resonance about ω0 with σω0 up to 10 μeV, both with and without disorder in |gk|. This type of inhomogeneity has a more significant influence on the system, resulting in the formation of a large number of weaker peaks in the LDOS depletion region around ω0, and substantial increases or reductions in the peak Fz. Both these effects grow with increasing σω0, with 10 μeV resulting in Δω0=5±1  μeV and a wide range of polariton peak Fz from 160 to 594. Relative to QD resonance fluctuations, additional QD inhomogeneity in coupling strength has little effect on the LDOS, as demonstrated on the right panel of Fig. 4. We also note that even with σω0Γ1 as well as other spectral features, the system properties remain largely unchanged and in particular the sharp LDOS peak is maintained. Indeed, even the instance with the lowest peak, with Fz=160, remains capable of achieving strong coupling with an external emitter, producing a |geff|=3.0  μeV (described in Section 6). We can thus conclude that the main features of our proposed polariton waveguides are highly robust with regard to the inevitable moderate fluctuations in QD properties in experimentally realized systems.

6. SINGLE QUANTUM DOT STRONG COUPLING

We now consider an important quantum optical application of these polariton waveguides, studying the behavior of a single external QD at rt=r51 of the N=101 polariton waveguide. The system LDOS enhancements, specifically the polariton peak at ω1, are possibly sufficient to strongly couple the waveguide to a realistic QD. We adopt a rigorous quantization procedure appropriate for lossy inhomogeneous systems [42], such as these polariton waveguides. The system Hamiltonian in the dipole approximation, consisting of a single target QD interacting with the electromagnetic environment of the polariton waveguide, is given by

H=ωtσ^+σ^+dr0dωlωlb^(r;ωl)·b^(r;ωl)(σ^++σ^)(dt·E^(rt)+H.c.),
where the external QD is treated as two-level atom (TLA) with exciton frequency ωt and transition dipole moment dt; b^(r;ωl) is the bosonic-field annihilation operator associated with the field frequency ωl, which generates the electric-field operator via E^(r)=iπε0dωldrIm{ε(r;ωl)}G(r,r;ωl)·b^(r;ωl) [42]. We Laplace transform the resulting operator Heisenberg equations of motion to construct the spontaneous emission spectrum at detector position rD: S(rD,ω)=E^(rD,ω)E^(rD,ω). Using the initial condition of a vacuum field and excited QD, we derive the exact detected spontaneous emission spectrum,
S(rD,ω)=|(ωt+ω)G(rD,rt;ω)·d/ε0ωt2ω2ωΣ(ω)iωΓt|2.
In the above, the self-energy is Σ(ω)=2dt·G(rt,rt;ω)·dtε0, and Γt is the polarization decay rate of the target TLA due to interactions with the environment (i.e., anything other than the LDOS given by the polariton waveguide GF). The spectrum depends not only on the GF at TLA position rt but also on the propagator to the detector G(rD,rt;ω). When we present spectra below, we show both S(rD,ω) and S0(ω), where S0(ω) is calculated from Eq. (7) by replacing the propagator in the numerator with G(rt,rt;ωt). This is done because S(rD,ω) is often distorted by the rich spectral features of the GFs of these waveguide structures (see Fig. 3). The bare spectrum S0(ω) more cleanly shows the system energy levels and dynamics of the target QD and is typically the only one calculated [43].

We assume a ez-aligned target QD with a polarization decay rate Γt=1  μeV and calculate spectra for target QD dipole moments of 10, 30, and 60 D, encompassing the range of QDs that could feasibly be coupled to the polariton waveguide. For a simple Lorentzian LDOS, the resultant system dynamics can be approximated with the Jaynes–Cummings (JC) Hamiltonian, where the interaction of a single photonic mode and TLA leads to an anti-crossing as they approach resonance (splitting of 2g on resonance); near ω1 we define an effective coupling constant |geff|=Γ1dt·Im{G(rt,rt;ω1)}·dt2ε0 [39]. The three target dipole moments produce geff of 1.24, 3.72, and 7.43 μeV, respectively, which meet the criteria for strongly coupling with the ω1 resonance: 2geff>Γt,Γ1 [44]. We assign a detector position rD=rt55aex, which is outside the polariton waveguide portion of the structure to reduce the filtering of the detected spectrum, while in the same position in the unit cell as the QD to take advantage of the strong output coupling demonstrated in Fig. 3.

