Strong Coupling and Entanglement of Quantum Emitters Embedded in

Article pubs.acs.org/journal/apchd5

Strong Coupling and Entanglement of Quantum Emitters Embedded in a Nanoantenna-Enhanced Plasmonic Cavity Matthias Hensen,*,† Tal Heilpern,‡ Stephen K. Gray,‡ and Walter Pfeiffer§ †

Institut für Physikalische und Theoretische Chemie, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany Center for Nanoscale Materials, Argonne National Laboratory, 9700 Cass Avenue, Lemont, Illinois 60439, United States § Fakultät für Physik, Universität Bielefeld, Universitätsstraße 25, 33615 Bielefeld, Germany

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‡

ABSTRACT: Establishing strong coupling between spatially separated and thus selectively addressable quantum emitters is a key ingredient to complex quantum optical schemes in future technologies. Insofar as many plasmonic nanostructures are concerned, however, the energy transfer and mutual interaction strength between distant quantum emitters can fail to provide strong coupling. Here, based on mode hybridization, the longevity and waveguide character of an elliptical plasmon cavity are combined with intense and highly localized field modes of suitably designed nanoantennas. Based on FDTD simulations a quantum emitter−plasmon coupling strength ℏg = 16.7 meV is reached while simultaneously keeping a small plasmon resonance line width ℏγs = 33 meV. This facilitates strong coupling, and quantum dynamical simulations reveal an oscillatory exchange of excited state population and a notable degree of entanglement between the quantum emitters spatially separated by 1.8 μm, i.e., about twice the operating wavelength. KEYWORDS: nanoscale quantum optics, plasmonic cavity, strong coupling limit, entanglement, quantum plasmonics, quantum dynamical Purcell factor fficient light−matter interaction in nanophotonic structures and the related quantum emitter (QE) coupling phenomena provide one possible route for downsizing quantum optics based devices to scales compatible with usual microelectronics. Quantum correlations such as entanglement1,2 or squeezing3 persist in plasmonic nanostructures and quantum phenomena for both matter and light play an increasingly important role.4,5 Therefore, strong coupling phenomena between a plasmonic mode and a QE ensemble6−9 or, especially, a single QE10−14 have received intense attention in nanophotonics.15 For a long time strong coupling between plasmons and a single QE was only subject to theoretical studies, but recently Chikkaraddy et al. have experimentally demonstrated strong coupling of a single dye molecule with a plasmonic mode.16 This observation marks a major step toward future applications of plasmonic nanostructures in quantum optical based devices since this demonstrates that QE excitation can be transferred on very short time scales, i.e., on a few 10−14 s, to a plasmonic resonator. Achieving this strong QE−plasmon coupling relies on the strong field enhancement and reduced electromagnetic mode volume in a gap plasmon formed between a gold nanoparticle and a gold film.16 The next essential steps are now to employ this strong coupling to a plasmonic resonator to efficiently transfer energy, strongly couple, and establish quantum entanglement between two or more spatially well separated QEs. Coupling multiple QEs and selective addressing (excitation, or read-out) of

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© 2017 American Chemical Society

individual QEs in an ensemble of many interacting QEs are essential ingredients for the realization of coherent control schemes needed for example for implementing quantum computation algorithms. Also in this respect plasmonics provides helpful functionality. Suitably designed plasmonic nanostructures can serve to channel energy launched by one QE to another QE that is spatially sufficiently separated to allow selective excitation, control, and read-out. In theoretical investigations plasmonic wedge waveguides17,18 and metal nanoparticle chains19 were investigated. In all these schemes the resulting coupling between QEs remains in the weak coupling limit; that is, the excitation energy leaves one QE, is transmitted to the other QE, and so efficiently dissipated either by radiative or nonradiative processes that no periodic energy exchange between both emitters is observed. Achieving efficient energy transfer between two separated QEs or even the strong coupling limit in quantum plasmonics is strongly affected by the lossy character of plasmonic resonators. In conventional cavity quantum electrodynamics (cQED), resonators with extremely high quality (Q)-factors in the range of 1012 are achievable20 and consequently strong coupling physics is reached even for large mode volumes. In plasmonics, Special Issue: Strong Coupling of Molecules to Cavities Received: July 3, 2017 Published: September 29, 2017 240

