Cavity Quantum Electrodynamics with Anderson-Localized Modes

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Science  12 Mar 2010:
Vol. 327, Issue 5971, pp. 1352-1355
DOI: 10.1126/science.1185080


A major challenge in quantum optics and quantum information technology is to enhance the interaction between single photons and single quantum emitters. This requires highly engineered optical cavities that are inherently sensitive to fabrication imperfections. We have demonstrated a fundamentally different approach in which disorder is used as a resource rather than a nuisance. We generated strongly confined Anderson-localized cavity modes by deliberately adding disorder to photonic crystal waveguides. The emission rate of a semiconductor quantum dot embedded in the waveguide was enhanced by a factor of 15 on resonance with the Anderson-localized mode, and 94% of the emitted single photons coupled to the mode. Disordered photonic media thus provide an efficient platform for quantum electrodynamics, offering an approach to inherently disorder-robust quantum information devices.

The interaction between a single photon and a single quantized emitter is the core of cavity quantum electrodynamics (QED) and constitutes a node in a quantum information network (1, 2). So far, cavity QED experiments have been realized with a wide range of two-level systems, including atoms (3), ions (4), Cooper-pair boxes (5), and semiconductor quantum dots (68) coupled to photons confined in a cavity. A common requirement for all these implementations is highly engineered cavities, in some cases requiring nanometer-scale accuracy (9). Surprisingly, multiple scattering of photons in disordered dielectric structures offers an alternative route to light confinement. If the scattering is very pronounced, Anderson-localized modes form spontaneously. Anderson localization (10) is a multiple-scattering wave phenomenon that has been observed for, e.g., light (11), acoustic waves (12), and atomic Bose-Einstein condensates (13). We have demonstrated cavity QED with Anderson-localized modes by efficiently coupling a single quantum dot (QD) to a disorder-induced cavity mode (14) in a photonic crystal waveguide.

Photonic crystals are composite nanostructures in which a periodic modulation of the refractive index forms a photonic band gap of frequencies where light propagation is fully suppressed. By deliberately introducing a missing row of holes in a two-dimensional photonic crystal membrane, the periodicity is broken locally and light is guided (Fig. 1A). Such photonic crystal waveguides are strongly dispersive, i.e., light propagation depends sensitively on the optical frequency and can be slowed down. Engineering the photonic crystal waveguide enables the enhancement of light-matter interaction, which is required for high-efficiency single-photon sources (15) for quantum information technology (1, 16). In the slow-light regime of photonic crystal waveguides, light propagation is very sensitive to unavoidable structural imperfections (17, 18) and multiple-scattering events randomize propagation (19). Although multiple scattering is commonly considered a nuisance for a device, leading to optical losses, here the influence of wave interference in multiple scattering stops light propagation and forms strongly confined Anderson-localized modes (10) (Fig. 1B). Anderson-localized modes in a photonic crystal waveguide appear as a result of the primarily one-dimensional nature of the propagation of light provided that the localization length is shorter than the length of the waveguide (20).

Fig. 1

Anderson localization in disordered photonic crystal waveguides. (A) Scanning electron micrograph of a photonic crystal membrane waveguide without engineered disorder, containing a single layer of QDs (yellow symbols, not to scale). (B) Photonic crystal waveguide with 6% engineered disorder. The red circles represent the hole positions in an ideal structure without disorder. The orange arrows depict the wavevectors of localized modes. (C) High-power photoluminescence spectra collected while the excitation and collection microscope objective was scanned along the waveguide at 10 K for a 3% disordered sample. (D) Measured probability distribution of the normalized photoluminescence intensity extracted from the data presented in (C). The black dashed line represents the Rayleigh distribution.

We deliberately created Anderson-localized modes by fabricating photonic crystal waveguides with a lithographically controlled amount of disorder (Fig. 1B). The hole positions in three rows above and below the waveguide were randomly perturbed with a standard deviation varying between 0 and 6% of the lattice parameter. We investigated the Anderson-localized modes by recording QD photoluminescence spectra under high-excitation power where the feeding from multiple QDs makes Anderson-localized modes appear as sharp spectral resonances (Fig. 1C). The observation of spectrally separated random resonances is a signature of Anderson localization of light (14), while the detailed statistics of the intensity fluctuations unambiguously verifies localization even in the presence of absorption (21). Figure 1D shows the intensity distribution from spectra recorded at different spatial and spectral positions, which allow us to average over different realizations of disorder. Clear deviations from the Rayleigh distribution predicted for nonlocalized waves are observed. Light is localized if the variance of the normalized intensity fluctuations exceeds the critical value of 7/3 (21), and we extract a variance of 5.3, which proves Anderson localization.