The emitted and detected spectra S0(ω) and S(rD,ω) at ωt=ω1 for the 30 D QD are shown in Fig. 5(a). Remarkably, substantial splitting is seen in both S0(ω) and S(rD,ω) despite the modest choice of dt. Furthermore, peak locations of ω=ω13.02  μeV and ω+=ω1+3.67  μeV agree well with the JC Hamiltonian when the Lamb shift is included; however, we stress that the widths and weighting of the spectral peaks, as well as propagation effects and polarization decay, are only captured through the formalism leading to Eq. (7). In particular, the reduced height of the ω peak is a result of the unique shape of Re{G} near ω1 producing a large positive Lamb shift. The importance of propagation effects can be clearly seen by comparing S0(ω) and S(rD,ω), where the relative LDOS depletion above ω1 reduces the height of the ω+ peak. In Fig. 5(b) we show the spontaneous emission spectra at ωt=ωFP for dt=60  D, where the QD–polariton-waveguide coupling is sufficiently strong that significant exchange occurs with both the ωFP and ωFP modes for ωt=ωFP. This results in a triplet forming in S0(ω), with peaks at ωFP0.44  μeV, ωFP1.81  μeV, and ωFP+2.16  μeV; the splitting is substantially stronger than ΓFP/2, so these peaks are seen at the detector position as well.

 figure: Fig. 5.

Fig. 5. S0(ω) in dashed blue and S(rD,ω) in solid orange for ωt=ω1, with dt=30  D in (a) and ωt=ωFP with dt=60  D in (b). Im{G(rt,rt;ω)} in arbitrary units is shown in dash-dotted gray.

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Finally, to demonstrate that the features in the above spontaneous emission spectra are a consequence of the strong coupling regime, the clearly resolvable peaks of S0(ω) as ωt is brought to resonance with ω1 are shown in Fig. 6 for a target QD with dt=30 and 60 D. The left plots depict anti-crossing with the primary resonance at ω1, while the peaks as ωt is swept through the FP region are shown on the right and markers denote the peaks of Fig. 5. Remarkably for a waveguide system, all three QDs demonstrate a clear anti-crossing as they approach ω1, showing conclusive evidence that they are strongly coupled to the polariton waveguide despite the complicated nature of this system. While similar strong coupling behavior has been theoretically predicted for coupled-cavity PC waveguides [32], a later study showed that this was inevitably spoiled by disorder [10]. Strong coupling has recently been observed with Anderson-localized cavities in disordered PC waveguides [6], although this is fundamentally different from strong coupling with a propagating waveguide mode as reported here. It is also notable that our predicted splitting is seen even for the 10D target QD, which is much weaker than that used in other studies. Of further interest is the system behavior in the FP region of the polarition waveguide, where the unusual shape of the system LDOS leads to multiple anti-crossings that are poorly described by a JC Hamiltonian. The interaction of the FP modes with the 60D target QD is particularly striking: up to four energy levels are seen for a given ωt, and the multimode nature of the QD-polariton waveguide has flattened these anti-crossing lines. Indeed, the effective coupling constant |geff(ω)| of the 60 D target QD with the FP peaks on the order of the FP peak spacing, indicating that this system has entered the exotic and highly sought-after regime of multimode strong coupling [45].

 figure: Fig. 6.