DOI: 10.1021/acsphotonics.7b00717 ACS Photonics 2018, 5, 240−248

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resonances having a large mode volume show indeed rather long lifetimes21 compared to strongly localized modes of nanoantennas. Unfortunately such strongly localized nanoantenna modes, i.e., the intense and short-lived gap plasmons, are required to achieve strong coupling between a single QE and a plasmon.16 In other words, it seems that properties such as longevity and intense optical modes mutually exclude each other in the field of plasmonics. We propose that hybridization of plasmonic modes22 could solve this dilemma: When two or more systems hybridize to a new entity, the properties of the constituents are imprinted on the new eigenmodes. Therefore, one needs to find a combination of different plasmonic structures that, on one hand, provide opposed features such as strongly localized modes and rather high Q-factors and, on the other hand, can be efficiently coupled. In this context a recently realized hybridized plasmonic device could serve well. Aeschlimann et al. demonstrated strong coupling of optical nanoantennas embedded in an elliptical plasmonic cavity.23 The antennas are placed in the two foci of the elliptical cavity, and periodic energy exchange between the antennas separated by about 1.6 μm was observed. Note that this large distance allows selective excitation of specific antennas from the far field. The corresponding hybridized plasmonic modes thus form an interesting platform for plasmon-assisted long-range QE coupling provided means are identified to strongly couple the QEs to the embedded nanoantennas. Here we make use of this experimentally demonstrated plasmonic coupling scheme23 in a theoretical study and predict strong coupling of two QEs that are spatially separated by more than 1.6 μm, as well as a short-lived meaningful entanglement, by using realistic quantum emitter properties. The article is organized as follows: After a brief presentation of the plasmonic structures and the finite-difference time-domain (FDTD)-based electric field simulations of plasmonic cavity and antenna structures the quantum mechanical coupling phenomena at the boundary of strong coupling are discussed. Finally, based on these coupling constants, strong coupling between widely separated quantum emitters is demonstrated and the dynamics of quantum entanglement is discussed.

Figure 1. Elliptical SPP cavity and QE−antenna coupling schemes. (a) Cross section of mode field amplitude distribution for the “bare” elliptical SPP cavity for excitation with a point dipole source located 4 nm above the right focal point. The inset shows the lateral mode pattern in the cavity. (b, c) Mode field amplitude cross sections for the WGM antenna (b) and the gaptenna (c) used to more efficiently couple the QE to the elliptical SPP cavity mode. Again point dipole sources were used for excitation, now placed 4 nm above the center mesa of the WGM antenna and in the center of the 8 nm gap of the gaptenna structure. All mode cross sections are shown on the same logarithmic scale. The inset in part (b) shows a 3D representation of the WGM antenna for clarity.

depth into the vacuum half-space is about 600 nm for Au substrates. Consequently, the SPP mode exhibits a rather high Q-factor of 137. The longevity of the cavity mode will be later exploited by the nanoantennas when they are placed in the focal field antinodes since all constituents then form a hybridized system.23 The nanoantenna simply consists of a circular groove (Figure 1b), and it can be resonantly excited when channel plasmon polaritons26 propagating inside those grooves form a standing wave.27 Here, the groove depth dgr was set to 76 nm and the radius was set to 70 nm so that this so-called whispering gallery mode (WGM) resonator exhibits the m = 0 mode, which resembles the field distribution of a dipole oscillating perpendicular to the substrate.28 This field distribution overlaps well with the spatial cavity mode near the focal spots, which is beneficial for energy transfer between antennas at opposite sites. Also, the interaction with attached quantum emitters is expected to be enhanced since far field radiation loss is suppressed due to the orientation of the modal dipole, and the Q-factor amounts to 24. To further boost light−matter interaction in our structure, we utilize the gap-plasmon approach of Chikkaraddy et al.16 and position a 100 nm nanoparticle on top of the WGM antenna so that the point dipole source is located in the center of an 8 nm wide gap between the antenna mesa and the nanoparticle surface (Figure 1c). Note that dgr is reduced to 46 nm to compensate for the nanoparticle-induced red-shift of the WGM antenna. This structure is labeled gaptenna in the following. After having discussed the response of cavity and antenna structures independently we now present results on the

