We theoretically investigate the interaction of a single quantum dipole with the modes of a fiber-coupled semiconductor waveguide. Through a combination of tight modal confinement and phase-matched evanescent coupling, we predict that of the dipole’s emission can be collected into a single mode optical fiber. We further show that the dipole strongly modifies resonant light transmission through the system, with over an order of magnitude change for an appropriate choice of fiber-waveguide coupler geometry.
Fiber-coupled semiconductor waveguides as an efficient optical interface to a single quantum dipole
Marcelo Davanço and Kartik Srinivasan
The interaction of a single quantum dipole with a strongly confined optical field is a central paradigm in quantum optics . The ability to collect a large fraction of the dipole’s emission or use it to modify an incident optical field lies behind a number of proposed applications in areas such as classical and quantum information processing [1, 2, 3, 4, 5] and single emitter spectroscopy . Such applications depend on the availability of efficient and accessible dipole excitation and emission channels. For instance, a single atom in free-space is exclusively excited by the dipole wave component of an illuminating field , and perfect reflection of an illuminating directional dipolar field is expected . Alternately, a single atom inside a Fabry-Perot cavity is strongly excited by, and radiates efficiently into, externally-accessible cavity modes and profoundly modifies the resonator transfer function . Here, we theoretically investigate a system in which a single emitter embedded in a fiber-coupled semiconductor channel waveguide is optically accessed with high efficiency, potentially yielding fluorescence collection into a single mode optical fiber. When resonantly interrogated, the dipole modifies the system’s transmission level by over an order of magnitude ( dB).
Our system (Fig. 1) is an emitter embedded in a suspended semiconductor channel waveguide (WG), evanescently coupled to an optical fiber taper WG. The taper is a single mode optical fiber whose diameter has been adiabatically and symmetrically reduced to a wavelength-scale minimum, resulting in a low-loss, double-ended device with standard fiber input and output. Fiber and channel WGs form a directional coupler (cross-section shown in Fig. 1(a)) of length , so power may be transferred between the two guides. This system serves as an efficient optical interface to a single dipole due to the availability of a small number of WG modes with highly effective coupling to the atomic transition (i.e., high -factors[8, 9]), and access to such modes via the fiber taper WG, which links on-chip nanophotonics and off-chip fiber optics. As depicted in Fig. 1(c), a signal launched into the fiber input, adiabatically reduced in size along the fiber taper, excites supermodes of the directional coupler. Guided supermodes illuminate the WG-embedded dipole at position along the coupler. Upon non-resonant excitation, the dipole emits coupler supermodes, at a red-shifted wavelength, in the directions. Emitted supermodes are converted into input and output fiber modes through the taper transition regions, after which emission is detected. The individual supermode contribution to the total photoluminescence (PL) collection efficiency, , is , where is the supermode emission rate, and the total emission rate. The fraction is supermode ’s -factor. Since emission in both directions is equally likely, . The fiber mode fraction, , is an overlap integral between the fundamental fiber mode and supermode [11, 12]. Its quantum mechanical operator analog is given here as Eq. (2).
We study a geometry modeling a suspended GaAs channel with an embedded self-assembled InAs quantum dot (modeled as a two-level atom with electric dipole moment on the plane) produced from the material used in . The channel WG, surrounded by air, has thickness , width , and refractive index =3.406 at a wavelength . The adjacent fiber has a radius and =1.45. For our parameter range, the directional coupler region supports a set of propagating supermodes named hE, hybrids of the E rectangular dielectric channel WG modes  and fundamental fiber mode. Supermodes hE and hE are excited by the - and -electric dipole moment components, respectively. Following , where fiber-based collection of emitters in membranes was studied, supermode field profiles (calculated with the finite element method) were used to find and , while the finite-difference time-domain (FDTD) method was used to calculate the total spontaneous emission rate . These quantities allowed us to determine the total fiber-collected PL efficiency () and individual supermode contributions . In addition, FDTD was used to obtain without use of supermodes.
Varying the channel width between and allows for significant modification of the supermode effective index . The real part of for the doublet available in this range, labeled I and II, is shown in Fig. 2(a). Both supermodes are guided, with . Field profiles for are shown in Fig. 1(b). Phase-matching between the fiber and E modes is apparent near , where in Fig. 2(b) are equal. As increases, supermode concentrates in the channel, resulting in reduced and increased (Fig. 2(c)). Note, for , approaches the upper limit of . For -oriented dipoles, two guided () supermodes are available, labeled I and III, with and plotted in Fig. 2(e)-(f). A third supermode, (leaky, ), , has the highest emission rate, though small . Since the -electric field component is dominant, for -dipoles (Fig. 2(e)) is small compared to the -dipole case. The highest contribution to is from the supermode, with .