Fig. 6. Peaks in S0(ω) in blue as a function of ωt and dt. Results for dt=10, 30, and 60 D are presented from bottom to top, with ωt near ω1 and in the FP region on the left and right, respectively. Crosses and circles denote peaks of Figs. 5(a) and 5(b), ωt is in dashed red, and LDOS peaks are in dashed gray.

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7. CONCLUSIONS

In conclusion, we have proposed a nano-engineered metamaterial system, a polaritonic waveguide, consisting of a PC waveguide with periodic embedded QDs. We developed two separate theoretical approaches to describe the optical physics of this system, studying both infinite and finite polariton waveguides and demonstrating that in both instances strong light–matter interactions lead to rich and dramatic LDOS enhancements that are robust to disorder and without the material losses typically associated with metallic metamaterials. We then considered the interaction of a finite-size structure with an external QD and showed that these LDOS enhancements can be exploited to strongly couple with a single external emitter and produce interesting spectral features clearly visible in the externally detected spectrum. Although we studied the specific material system of a NW PC with embedded QDs, similar results can be found for chains of in-plane–polarized QDs coupling the waveguide mode of a slab PC, and our proposal can be readily extended to other periodic material systems if it proves more experimentally feasible. Furthermore, chiral PC waveguides have recently demonstrated directional emission for QDs embedded at chiral points of circular polarization [46], allowing for interesting collective effects with spin-dependent quantum emitters [47,48]. Polariton waveguides can be readily extended to these chiral systems and potentially exhibit novel and useful properties.

These polariton structures are capable of solving the outstanding problem of strong coupling and controlled LDOS enhancement in PC waveguides, using physics fundamentally different from the usual slow-light regime at a mode edge, which is strongly influenced by disorder-induced scattering losses. Polariton waveguides could be used to design next-generation devices for quantum information and explore new regimes of open-system waveguide QED. As an example application, such structures have the potential to act as a single photon switch with greater input/output coupling efficiency and operating bandwidth than cavity-based devices [7,49].

Funding

Natural Sciences and Engineering Research Council of Canada (NSERC); Queen’s University, Canada.

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2. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008). [CrossRef]  

3. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006). [CrossRef]  

4. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990). [CrossRef]  

5. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef]  

6. J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

7. R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012). [CrossRef]  

8. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005). [CrossRef]  

9. M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009). [CrossRef]  

10. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008). [CrossRef]  

11. T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012). [CrossRef]  

12. V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009). [CrossRef]  

13. H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011). [CrossRef]  

14. N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014). [CrossRef]  

15. G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014). [CrossRef]  

16. G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015). [CrossRef]  

17. M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013). [CrossRef]  

18. S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011). [CrossRef]  

19. J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009). [CrossRef]  

20. J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015). [CrossRef]  

21. A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015). [CrossRef]  

22. D. S. Citrin, “Coherent transport of excitons in quantum-dot chains: role of retardation,” Opt. Lett. 20, 901–903 (1995). [CrossRef]  

23. G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 1–7 (2009).

24. P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009). [CrossRef]  

25. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003). [CrossRef]  

26. M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012). [CrossRef]  

27. J. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30, 2001–2003 (2005). [CrossRef]  

28. F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).

29. A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001). [CrossRef]  

30. Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005). [CrossRef]  

31. S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

32. D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007). [CrossRef]  

33. M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004). [CrossRef]  

34. S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012). [CrossRef]  

35. Y. M. Niquet and D. C. Mojica, “Quantum dots and tunnel barriers in InAs InP nanowire heterostructures: electronic and optical properties,” Phys. Rev. B 77, 1–12 (2008). [CrossRef]  

36. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

37. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of shallow photonic crystal waveguides: semi-analytic treatment,” Opt. Express 17, 19629–19643 (2009). [CrossRef]  

38. M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81, 245321 (2010). [CrossRef]  

39. G. Angelatos, “Theory and applications of light-matter interactions in quantum dot nanowire photonic crystal systems,” Master’s thesis (Queen’s University, 2015).

