We present and apply a medium-dependent quantum optics formalism for describing the exciton dynamics of two spatially-separated quantum dots on-chip, in the regime of coupled-cavity quantum electrodynamics. With each dot placed in a spatially-separated cavity and coupled through a periodic waveguide channel, the quantum dot excitons behave as a composite entangled pair, exhibiting pronounced entanglement over distances of 300µm and more. The computed light spectra above the two cavities show clear signatures of pronounced photon coupling including increased vacuum Rabi splitting and cavity-induced transmission and absorption. The macroscopic entanglement is confirmed by investigating the Bell inequality, which is shown to be violated for hundreds of picoseconds.
© 2009 Optical Society of America
The formation of entanglement between quantum bits (qubits) can produce entangled states, namely non-factorizable composite wave-functions whose separated parts may be correlated over large distances, resulting in what Einstein termed: “spooky action at a distance.” Besides fundamental interest, entangled states may be used to realize quantum information protocols such as the transmission of secret messages via quantum key distribution , teleportation of quantum information , and dense coding . With regard to solid-state qubits, individual semiconductor quantum dots (QDs) have generated much interest as they behave like “artificial atoms” due to their characteristic discrete energy spectra and long excitonic lifetimes. Quantum dots also have a number of other attractive features for generating entangled states: i) they can be fixed in position (unlike atoms), ii) discrete excitons have large optical dipole moments, iii) exciton frequencies are compatible with telecom components, and iv) QDs can be integrated with on-chip cavity structures such as those facilitated by planar photonic crystals (PCs) [4–6] [see Fig. (1)].
Significant progress has been made on nanophotonic systems based on single QD-cavity systems, such as the enhancement and suppression of single photon spontaneous emission , strong coupling cavity-QED [6, 8–10], and single exciton control of nanocavity reflectivity [11, 12]. Cavity-QED coupling between two QDs in a single PC cavity  has also been shown to lead to sizable entanglements for QDs that are coupled nearby to the same cavity mode; while an interesting scheme, the experimental verification of measuring and using parts of the same composite pair for coupled QDs and proving non-local behavior such as the violation of Bell’s inequalities is very difficult. Nevertheless, the proof-of-principle entanglement between spatially separated quantum states in a pair of vertically aligned, self-assembled QDs has been indirectly demonstrated by Bayer et al. through investigating the QD emission spectra ; the maximum entanglement of the coupled excitons was found to be around 0.8 at an inter-dot separation of only 8.5 nm, which quickly diminishes for larger and smaller inter-dot distances . A major disadvantage of nearby QDs is that it is not possible to individually tune/excite the QD, nor probe the quasi-degenerate excitons separately.
In this work, we propose, and develop the theory for, an integrated solid-state device that enables pronounced entanglement between macroscopically separated QDs on-chip. We demonstrate that, with suitably excited QDs in a planar PC medium, long lived entangled states can be maintained, showing the violation of the Bell inequality over hundreds of picoseconds. More fundamentally, this coupled QD system gives rise to rich coupling effects including modified strong coupling regimes of cavity-QED. The structure of interest is depicted above in Fig. 1, which includes two single-mode nanocavities and one coupling waveguide. The nanocavities can be formed by localized defects created within the PC, for example by removing and rearranging two of the air holes . The distance between the two nanocavities is L and no direct coupling is assumed. The PC waveguide and cavity resonances are considered to be deep inside the PC photonic bandgap and the propagating waveguide mode is below the light line (lossless).
To establish a quantitative theory, we closely follow the approach introduced by Wubs et al. . Specifically, we adopt a canonical Hamiltonian that quantizes the macroscopic electromagnetic fields, and use the dipole approximation for the QD-medium coupling. We label the two QDs a and b, which are embedded in cavities 1 and 2, respectively. Subsequently, we obtain the multipolar-coupling Hamiltonian:
where âλ represents the field mode operators and m +/- are the Pauli operators of the QD excitons; we assume one exciton in the spectral region of interest per QD. In addition, Ωm is the resonant frequency of each QD, and ωλ is the eigenfrequency corresponding to the transverse modes of the system (f λ (r)), excluding the dots; g mλ is the field-dot coupling coefficient, defined through , where µm=n mµm is the optical dipole moment of the QD electron-hole pair.