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DISCUSSION AND RESULTS We start the discussion with an introduction of the single building blocks of the hybridized plasmonic system. All structural parameters were chosen in such a way that the various building blocks exhibit resonant behavior at νs = 375 THz (wavelength 800 nm, photon energy 1.55 eV). The corresponding electric field distributions of the resonant modes, which are shown in Figure 1a−c, were retrieved from FDTD simulations by placing a point electric dipole source close to the respective mode maximum but 4 nm separated from the nearest metal surface (see Methods section for details). The elliptic surface plasmon polariton (SPP) cavity24 and its characteristic mode pattern, which is dominated by field components pointing perpendicular to the cavity floor, are presented in Figure 1a. In general, the problem of a vibrating elliptical membrane is solved by Mathieu functions,25 and the prevailing mode is determined by the size of the major and minor axis sizes a and b, respectively. Here, a is set to 2072 nm and b is set to 1450 nm so that injected SPPs that are reflected at the cavity walls form a mode that shows two pronounced field antinodes near the geometrical focal spots. The wall height of 800 nm prevents leakage loss since the SPP penetration 241


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peak, while the function of the gaptenna shows a broad peak corresponding to a 10 times faster decay time, directly reflected in the different spectral Purcell functions. In the combined system the three modes, i.e., the two antenna modes and the cavity mode, hybridize, resulting in the spectral Purcell function f(ω) shown in Figure 2c (dashed line): Three clearly separated peaks are visible, and the maximum splitting, defined by the outer modes, amounts to 37 THz (153 meV), i.e., 10% of the operating frequency. Note that the clearly resolved mode splitting shows that the gaptenna modes are strongly coupled via the cavity mode. Interestingly, the coupling strength of the plasmonic structures is increased by more than a factor of 1.5 when gaptennas are used instead of WGM antennas, where a splitting of 23 THz (95 meV) was reported.23 This increase is attributed to the presence of the nanoparticles which also interact with the SPP mode. The impact of the cavity mode on this hybridized system is made visible by fitting the spectral Purcell function with a sum of Lorentzian line-shape functions (solid black and red lines in Figure 2c): The line width of the outermost peaks corresponds to a decay time that is only about 3 times larger than the decay time of the “bare” cavity despite the presence of gaptennas. Still, the gaptennas contribute to the new eigenmodes with their large Purcell enhancement leading to an even higher peak in f(ω) for the red-shifted mode, located at 356 THz (842 nm), compared to the gaptenna case alone (Figure 2b). The hybridization scheme thus indeed combines favorable properties of both modes and enhances coupling between the two gaptennas. Below we will utilize this particular mode to study the plasmon-mediated formation of quantum entanglement between two widely separated QEs since it offers, besides the cavity-induced reduction of line width, the highest Purcell enhancement among the three hybridized modes. For a quantitative assessment of the achieved QE−plasmon and QE−QE coupling in the different schemes considered here and for performing quantum dynamics simulations we now determine the corresponding coupling constants g. According to Fermi’s golden rule, i.e., in the weak coupling limit, the strength of light−matter interaction depends on the photon density of states (PDOS) in which the QE is embedded.29,30 Plasmonic resonances modify the local PDOS and thus affect g. In the following this effect is accounted for by the spectral density |g(ω)|2, which is proportional to the PDOS. |g(ω)|2 is expressed using the Green’s function formalism12 yielding

combined system consisting of two gaptennas positioned at the focal antinodes of the elliptic SPP cavity (Figure 2a). Note that