40. V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007). [CrossRef]  

41. P. T. Kristensen, J. Mørk, P. Lodahl, and S. Hughes, “Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs,” Phys. Rev. B 83, 075305 (2011). [CrossRef]  

42. L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004). [CrossRef]  

43. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).

44. P. Meystre and M. Sargent, Elements of Quantum Optics (Springer, 1999).

45. D. O. Krimer, M. Liertzer, S. Rotter, and H. E. Türeci, “Route from spontaneous decay to complex multimode dynamics in cavity QED,” Phys. Rev. A 89, 1–8 (2014). [CrossRef]  

46. B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015). [CrossRef]  

47. I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015). [CrossRef]  

48. A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

49. T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012). [CrossRef]  

References

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  1. P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
    [Crossref]
  2. J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
    [Crossref]
  3. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
    [Crossref]
  4. S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
    [Crossref]
  5. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
    [Crossref]
  6. J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).
  7. R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
    [Crossref]
  8. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
    [Crossref]
  9. M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
    [Crossref]
  10. D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
    [Crossref]
  11. T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
    [Crossref]
  12. V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
    [Crossref]
  13. H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
    [Crossref]
  14. N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
    [Crossref]
  15. G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014).
    [Crossref]
  16. G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015).
    [Crossref]
  17. M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
    [Crossref]
  18. S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
    [Crossref]
  19. J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
    [Crossref]
  20. J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
    [Crossref]
  21. A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
    [Crossref]
  22. D. S. Citrin, “Coherent transport of excitons in quantum-dot chains: role of retardation,” Opt. Lett. 20, 901–903 (1995).
    [Crossref]
  23. G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 1–7 (2009).
  24. P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
    [Crossref]
  25. A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
    [Crossref]
  26. M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
    [Crossref]
  27. J. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30, 2001–2003 (2005).
    [Crossref]
  28. F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).
  29. A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
    [Crossref]
  30. Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
    [Crossref]
  31. S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).
  32. D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007).
    [Crossref]
  33. M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
    [Crossref]
  34. S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
    [Crossref]
  35. Y. M. Niquet and D. C. Mojica, “Quantum dots and tunnel barriers in InAs InP nanowire heterostructures: electronic and optical properties,” Phys. Rev. B 77, 1–12 (2008).
    [Crossref]
  36. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
  37. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of shallow photonic crystal waveguides: semi-analytic treatment,” Opt. Express 17, 19629–19643 (2009).
    [Crossref]
  38. M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81, 245321 (2010).
    [Crossref]
  39. G. Angelatos, “Theory and applications of light-matter interactions in quantum dot nanowire photonic crystal systems,” Master’s thesis (Queen’s University, 2015).
  40. V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
    [Crossref]
  41. P. T. Kristensen, J. Mørk, P. Lodahl, and S. Hughes, “Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs,” Phys. Rev. B 83, 075305 (2011).
    [Crossref]
  42. L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
    [Crossref]
  43. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).
  44. P. Meystre and M. Sargent, Elements of Quantum Optics (Springer, 1999).
  45. D. O. Krimer, M. Liertzer, S. Rotter, and H. E. Türeci, “Route from spontaneous decay to complex multimode dynamics in cavity QED,” Phys. Rev. A 89, 1–8 (2014).
    [Crossref]
  46. B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015).
    [Crossref]
  47. I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
    [Crossref]
  48. A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).
  49. T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
    [Crossref]

2015 (6)

G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
[Crossref]

B. le Feber, N. Rotenberg, and L. Kuipers, “Nanophotonic control of circular dipole emission,” Nat. Commun. 6, 6695 (2015).
[Crossref]

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
[Crossref]

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

2014 (4)

D. O. Krimer, M. Liertzer, S. Rotter, and H. E. Türeci, “Route from spontaneous decay to complex multimode dynamics in cavity QED,” Phys. Rev. A 89, 1–8 (2014).
[Crossref]