The Heisenberg equations of motion for the operators can be derived from i=-ih̄ -1[Ôi,Ĥ], yielding dynamical equations for âλ, â † λ, m +/-, and mz= + m - m--m - m +. In an electron picture, +=cĉ † e ĉg creates an electron in the excited state (conduction band) and -=ĉe ĉ † g destroys an electron in the excited state [or creates an electron in the ground state (valence band)]; similarly, + -=ĉ † e ĉe, where ρ aa/bb=〈 + aa/bb - aa/bb〉 is the excited state population in each QD. After carrying out the Laplace transform of operators , where Ôi(ω)=∫∞ 0 Ô(t)eiωtdt, then
Here, Êµ (r m,ω)=Ê(r m,ω) · n m, and the symbol ‘∗’ represents a convolution operator. Using the formal definition of electric-field operator, , we obtain the “exact” electric-field operator in frequency space:
where αm(ω)=2Ωm µ 2 m/[h̄ε 0(Ω2 m-ω 2)] is a bare polarizability term, and K(r,r′;ω)=∑λ ω 2 λ/(ω 2 λ-ω 2)f λ (r)[f λ (r′)]* is a medium Green function [17, 18] in the absence of QDs. To derive the above electric-field operator, we have assumed that the dipole moments are real, and have exploited the fact that, for real ε media, both f λ and f*λ are used per λ in the mode summation. The K Green function is then obtained only through the transverse modes of the system and is related to the more familiar G=G T +G L (including both transverse and longitudinal modes), through  K=G+1 δ (r-r′)/ε (r), with 1 the unit tensor, and ε (r) is the spatially-dependent dielectric constant. These Green functions are classical and can be defined in the usual way from Maxwell’s equations, solved with a polarization dipole. Under the weak excitation approximation, where we consider at most one excitation in the system, and assuming that the ensuing dynamics will be driven by the initially excited QD(s), then the electric-field operator can be expressed explicitly as
where QD a/b is located at r a/b, and we have introduced the quantum dipole source , and a QD-modified Green function for the medium:
where K̃(r,r m;ω)=K(r,r m;ω)/[1-n m ·K(r m,r m;ω) ·n m αm(ω)].We stress that K (2) exactly includes the full dynamical coupling effects between the two QDs, and fully covers both weak and strong coupling regimes in a self-consistent way. Similar expressions have been derived classically for the medium-dependent QD polarizabilities , and quantum mechanically for coupled atoms (using a harmonic oscillator model) .
Next, we require the classical Green functions for the coupled-cavity planar PC medium. These have been derived before , so here we just introduce the main points, and extend the theory for the problem at hand. The analytic Green function for the waveguide-plus-two-cavity medium (K wcc) can be derived by assuming weakly coupled cavity-to-waveguide channels and employing a matrix inversion technique [11, 18, 20]. We can conveniently choose different modal components through K(r,r′;ω)=∑αβ 〈r|K αβ|r′〉, where K αβ is the projection of K on |E α〉 from the left and 〈E β| from the right, and we use the notation f α (r)=〈r|E α〉 and f*α(r)=〈E α|r〉. Assuming a positive group velocity for the waveguide mode (vg>0), example projections include 
where r 1(ω)=iωΓw 1/[ω 2 1-ω 2-iω(Γ0 1+Γw 1)] is the waveguide mode reflection coefficient from cavity 1, Γwc is the cavity-to-waveguide coupling rate whose explicit form depends on the overlap between the cavity mode and the waveguide mode , and Γ0≪Γwc is the bare cavity coupling rate caused by coupling to radiation modes above the light line. We note that these functions are not simple Lorenzian lineshapes, and the cavity decay process is a complicated function of frequency and waveguide length L.
