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
The design of nanoscale solid-state devices to control light–matter interactions is highly desirable, both for applications in quantum information science  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 [4–7]. However, these PC slab structures have been partly hindered by fabrication issues including surface roughness [8–10] and limited control of the QD properties . Arrays of nanowires (NWs) grown through a molecular beam epitaxy (MBE) technique [12–14] 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 [17–19]. 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 ; 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  and metal nanoparticle chains coupled to traditional waveguides  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 ; 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. PROPOSED STRUCTURE
Our proposed structure exploits the elevated NW PC waveguide design of Ref. , where a PC is formed from a square lattice array of GaAs NWs extended from an AlO substrate . The waveguide channel is introduced by reducing the radius of a row of NWs from to , which we define to be along . 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 , contains a single vertically polarized () 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 ), while the AlO layer separates them from the substrate and an array width of 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 . 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 , where is an integer. We take the embedded QDs to have a Lorentzian polarizability , with a realistic dipole moment (0.626e-nm) and polarization decay rate (including both nonradiative decay and coupling into nonwaveguide modes); these are similar to experimental parameters for InAs QDs at 4 K . The QD exciton line is taken to be , which is in the moderately slow-light regime of the waveguide band corresponding to a group velocity and (relative LDOS enhancement, ). 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 , which describes the system electric-field response at to a point source at (fully including all light-scattering events):1,16]. These GFs are projected along the relevant component for vertically polarized QDs, , and given in units of the imaginary part of free-space GF  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 , where 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 , which shifts the waveguide resonance for a given to . We obtain from perturbation theory [37–39], where is the unit-cell volume and is the waveguide unit-cell function. The unit-cell function is normalized through , and corresponds to waveguide mode , where 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 at each , with1(b), where the strong mode splitting () flattens the dispersion as and losses remain low: . 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 where perturbation theory naturally breaks down we find that only the waveguide Bloch-mode contribution is significant and . 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, . Converting this sum to an integral in the complex plane , we arrive at an analytic expression similar in form to that for a regular PC waveguide [cf. Eq. (5)]: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 , the original PC waveguide GF is quickly recovered. At the above minimum detuning, these polariton waveguides yield —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 is that of the underlying PC waveguide, given by [1,40]41] . 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, .
Figure 2 shows the for increasing , highlighting the 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) ( for the 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 at ; however, with increasing the buildup of off-resonant enhancement from QD scattering leads to the formation of a strong resonance in , which exceeds for . This resonance is red-shifted from , with a peak and FWHM at for the 101 QD case. As more QDs are added, this polariton peak grows, narrows, and blue-shifts toward while additional weaker red-shifted resonances begin to appear. Above , resonances also form, which grow and become more numerous with increasing , indicating that they arise from QD chain Fabry–Perot (FP) modes. The higher two FP modes for the structure are at and . Away from , 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 that remains substantial over a broad frequency region, particularly for larger . These findings were verified to hold for a range of QD decay rates , with the dip extending deeper and resonances initially appearing at lower and closer to as is reduced. Figure 2(b) further highlights the profound influence of the QD array on the PC LDOS: the strong resonances and depletion around 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 3 shows the propagator , 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, and are found for . One can produce any arbitrary combination of and through careful choice of separation and frequency, as acquires a phase shift of as and are varied. The interaction between a pair of dipole emitters in a photonic medium at and is governed by both the real and imaginary portions of ; 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 has a peak of at (), more than double the bare PC waveguide result: . 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.
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 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 . This field amplitude only appears in in the form , 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 for each QD with a normal distribution and standard deviation of up to . This corresponds to a shift in position of up to , yielding a 3 D reduction in , 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 ) and the shift in this peak , we find that yields a mean and small fluctuations in peak height from -20 to 5, and increasing to 10% has little influence on but increases to . 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.
We then considered fluctuations in QD resonance about with up to 10 μeV, both with and without disorder in . 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 , and substantial increases or reductions in the peak . Both these effects grow with increasing , with 10 μeV resulting in and a wide range of polariton peak 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 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 , remains capable of achieving strong coupling with an external emitter, producing a (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 of the polariton waveguide. The system LDOS enhancements, specifically the polariton peak at , are possibly sufficient to strongly couple the waveguide to a realistic QD. We adopt a rigorous quantization procedure appropriate for lossy inhomogeneous systems , 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 by42]. We Laplace transform the resulting operator Heisenberg equations of motion to construct the spontaneous emission spectrum at detector position : . Using the initial condition of a vacuum field and excited QD, we derive the exact detected spontaneous emission spectrum, 7) by replacing the propagator in the numerator with . This is done because is often distorted by the rich spectral features of the GFs of these waveguide structures (see Fig. 3). The bare spectrum more cleanly shows the system energy levels and dynamics of the target QD and is typically the only one calculated .
We assume a -aligned target QD with a polarization decay rate 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 on resonance); near we define an effective coupling constant . The three target dipole moments produce of 1.24, 3.72, and 7.43 μeV, respectively, which meet the criteria for strongly coupling with the resonance: . We assign a detector position , 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 and at for the 30 D QD are shown in Fig. 5(a). Remarkably, substantial splitting is seen in both and despite the modest choice of . Furthermore, peak locations of and 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 peak is a result of the unique shape of near producing a large positive Lamb shift. The importance of propagation effects can be clearly seen by comparing and , where the relative LDOS depletion above reduces the height of the peak. In Fig. 5(b) we show the spontaneous emission spectra at for , where the QD–polariton-waveguide coupling is sufficiently strong that significant exchange occurs with both the and modes for . This results in a triplet forming in , with peaks at , , and ; the splitting is substantially stronger than , so these peaks are seen at the detector position as well.
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 as is brought to resonance with are shown in Fig. 6 for a target QD with and 60 D. The left plots depict anti-crossing with the primary resonance at , while the peaks as 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 , 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 , a later study showed that this was inevitably spoiled by disorder . Strong coupling has recently been observed with Anderson-localized cavities in disordered PC waveguides , 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 , and the multimode nature of the QD-polariton waveguide has flattened these anti-crossing lines. Indeed, the effective coupling constant 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 .
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 , 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].
Natural Sciences and Engineering Research Council of Canada (NSERC); Queen’s University, Canada.
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