N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
[Crossref]

G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014).
[Crossref]

S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

2013 (2)

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

2012 (5)

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
[Crossref]

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

S. Weiler, A. Ulhaq, S. M. Ulrich, D. Richter, M. Jetter, P. Michler, C. Roy, and S. Hughes, “Phonon-assisted incoherent excitation of a quantum dot and its emission properties,” Phys. Rev. B 86, 241304 (2012).
[Crossref]

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
[Crossref]

2011 (3)

P. T. Kristensen, J. Mørk, P. Lodahl, and S. Hughes, “Decay dynamics of radiatively coupled quantum dots in photonic crystal slabs,” Phys. Rev. B 83, 075305 (2011).
[Crossref]

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
[Crossref]

2010 (3)

P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
[Crossref]

F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).

M. Patterson and S. Hughes, “Interplay between disorder-induced scattering and local field effects in photonic crystal waveguides,” Phys. Rev. B 81, 245321 (2010).
[Crossref]

2009 (6)

S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of shallow photonic crystal waveguides: semi-analytic treatment,” Opt. Express 17, 19629–19643 (2009).
[Crossref]

G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 1–7 (2009).

P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
[Crossref]

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
[Crossref]

V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
[Crossref]

2008 (3)

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Y. M. Niquet and D. C. Mojica, “Quantum dots and tunnel barriers in InAs InP nanowire heterostructures: electronic and optical properties,” Phys. Rev. B 77, 1–12 (2008).
[Crossref]

2007 (3)

D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

V. S. C. Manga Rao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75, 205437 (2007).
[Crossref]

2006 (1)

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

2005 (3)

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005).
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Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
[Crossref]

J. Shen and S. Fan, “Coherent photon transport from spontaneous emission in one-dimensional waveguides,” Opt. Lett. 30, 2001–2003 (2005).
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2004 (2)

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
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L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
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2003 (1)

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
[Crossref]

2001 (1)

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
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1995 (1)

1990 (1)

S. John and J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
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M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Akopian, N.

J.-C. Harmand, L. Liu, G. Patriarche, M. Tchernycheva, N. Akopian, U. Perinetti, and V. Zwiller, “Potential of semiconductor nanowires for single photon sources,” Proc. SPIE 7222, 722219 (2009).
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Androvitsaneas, P.

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

Angelatos, G.

G. Angelatos and S. Hughes, “Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system,” Phys. Rev. A 91, 051803(R) (2015).
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G. Angelatos and S. Hughes, “Theory and design of quantum light sources from quantum dots embedded in semiconductor-nanowire photonic-crystal systems,” Phys. Rev. B 90, 205406 (2014).
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G. Angelatos, “Theory and applications of light-matter interactions in quantum dot nanowire photonic crystal systems,” Master’s thesis (Queen’s University, 2015).

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M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
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Arakawa, Y.

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
[Crossref]

Atatüre, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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Ba Hoang, T.

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

Badolato, A.

T. Volz, A. Reinhard, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoğlu, “Ultrafast all-optical switching by single photons,” Nat. Photonics 6, 607–611 (2012).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
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Bakkers, E. P. A. M.

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
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Balet, L.

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

Baur, M.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Beetz, J.

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
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Beggs, D. M.

A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton, “Polarization engineering in photonic crystal waveguides for spin-photon entanglers,” Phys. Rev. Lett. 115, 1–5 (2015).

Bianchetti, R.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Binder, R.

Z. Yang, N. Kwong, R. Binder, and A. Smirl, “Stopping, storing and releasing light in quantum well Bragg structures,” J. Opt. B 22, 2144–2156 (2005).
[Crossref]

Blais, A.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Blaize, S.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
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Bose, R.