Figure 2. Strong coupling of gaptennas embedded in an elliptical SPP cavity. (a) Cross section of mode field amplitude distribution for an elliptical SPP cavity with gaptennas embedded at the two foci of the cavity mode. The inset shows a close-up of the gap on the right, in which the placement of the point dipole source is indicated by a white arrow. (b) Frequency-dependent or spectral Purcell functions f(ω) for gaptenna and “bare” cavity as a solid and dotted line, respectively. (c) f(ω) for the hybridized modes (dotted line) in the combined structure shown in part (a) together with the best fit of a sum of Lorentzian lineshape functions (red solid line) and their individual contributions (solid black lines). The low-frequency resonance (gray-shaded peak) is used in the following to mediate efficient coupling between a QE placed in the gaps of both gaptennas. The resonance width is ℏγs = 30 meV using γs/2π = 7.5 THz from (c).

|g (ω)|2 =

1 ω2 T μ Im[Gs(ω , rq , rq)]μq ℏπε0 c 2 q

(1)

where the dyadic Green’s function Gs(ω, rq, rq) reflects the local field effects at the position rq of the QE induced by environmental effects, i.e., the presence of a plasmonic resonator. In addition, the dipole moment μq of the actually chosen QE must be known to determine the coupling strength. Here, we assume that the QE exhibits a decay width of ℏγq = 0.66 μeV (corresponding to a 1 ns lifetime or γq = 1 GHz) at its transition frequency ωq that is chosen in resonance with the plasmon resonance ωs. From γq the dipole moment μq was derived using the relation30

the structure is excited by a dipole source that is located at the same position as in the case of the pure gaptenna system; that is, the source is located 4 nm from both the WGM inner mesa and the nanoparticle. The obtained field strength is increased by 1 order of magnitude in the cavity center of the combined system compared to the case of an empty cavity (Figure 1a), and therefore the mode exhibits a delocalized character, as expected from a hybridized system. To discuss the spectral properties of the combined system, we first show in Figure 2b the spectral Purcell function f(ω) of the “bare” cavity and the gaptenna as retrieved from a transmission monitor that measures the power radiated by the dipole source (see Methods section for details). According to a Q-factor of 137, f(ω) for the “bare” cavity exhibits a sharp

γq =

ωq3 |μq |2 3πε0ℏc 3

(2)

The spectral density |g(ω)| according to eq 1 is retrieved from FDTD simulations and the assumed μq. For the various 2