Examples of Anderson-localized modes are shown in Fig. 2 as peaks appearing at random spectral positions, although limited to the slow-light regime of the photonic crystal waveguide. The latter property is due to the strongly dispersive behavior of the localization length that is considerably shortened in the slow-light regime. We tuned the spectral range of Anderson-localized modes by controlling the amount of disorder. Even in samples without engineered disorder, intrinsic and thus unavoidable imperfections, such as surface roughness, are sufficient to localize light (22).

Fig. 2

Spectral signature of Anderson-localized modes. Photoluminescence spectra collected as in Fig. 1C, for various degrees of engineered disorder. Each spectrum was collected with the excitation and collection microscope objective at a fixed position on the waveguide and was vertically shifted for visual clarity. The gray area highlights the calculated waveguide (WG) cut-off region, assuming a refractive index of GaAs of 3.44. Embedded Imageis the calculated group velocity slow-down factor for an ideal structure without disorder, where c is the vacuum light speed, ω is the frequency, and k is the wave number.

The important cavity figures-of-merit are the mode volume V and the Q factor. Decreasing V leads to an enhancement of the electromagnetic field and thus improves light-matter coupling. The Q factor is proportional to the cavity storage time of a photon that needs to be increased for cavity QED applications. High Q factors ranging between 3000 and 10,000 are obtained for different degrees of disorder (Fig. 2) and are comparable to state-of-the-art values obtained for traditional photonic crystal nanocavities containing QDs (7). Anderson-localized cavities thus offer a fundamentally new route to cavity QED that is inherently robust to fabrication imperfections, as opposed to traditional cavities (9).

Pumping the sample at low-excitation power allowed us to resolve single QD lines and therefore to enter the regime of cavity QED. Figure 3A shows an example of a photoluminescence spectrum displaying single QD peaks and Anderson-localized cavities. QDs and cavity peaks can be easily distinguished from their different temperature dependences (Fig. 3B) that also enable the spectral tuning of single QDs into resonance with an Anderson-localized cavity. Figure 3C displays the crossing between a QD and an Anderson-localized cavity, demonstrating that the cavity-QD system is in the Purcell regime where the cavity promotion of vacuum fluctuations enhances the QD decay rate (6).

Fig. 3

Temperature tuning of single QDs into resonance with Anderson-localized cavities. (A) Low-power photoluminescence spectrum of a sample with 3% disorder at 10 K. (B) Photoluminescence spectra collected while varying the sample temperature in steps of 5 K. The dotted (dashed) lines are guides to the eye of the wavelength displacement of selected QD emission lines (localized modes). (C) Enlargement of the spectra displaying the QD-cavity crossing. The spectra are fitted to two Lorentzians (solid lines) representing the QD and the cavity peak.

The Purcell enhancement is studied by means of time-resolved photoluminescence spectroscopy: A QD is repeatedly excited with a short optical pulse and the emission time is measured. Collecting many single-photon events allowed us to record a decay curve representing a histogram of detection events versus time. Two examples of decay curves for the QD tuned on- and off-resonance with an Anderson-localized cavity are presented in Fig. 4A. Off resonance, the QD decay rate is inhibited due to the two-dimensional photonic band gap, leading to an emission rate of 0.5 ns−1. A pronounced enhancement by a factor of 15 is observed on resonance where a fast decay rate of 7.9 ns−1 is extracted. An important figure-of-merit for, e.g., single-photon sources or nanolasers is the β factor, which expresses the fraction of photons emitted into a cavity mode. By comparing the emission rates on and off resonance, we extract β = 94%, which represents a lower bound because even for large detuning, residual coupling to the waveguide can persist. The high β factor competes with results obtained on standard photonic crystal nanocavities with carefully optimized cavity design and QD density (7). Our results demonstrate that distributed photonic disorder provides a powerful way of enhancing the interaction between light and matter, enabling cavity QED.