R. Bose, D. Sridharan, H. Kim, G. S. Solomon, and E. Waks, “Low-photon-number optical switching with a single quantum dot coupled to a photonic crystal cavity,” Phys. Rev. Lett. 108, 227402 (2012).
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Botten, L. C.

Boulanger, J.

N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
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Busch, K.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, 1999).

Chang, D. E.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

Chauvin, N.

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

Chelnokov, A.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Chia, A.

N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
[Crossref]

Christ, A.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
[Crossref]

Cirlin, G. E.

V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
[Crossref]

Citrin, D. S.

Cohen, J. D.

S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

Cole, J. H.

A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006).
[Crossref]

Combrie, S.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Combrié, S.

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

Dagens, B.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Davies, D. G.

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

De Rossi, A.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

de Sterke, C. M.

Delacour, C.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Dhindsa, N.

N. Dhindsa, A. Chia, J. Boulanger, I. Khodadad, R. LaPierre, and S. S. Saini, “Highly ordered vertical GaAs nanowire arrays with dry etching and their optical properties,” Nanotechnology 25, 305303 (2014).
[Crossref]

Diedenhofen, S. L.

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
[Crossref]

Dignam, M. M.

P. Yao, C. P. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009).
[Crossref]

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007).
[Crossref]

Dossou, K. B.

Douglas, J. S.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

Dubrovskii, V. G.

V. G. Dubrovskii, G. E. Cirlin, and V. M. Ustinov, “Semiconductor nanowhiskers: synthesis, properties, and applications,” Semiconductors 43, 1539–1584 (2009).
[Crossref]

El-Ella, H.

I. Söllner, S. Mahmoodian, S. L. Hansen, L. Midolo, A. Javadi, G. Kiršanskė, T. Pregnolato, H. El-Ella, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Deterministic photon-emitter coupling in chiral photonic circuits,” Nat. Nanotechnol. 10, 775–778 (2015).
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Evers, J.

G. X. Li, J. Evers, and C. H. Keitel, “Spontaneous emission interference in negative-refractive-index waveguides,” Phys. Rev. B 80, 1–7 (2009).

Fält, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Fan, S.

Février, M.

M. Février, P. Gogol, A. Aassime, R. Mégy, C. Delacour, A. Chelnokov, A. Apuzzo, S. Blaize, J. M. Lourtioz, and B. Dagens, “Giant coupling effect between metal nanoparticle chain and optical waveguide,” Nano Lett. 12, 1032–1037 (2012).
[Crossref]

Fink, J. M.

J. M. Fink, M. Göppl, M. Baur, R. Bianchetti, P. J. Leek, A. Blais, and A. Wallraff, “Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system,” Nature 454, 315–318 (2008).
[Crossref]

Fiore, A.

T. Ba Hoang, J. Beetz, L. Midolo, M. Skacel, M. Lermer, M. Kamp, S. Höfling, L. Balet, N. Chauvin, and A. Fiore, “Enhanced spontaneous emission from quantum dots in short photonic crystal waveguides,” Appl. Phys. Lett. 100, 061122 (2012).
[Crossref]

Foster, A. P.

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

Fry, P. W.

M. N. Makhonin, A. P. Foster, A. B. Krysa, P. W. Fry, D. G. Davies, T. Grange, T. Walther, M. S. Skolnick, and L. R. Wilson, “Homogeneous array of nanowire-embedded quantum light emitters,” Nano Lett. 13, 861–865 (2013).
[Crossref]

Fujita, T.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, “Polariton effect in distributed feedback microcavities,” J. Phys. Soc. Jpn. 70, 1137–1144 (2001).
[Crossref]

Fussell, D. P.

D. P. Fussell, S. Hughes, and M. M. Dignam, “Influence of fabrication disorder on the optical properties of coupled-cavity photonic crystal waveguides,” Phys. Rev. B 78, 144201 (2008).
[Crossref]

D. P. Fussell and M. M. Dignam, “Quantum-dot photon dynamics in a coupled-cavity waveguide: observing band-edge quantum optics,” Phys. Rev. A 76, 053801 (2007).
[Crossref]

Gabet, R.