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considered here. It is more than 50 times higher compared to a “bare” cavity and even 6 times higher than for the WGM antenna. Interestingly it is approaching a value that gets close to thermal energies at room temperature, indicating that the design concept can become relevant for room-temperature applications. For the hybridized system combining two gaptennas with the elliptical cavity the coupling strength is reduced by about a factor of 2 compared to the gaptenna case. This reduced coupling can be understood qualitatively: In the case of the hybridized system the corresponding plasmonic mode is delocalized between both gaptennas, and consequently the mode volume in the gap between the nanoparticle and the WGM mesa increases by a factor of about 2 since the fields are highly localized in the gaps. This reduction in the coupling constant because of the delocalization between two gaptennas can be compensated by reducing the gap size by a factor of 2. With a gap width of 4 nm we get close to the limits of the used FDTD method. Nevertheless it could make sense to consider even higher possible coupling constants g, as they have been recently demonstrated experimentally for a single dye molecule embedded in a 1 nm wide gap plasmon mode.16 However, for such dimensions simple FDTD calculations are no longer trustworthy, and we have therefore kept the gap width to values larger than 4 nm. In the weak coupling limit F is a well-established quantity to quantify the coupling strength. For increasing coupling this simple concept breaks down and the nonlinearity introduced by the back-reaction of the system on the QE has to be considered. In the following we investigate the transition between the weak and strong coupling regime for QE−plasmon interaction. Based on quantum dynamics calculations (see Methods) with various values of the emitter and plasmon decay constants and QE−plasmon coupling constants discussed above we obtain further insight into the weak and strong coupling regimes, as well as to conveniently compare the various systems under study here. The approach we use is to consider a single QE−plasmon system (i.e., include only k = 1 in eq 7) and numerically solve for the time evolution of the system. With the same initial condition of only QE1 (now the only QE) initially excited, we obtain the probability P1(t) of QE1 being in the excited state as a function of time and infer an effective decay constant, γeff, by fitting P1(t) to an exponential. In the weak coupling limit this probability function will be exponential in time, and γeff can be obtained from a fit of the ln[P1(t)] to a line. For larger values of the coupling parameter P1(t) will exhibit damped oscillations, but such a linear fit to ln[P1(t)] will still yield the underlying decay constant. These damped oscillations reflect the degree of coupling g in each system; in the limit of strong coupling they have frequency 2g or period π/g. We normalize this effective QE decay rate in the presence of the plasmonic system by the isolated QE decay rate to obtain what we refer to as the dynamical Purcell factor: FQD = γeff/γq. Figure 3a shows plots of P1(t) on a logarithmic scale for systems B−F that we have introduced (Table 1). Note that to completely define the quantum system, we also specify a value of the QE dephasing parameter, γd. This parameter, just as long as it is small compared to the coupling parameter g and the plasmon resonance decay parameter γs, does not significantly impact the dynamics, and we have made the arbitrary choice here that γd = 0.1g. Unless otherwise stated, we are always assuming this for the QE dephasing. Note also that QE and plasmonic transition frequencies are assumed to be resonant,

QE−plasmon coupling schemes the Green’s functions are calculated using prebuilt scripts.31 In the present case Gs and thus also |g(ω)|2 are dominated by distinct plasmon resonances, and hence |g(ω)|2 is approximated by a Lorentzian given by12 |g (ω)|2 =

γs g2 2π (ω − ω )2 + s

γs2

(3)

4

with the coupling constant g between QE and the surface plasmon resonance and the plasmon resonance width γs. By fitting |g(ω)|2 using eq 3 the corresponding QE−plasmon coupling constants g, as listed in Table 1, are obtained. Table 1. Quantum Emitter Coupling Parameter g, Plasmon Resonance Width γs, and Purcell Factor F for Various Configurationsa

“bare” cavity (A) WGM (B) gaptenna (C) hybrid, 8 nm gap (D) hybrid, 4 nm gap (E) gap plasmon16 (F)b)

ℏg/meV eq 3

ℏg/meV eq 4

ℏγs/ meV

F (FDTD)

0.3 2.5 16.1 9.0 16.7 NA

0.3 2.5 15.8 8.6 15.6 90

11 64 135 31 33 147

47.6 597 11.3 × 103 14.3 × 103 45 × 103 3.5 × 106

a

The capital letters A to F serve in the following as labels for the different QE coupling schemes. With the exception of the gap plasmon case ℏγq = 0.66 μeV, i.e., a 1 ns lifetime of the QE in vacuum, is assumed. bFor the gap plasmon eq 4 is used to determine F based on the measured Rabi splitting for a single molecule that directly yields g, ℏγs extracted from the upper part in Figure 3a in ref 16, and ℏγq = 0.07 μeV as reported in ref 16.