Figure 3 shows total collection efficiency for - and -oriented dipoles (including emission in both direcions, which is experimentally realizable), obtained with FDTD and the supermode expansion method of . Since in each case multiple supermodes with differing propagation constants are excited, the collection efficiency oscillates along , evidence of the power exchange between channel WG and fiber. Collection maxima for are plotted for each . The collection efficiency for -dipoles is maximized, nearing , for . Near the optimal point, most () of the emitted power is coupled into propagating supermodes; fiber and slab are phase-matched, with equal fiber fractions of , so the collection contributions are maximized. For -dipoles, a more modest is achieved, due to lower and (Figs. 2(f) and (g)). For , reaches , however the total rate is only of that in the -dipole case.
We next show that the transmission through this directional coupler can be significantly affected by the presence of the embedded dipole. Dipole-mediated control of light transmission has been considered in studies involving free-space [2, 6, 7] and guided mode excitation [3, 4, 5], and also in cavity QED (e.g., [1, 13]). We start by describing spontaneous emission as an electric dipole-type interaction between the emitter and a vacuum field reservoir, given in terms of the coupler supermodes [11, 15, 2]. With the input-output formalism of , under the Markoff approximation, we obtain the following steady-state, positive-frequency, output multimode field operator for (i.e., past the dipole location):
Here, is the atomic lowering operator, is supermode ’s input field annihilation operator, is the electric field distribution, the propagation constant, the phase index, and , with the plane. The expression in brackets is a well-known result of the input-output formalism, with explicit input (or ”free”) field and radiated (”source”) field contributions . Next, we assume the percentages of incident fiber mode power transferred to coupler supermodes at are given by the fiber-mode fractions , and that the power coupled into the output fiber at is approximated by an overlap integral between the field at this position and the fiber mode (Eq. (2) in ). This expression is translated into the fiber power operator
where and are the fiber mode electric and magnetic field distributions, and . Photon flux and higher order correlation functions at the output fiber may be obtained with . Using Eq. (1) into Eq. (2) and assuming a coherent state illumination source, an expression for the output fiber photon flux expectation value is obtained in the low-excitation limit (far below saturation), and normalized to the input photon flux to produce the transmission level . The resulting expression consists of a sum of terms proportional to , and is used to calculate the transmission contrast through the fiber, defined as , where and are the transmission levels on and off resonance with an -polarized dipole. The transmission contrast is significant over a bandwidth of the order of the transition linewidth (the Purcell enhancement is small in these structures), which is much smaller than the coupler transmission bandwidth. In Fig. 4, we plot , , and for a coupler with (phase-matched channel and fiber WGs). As expected for a directional coupler, oscillates along between close to zero and close to unity, with beat length . The coupler 3 dB transmission bandwidth is for . For a dipole located at , can be significantly enhanced or suppressed relative to , depending on : at (), is nearly 30 times larger than ; at m, is 2.4 times smaller than . A judicious choice of coupler length thus produces structures in which a single dipole strongly affects transmission. This could enable, e.g., measurements of emitter spectral diffusion, or, with AC or DC Stark effect emitter frequency control, dipole-controlled light modulation. We note that phase matching is crucial in such devices, as is limited by incomplete power transfer in phase-mismatched fiber and channel WGs. Figure 4(b) shows more modest results for , due to phase mismatch ( may still be achieved).
If a single coupler supermode is accessed by the fiber, i.e., for all but one supermode, we find , which illustrates the essential role of in extinction measurements . Perfect extinction is predicted for , or exclusive -supermode emission. As emission is in both directions, this is equivalent to perfect reflection. In Fig. 2(c), it is apparent that extinction near may be achieved for , provided only supermode is accessible. This situation can be approximated with WG mode conversion structures (e.g., lateral or vertical tapers) that favor coupling between the fiber mode and specific coupler supermodes . For example, for , a modest 80:20 coupling ratio to the type I and II supermodes (i.e., , , ) would lead to extinction, independent of dipole position and coupler length.
In summary, we have investigated a hybrid waveguide structure in which strong dipole excitation is combined with efficient optical access through an evanescently-coupled optical fiber-based waveguide. These devices may have application in areas such as quantum information processing and single emitter spectroscopy.
This work was partly supported by the NIST-CNST/UMD-NanoCenter Cooperative Agreement.
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