M. Patterson, S. Hughes, S. Combrié, N.-V.-Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouën, “Disorder-induced coherent scattering in slow-light photonic crystal waveguides,” Phys. Rev. Lett. 102, 253903 (2009).
[Crossref]

Gao, J.

J. Gao, S. Combrie, B. Liang, P. Schmitteckert, G. Lehoucq, S. Xavier, X. Xu, K. Busch, D. L. Huffaker, A. De Rossi, and C. W. Wong, “Strongly coupled slow-light polaritons in one-dimensional disordered localized states,” Sci. Rep. 3, 1994 (2013).

Gao, Q.

H. J. Joyce, Q. Gao, H. Hoe Tan, C. Jagadish, Y. Kim, J. Zou, L. M. Smith, H. E. Jackson, J. M. Yarrison-Rice, P. Parkinson, and M. B. Johnston, “III-V semiconductor nanowires for optoelectronic device applications,” Prog. Quant. Electron. 35, 23–75 (2011).
[Crossref]

García-Meca, C.

F. J. Rodríguez-Fortuño, B. Tomás-Navarro, C. García-Meca, R. Ortuño, J. Martí, and A. Martínez, “Zero-bandwidth mode in a split-ring-resonator-loaded one-dimensional photonic crystal,” Phys. Rev. B 81, 2–5 (2010).

Gerace, D.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Giessen, H.

A. Christ, S. G. Tikhodeev, N. A. Gippius, J. Kuhl, and H. Giessen, “Waveguide-plasmon polaritons: strong coupling of photonic and electronic resonances in a metallic photonic crystal slab,” Phys. Rev. Lett. 91, 183901 (2003).
[Crossref]

Gippius, N. A.

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A. Goban, C. L. Hung, J. D. Hood, S. P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” Phys. Rev. Lett. 115, 1–5 (2015).
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ACS Nano (1)

S. L. Diedenhofen, O. T. A. Janssen, M. Hocevar, A. Pierret, E. P. A. M. Bakkers, H. P. Urbach, and J. Gómez Rivas, “Controlling the directional emission of light by periodic arrays of heterostructured semiconductor nanowires,” ACS Nano 5, 5830–5837 (2011).
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Appl. Phys. Lett. (3)

M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-mm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab,” Appl. Phys. Lett. 84, 4298–4300 (2004).
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[Crossref]

S. P. Yu, J. D. Hood, J. A. Muniz, M. J. Martin, R. Norte, C. L. Hung, S. M. Meenehan, J. D. Cohen, O. Painter, and H. J. Kimble, “Nanowire photonic crystal waveguides for single-atom trapping and strong light-matter interactions,” Appl. Phys. Lett. 104, 2012–2017 (2014).

Europhys. Lett. (1)

L. G. Suttorp and A. J. V. Wonderen, “Fano diagonalization of a polariton model for an inhomogeneous absorptive dielectric,” Europhys. Lett. 67, 766–772 (2004).
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J. Opt. B (1)

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J. Phys. Soc. Jpn. (1)

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Laser Photon. Rev. (1)

P. Yao, V. S. C. Manga Rao, and S. Hughes, “On-chip single photon sources using planar photonic crystals and single quantum dots,” Laser Photon. Rev. 4, 499–516 (2010).
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Figures (6)