The quantum dynamics of the spatially separated qubits can be obtained from the inverse Laplace transform of the QD operators, which in the weak excitation regime has the form
Exploiting the definition of the time-dependent wave function, |ψ(t)〉=∑m=a,b Cum (t)|aem,0〉+∑m=a,b C g m,λ (t) |ag m,λ〉, we define the amplitude coefficient of the QD upper excited state as Cu a/b(t)=∫∞ -∞〈0| - a/b(ω)|ψ(t=0)〉e -iωt dω, with |0〉 the lower level eigenstate of the QDs. The upper state population of each QD is then obtained from ρmm=|Cum|2.
Finally, we also derive exact analytical expressions for the light spectrum and the entanglement of formation . From the first-order quantum correlation function, the spectrum is defined as S(r,ω)=〈(Ê(ω))† Ê(ω)〉, yielding
which can be spatially projected to separately obtain the emitted spectrum detected above cavity 1 and (or) cavity 2. To compute the quantum correlations between the coupled QD qubits, the entanglement of formation is obtained from the concurrence , Con(t), through , where Con=|〈ψ|| (tilde represents a spin flip) . For this we need the reduced density matrix of our two-qubit system ρ, which is obtained by tracing out the electric field modes, where the elements of the wave function |ψ〉 are obtained from the QD operators, i.e., Eqs. (12) and (8). For our system of interest, where the initial excitation resides in the QDs, the concurrence becomes simply Con=2|Cua‖Cub|.
3. Quantum dynamics, entanglement of formation, and cavity-emitted spectra
For calculations, we use bare quality factors of cavity Q 0 1/2(=ω 1/2/Γ0 1/2)=20000, and a cavity-waveguide quality factor of Qw 1/2(=ω 1/2/Γw 1/2)=500 for both cavities. Increasing Q 0 1/2 actually has little influence on our results because the decay of PC system is dominated by the coupling between the cavities and the waveguide; and for the regime of coupled-cavity QED , the most important parameter is the ratio Γw/Γ0 or Q 0/Qw. In addition, we choose ω 1/2=Ωa/b, h̄ω 1=h̄Ωa=0.95eV, effective cavity mode volume V c1/c2=0.07µm3, and a background dielectric constant of the PC slab, εb=12; for the QD dipole moments we use parameters similar to those in experiments [22, 23], and first use an effective dipole moment µa=30D(Debye) and µb=60D (this asymmetry helps to ensure maximum entanglement for QD a initially excited). In practise, the effective dipole moments can be achieved from different orientations, or, equivalently, by different spatial positioning with respect to the cavity field antinode positions; though we could also easily change other parameters within the formalism. We note that, according to the recent fabrication improvements , the error of precise spatial positioning of the QD to a cavity field antinode can be less than 10% (so g exp c>0.9g max c), and to within this accuracy, we have tested that our proposed scheme is robust. Necessarily taking causality into consideration, the chromatic dispersion of the waveguide is introduced linearly and the group velocity vg=dω/dk=c/10; also, unless stated otherwise, the cavity-cavity separation L=3-300µm, which yield waveguide retardation times of 0.1 ps to 10 ps.
Experimentally, the QDs are usually optically excited far off resonance, and the initial excitons are created incoherently. Thus, a suitable initial condition for the two QD system is |ψ(t=0)〉QD=cos(θ)|10〉+eiϕ sin(θ)|01〉, corresponding to an exciton shared between the two dots with the field in vacuum. With only one QD in the system, we assume |ψ(t=0)〉QD=|1〉.
For a reference calculation, we first perform simulations with only 1 QD in the system, with QD b removed, and set ω 1=ω 2=Ωa. In Figs. 2(a,c,e), we show the excited QD dynamics for various cavity separation distances of L=3,30, and 300µm, respectively. In Fig. 2(a), we clearly see strong coupling phenomena and vacuum Rabi oscillations, which is enhanced over the one cavity result because cavity 2 acts to increase the local density of photon states at the position of QD a (in cavity 1) . Interestingly, for increasing L, the Rabi oscillation period increases (smaller effective cavity coupling), and we eventually recognize weak retardation oscillations in the excited state population dynamics; these oscillations are caused by the single photon being recycled between the two cavities with a period that corresponds to the round trip time of the coupled cavity system. The corresponding emitted spectra from cavity 1 and cavity 2 are shown in Figs. 2(b,d,f), which show that the vacuum Rabi doublet decreases in width as a function of waveguide length L. In this latter case, we highlight that the length L=300µm would correspond to around 700–800 PC unit cells. A clear advantage of our formalism is the ease with which we can study various structural parameters such as waveguide length, where direct numerical approaches would be impractical and cumbersome.