Fig. 4

Detuning dependence of single QD decay rates. (A) Decay curves of QD5 for two values of detuning Δ relative to the localized mode C1. (B) Decay rates of QD4 and QD5 versus detuning and cavity emission spectrum. (C) Decay rates of QD1, QD2, QD3 versus detuning. The dashed line is the calculated slow-down factor for the unperturbed photonic crystal waveguide. The enhancement at Δ = −4 nm stems from the coupling of QD2 to a weak Anderson-localized cavity mode (C3 in Fig. 3A).

The decay rates of two individual QDs tuned across an Anderson-localized cavity are plotted in Fig. 4B. Different enhancement factors (15 and 9 at temperature T = 25 and 55 K, respectively) are observed on resonance due to the different positions and dipole orientations of the QDs that influence their coupling to the cavity mode. The presence of an additional Anderson-localized cavity gives rise to the asymmetric detuning dependence of the decay rate. Assuming a perfect spatial match between the QD and the cavity mode, we can extract an upper bound on the mode volume of the Anderson-localized cavity of V ~ 1 μm3 from the observed rate on resonance. By estimating the extension of the localized modes in the two directions orthogonal to the waveguide (23), we derive a cavity length of 25 μm for cavity C1. Establishing the fundamental lower limit of the cavity length relates to the fundamental question of determining the localization length of Anderson-localized modes. It is predicted that the localization length can be reduced below the wavelength of light, and it was even suggested that no fundamental lower boundary exists (24). Consequently, engineered disorder might pave a way to subwavelength confinement of light in dielectric structures.

Figure 4B shows that Purcell enhancement is observed mainly within the cavity linewidth, which is opposed to the surprisingly far-reaching coupling reported for standard photonic crystal cavities under nonresonant excitation (7). Consequently, the extracted QD decay rates are sensitive probes of the local photonic environment of disordered photonic crystal waveguides. Photon emission in disordered photonic structures was predicted to lead to a new class of infinite-range correlations manifested as fluctuations in the decay rate of embedded emitters (25). Thus, the Purcell enhancement stems from the local enhancement of the photonic density of states in the Anderson-localized regime that promotes spontaneous emission of photons.

QDs detuned from Anderson-localized cavities may couple to the slowly propagating mode of the photonic crystal waveguide. In this case, the QD decay rate is expected to scale proportional to the group velocity slow-down factor ng (26). This behavior is observed for three different QDs at large detunings Δ from the dominating Anderson-localized cavity mode (Fig. 4C), i.e., here the radiative coupling is well described by the local photonic density of states of the unperturbed photonic crystal waveguide. This interesting coexistence of ordered and disordered properties occurs because relatively few periods of the photonic crystal lattice are required to build up the local environment determining the QD decay rate. Thus, the length scale on which the local photonic density of states builds up is mostly shorter than the localization length, which accounts for the success of photonic crystals despite ubiquitous disorder for, e.g., nanocavities (9), single-photon sources (15), or spontaneous emission control (27).

Our experiments demonstrate that disorder is an efficient resource for confining light in nanophotonic structures, thereby opening a new avenue to all-solid-state cavity QED that exploits disorder as a resource rather than a nuisance. Exploring disorder to enhance light-matter interaction and establishing the ultimate boundaries for this new technology provide exciting research challenges for the future, of relevance to not only QED but also other research fields that rely on enhanced light-matter interaction, such as energy harvesting or biosensing (28). Coupling several cavities is a potential way of scaling cavity QED for quantum information technology and represents one of the major challenges for engineered nanocavities. Controlled disorder might offer an interesting route to coherently couple cavities using so-called necklace states that are naturally occurring coupled Anderson-localized modes (29, 30).

Supporting Online Material

Materials and Methods


References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. We gratefully acknowledge T. Schlereth and S. Höfling for quantum dot growth, J.M. Hvam for discussions, and the Council for Independent Research (Technology and Production Sciences and Natural Sciences) and the Villum Kann Rasmussen Foundation for financial support.
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