Fig. 1.
Fig. 1. (a) Proposed polariton waveguide, with yellow QDs embedded inside the PC waveguide channel, and a red external QD coupling to the structure. Red arrows denote the waveguide direction ( e x ) and NWs are cut out so embedded QDs can be seen. (b) Slow-light region of the band structure on top, and Im { G ( r n , r n ) } in units of ρ h (see text) on bottom; the infinite polariton waveguide in solid blue is compared with the original PC waveguide in dashed orange.
Fig. 2.
Fig. 2. (a) Imaginary and real components of G ( N ) ( r t , r t ; ω ) on right and left, respectively. G ( 101 ) ( r 51 , r 51 ; ω ) in solid blue is compared with G ( 51 ) ( r 26 , r 26 ; ω ) in solid orange, G ( 21 ) ( r 11 , r 11 ; ω ) in dashed red, and N = 0 results in dash-dotted gray. (b)  Im { G ( 101 ) ( r n , r n ; ω ) } near ω 0 , plotted on a logarithmic scale due to the narrowness of the primary resonance. Δ x = n 51 and the gray dashed lines denote the terminus of the QD array at Δ x = ± 50 . All values are in units of ρ h ( ω ) .
Fig. 3.
Fig. 3. | G ( 101 ) ( r n , r 51 ; ω ) | in units of ρ h ( ω ) for the N = 101 polariton waveguide, where the termination of the QD array at Δ x = ± 50 is again denoted with dashed gray lines.
Fig. 4.
Fig. 4. Im { G ( 101 ) ( r n , r n ; ω ) } in units of ρ h ( ω ) under various instances of disorder compared with the disorder-free result in dashed–dotted gray. On left, sample results for coupling-strength reductions σ g = 5 % and 10% are plotted in dashed red and solid blue. On right, the influence of fluctuations in ω 0 is seen, with a σ ω 0 = 5    μeV result in solid orange compared with a σ ω 0 = 10    μeV result in solid blue. An instance of the same set of σ ω 0 = 10    μeV frequencies and random σ g = 10 % reductions is plotted in dashed red.
Fig. 5.
Fig. 5. S 0 ( ω ) in dashed blue and S ( r D , ω ) in solid orange for ω t = ω 1 , with d t = 30    D in (a) and ω t = ω FP with d t = 60    D in (b). Im { G ( r t , r t ; ω ) } in arbitrary units is shown in dash-dotted gray.
Fig. 6.
Fig. 6. Peaks in S 0 ( ω ) in blue as a function of ω t and d t . Results for d t = 10 , 30, and 60 D are presented from bottom to top, with ω t near ω 1 and in the FP region on the left and right, respectively. Crosses and circles denote peaks of Figs. 5(a) and 5(b), ω t is in dashed red, and LDOS peaks are in dashed gray.

Equations (7)

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[ × × ω 2 c 2 ε ( r ) ] G ( r , r ; ω ) = ω 2 c 2 1 δ ( r r ) ,
ω k , ± = ω 0 + ω k 2 ± 1 2 ( ω 0 ω k ) 2 + 4 g k 2 ,
Γ k , ± = Γ 0 ω k , ± 2 ω k 2 ( ω k , ± 2 ω k 2 ) + ( ω k , ± 2 ω 0 2 ) .
G P ( r , r ; ω ) = i a ω 2 v ˜ g [ Θ ( x x ) u k ω ( r ) u k ω * ( r ) e ( i k ω κ ω ) ( x x ) + Θ ( x x ) u k ω * ( r ) u k ω ( r ) e ( i k ω κ ω ) ( x x ) ] ,
G w ( r , r ; ω ) = i a ω 2 v g [ Θ ( x x ) u k ω ( r ) u k ω * ( r ) e i k ω ( x x ) + Θ ( x x ) u k ω * ( r ) u k ω ( r ) e i k ω ( x x ) ] .
H = ω t σ ^ + σ ^ + d r 0 d ω l ω l b ^ ( r ; ω l ) · b ^ ( r ; ω l ) ( σ ^ + + σ ^ ) ( d t · E ^ ( r t ) + H.c. ) ,
S ( r D , ω ) = | ( ω t + ω ) G ( r D , r t ; ω ) · d / ε 0 ω t 2 ω 2 ω Σ ( ω ) i ω Γ t | 2 .

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