Next, we explore the two QD dynamics by adding in the second dot b. We study the resonant condition of ω1(=Ωa)=ω 2(=Ωb), and also the case with detuning, ω 1(=Ω1)≠ω 2(=Ωb). For both cases, only QD a is initially excited (Cua(0)=1, Cub(0)=0). The resonant case is shown in Fig. 3, which demonstrates that pronounced entanglement values can be realized (see Fig. 3(a), green dashed curve), reaching a maximum value of around 0.67 ebits for L=3µm. The oscillations in the QD dynamics are caused by both the dot-dot photon exchange interactions and the cavity-dot interactions, and we find that the vacuum Rabi oscillations are much faster than the one QD scenario (cf. red curves in 2(a,c) and 3(a,c)). For the emitted spectra, in addition to the expected vacuum Rabi doublet (now with a significantly larger splitting than before), we also obtain a pronounced constructive or destructive center peak that originates from the photon emission from the excited dot in the opposite cavity propagating back and resonantly scattering from the cavity (e.g., the dot in cavity 2 can act as a source for pumping the dot in cavity 1); this is analogous to dipole-induced transparency or dipole-induced absorption [11, 24], where one obtains interference between the excitation field and the field emitted from the excited QD. The lineshape from cavity 1 is also similar to the well-known Mollow triplet . More interestingly, in the context of coupled QDs, we also see an additional doublet (larger splitting of around 0.4 meV for L=300µm) which is caused by the dynamic photon coupling that takes place from cavity to cavity, as discussed above. The latter spectral doublet give clear evidence to the formation of a macroscopic molecule. Again for the longer cavity separations, the vacuum Rabi oscillations (coherent oscillations) are suppressed, and for L=300µm, a clear retardation feature of around 10 ps is observed. When L is increased to a larger value of 300µm, the entanglement peak is still reasonable (~0.3). While the usual coupled-QD entanglement distance between excitons is only about 10 nm, here we obtain a macroscopic entanglement by indirect cavity-cavity coupling, via the integrated waveguide. Furthermore, this macroscopic entanglement has the added advantage that the coupling phenomena can be probed by measuring the individual cavity-emitted light spectra.
These coherent coupling phenomena naturally become less pronounced in the presence of non-radiative exciton decay and spectral detuning between the cavities and QDs, though significant qubit coupling is still be apparent for a wide range of parameters. For single QDs, at low temperatures of T=4K, even for regular (non-cavity) structures, the exciton relaxation is typically dominated by radiative decay, and the non-radiative decay times (rates) are around 2 ns (~2µeV) ; thus, for studying the early time dynamics of the first 500 ps or so, their influence can be approximately neglected. A more significant effect on the QD photon transfer mechanisms would result from detuning between the two QD-cavity sections, though these could be individually tuned to ensure resonance with each QD-cavity pair. As an example, in Fig. 4 we add a detuning of Ωa-Ωb=25µeV and repeat the same investigation as shown in Fig. 3. While the entanglement and photon coupling effects are reduced, we still obtain significant coupling between the QDs via the coupling waveguide.
4. Bell inequalities
Finally, we address the non-local coupling of our proposed PC system by analyzing the Bell inequality. The Bell inequality , if violated, is a proven way to test the EPR paradox  and the non-local realism of a coupled quantum state. For quantum spin systems, which mathematically map on to our two-coupled QD excitons, the Bell inequality of interest : Bs=|E(θa,θb)-E(θa,θ′b)+E(θ′a,θb)-E(θ′a,θ′b)|≤2, where the correlation function , , mx= + m+ m -, my=-i + m +im - and θm(θ′m) is a real-angle parameter. Thus, if one chooses θ=θa -θb=θb-θ′a=θ′a-θ′b, then above inequality will be simplified to Bs=|3E(θ,0)-E(3θ,0)|≤2, where . The Bell inequality is therefore violated if Bs>2.
One can consider the evolution of an initial maximally-entangled two-qubit state . The Bell states can in principle be generated by a single photon source, where the single photons are first split by a 50%-50% beam splitter, and then undergo a different phase delay (0 or π), which then excite the pair of QDs. More elaborate schemes could, in principle, send in an incident field through a leaky waveguide mode, that could prepare the QDs pair in the correct superposition state. Inspecting the symmetry properties of Eq. (12), is can be recognized that Cua(t)=±Cub(t) if cavity 1 is symmetric with cavity 2 and QD a is resonant with QD b, and their evolution will be governed by the term K(r a/b,r a/b;ω)±K(r b/a,r a/b;ω). The decay of the state will be faster than if K(r a,r a;ω) is in phase (interferences constructively) with K(r a,r b;ω). On the contrary, if K(r a,r a;ω) is out of phase (interferences destructively) with K(r a,r b;ω), then the decay of the state will be faster than . For our waveguide example, the violation time of the Bell inequality for the initial condition |ψ〉- QD is shorter than that of the initial condition |ψ〉+ QD, which is caused by the destructive interference between K(r a,r a;ω) and K(r a,r b;ω). Thus, we will only show the initial condition of |ψ〉+ QD.
The Bell inequality simulations are summarized in Fig. 5, which display a selection of population and Bell parameter dynamics, for various L and QD dipole moments. To better understand these results, we note that the violation of the Bell inequality for the initial condition |ψ〉+ QD is largely insensitive to the distance L, but more sensitive to the dipole moments (or effective dipole moments, as ultimately the coupling is controlled through gλ). In Fig 5(a–b), we show the QD population dynamics and Bs for a QD dipole of d a/b=30D, with L=3µm (red curve) and L=300µm (blue curve); the Bell inequality is found to be violated for both waveguide lengths for a timescale of up to 50 ps, with the longer system simply exhibiting more pronounced retardation dynamics (oscillations). When the effective dipole moment reduces from 30D to 8D, the violation time increases from around 50ps to more than 800ps (see Fig. 5(d)).
For a detuning of 25µeV, then this violation is reduced to around 200–300 ps (see Fig.5(f)), though can be increased by having a larger L. There is also an interesting entanglement revival for the longer waveguide, where the Bs increases about 400 ps. Note that for sufficiently large L, one must be careful of disorder-induced losses in the PC waveguide , though an increased entanglement duration can also be achieved by using smaller dipole moments or dots that are not maximally positioned on the cavity antinode, since this reduces the radiative decay; in this case, the ultimate limit will likely come from the non-radiative decay processes such as electron-phonon scattering, which we have neglected in this study. With the appropriate PC medium coupling, we expect that the entanglement dynamics will be robust for up to several hundred picoseconds using experimentally accessible parameters.
We have introduced a novel coupled-cavity QED scheme that can entangle two spatially separated QDs over macroscopic distances. By first developing an essentially exact photon Green-function technique, we have presented analytical formulas for the important system operators and experimental observables, allowing one to investigate a wide range of QD coupling scenarios in a remarkably simplistic and intuitive way – including the full non-Markovian dynamics. We find that effective “QD molecules” can be formed over several hundred microns and more, and that rich dot-cavity and dot-dot coupling dynamics occurs that can be observed indirectly by looking at the emitted light spectrum above each dot-cavity pair. The coupled-cavity QED technique also facilitates a way to test the violation of the Bell inequality using planar PC chips and two QDs that can be fixed in position, locally tuned, and separately probed experimentally.
This work was supported by the National Sciences and Engineering Research Council of Canada and the Canadian Foundation for Innovation. We thank Martijn Wubs and Jeff Young for useful discussions.
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