To check for consistency, a second method to determine g is applied. FDTD simulations provide directly the spectral Purcell function f(ω) (see Methods for more details). Fitting f(ω) with Lorentzians yields the Purcell factor F, i.e., the resonance peak height in f(ω), which is commonly used to characterize the enhanced decay rate of a QE because of the presence of local PDOS effects30 for the plasmon resonances in f(ω) (last column in Table 1). In the weak coupling limit, i.e., if the coupling constant g is much smaller than other decay constants, F can be expressed as32 F=

4g 2 γsγq

(4)

providing a simple relation between F and g. Hence, g can directly be inferred from F using eq 4. Note (see discussion below) that g so-obtained also depends on other system decay constants. While not as accurate as the spectral density approach outlined above, we find that g values obtained with eq 4 differ by less than 10% even in the moderate−strong coupling limit. We should note, however, that neither the spectral density approach nor the weak coupling limit formula properly accounts for the back-reaction effects, i.e., energy transfer from the plasmon back to the emitter, which is at the heart of strong coupling. This is because the underlying classical electrodynamics calculations to yield the spectral density and Purcell factor do not allow such back-reaction to occur. Thus, one should always exercise some caution with such coupling constant estimates. As expected, the gaptenna features the highest coupling strength of 16 meV for the QE−plasmon coupling schemes 243


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In the following we consider the quantum dynamics in a bipartite qubit system coupled via the plasmonic modes introduced above. We use the coupling constants, g, and decay constants inferred for our systems in quantum dynamics calculations (see Methods) to assess how effectively one QE about 1.6 μm separated from the other QE can nonetheless be coupled to it. The time evolution of two QEs in the hybridized cavity−gaptenna system is presented in Figure 4a−c, where in order to clearly see the added effects of the WGM antenna and the gaptenna, the evolutions of the two emitters in the “bare” elliptic SPP cavity are plotted in Figure 4d−f. In all cases the system is initiated with one QE excited and with both the other QE and the plasmonic system in their ground states. The time evolution is computed from eqs 5−7 (see Methods), where now k = 1, 2 in eq 7, using the coupling and decay parameters taken from Table 1. Note that again the QE dephasing γd is chosen as 0.1g, resulting in ℏγd ≈ 1 and 0.03 meV for the gaptenna and bare cavity, respectively. Whereas the value for the bare cavity might only be reached at low temperature, a line width in the range of 1 meV is a reasonable estimate for room temperature.33,34 In both cases the initially excited QE decay and some energy are transferred to the other QE. Whereas the dynamics in the “bare” cavity occur roughly on a 30 ps time scale (Figure 4d), this process occurs much faster. In the “bare” cavity case the dynamics are dominated by an almost exponential decay of the QE1 excitation and very little transfer to QE2. In contrast, the population dynamics in the hybrid system are much more complicated: Within about 100 fs almost 30% of the QE1 excitation is transferred to QE2 and after a small overshooting both excitations decay on a ps time scale. As discussed below, the small overshooting is a first indication of the onset of strong coupling effects between both QEs. Figure 4b,e show the quantum dynamics of both configurations projected on the symmetric and antisymmetric basis vectors that serve to better convey the underlying physics of this behavior.18,19 In the hybridized system the symmetric state exhibits a fast almost complete initial decay, whereas the antisymmetric state decays with a much longer time constant.18 In the latter case the plasmon-related dissipation cancels at least partially, and thus the decay rate for this state is smaller. This behavior is analogous to the radiative damping of symmetrically and antisymmetrically coupled oscillating dipole emitters. In this case the antisymmetric mode, often denoted as the dark mode, decays less efficiently since it does not couple to transverse modes of the electromagnetic field. This behavior is closely related to the dynamics of the bipartite entanglement in the present situation. Here we employ the concurrence C as a measure of entanglement between both QEs.35 C starts from an unentangled initial state and increases with the rapid decay of the symmetric state to about 0.4 in the case of the hybrid system. Afterward the entanglement decays with the same time constant as the antisymmetric state. That is, while the initial state is a separable product of the QE states, it can be decomposed as an equal amplitude superposition of completely entangled symmetric and antisymmetric states. The more rapid decay of the symmetric state leads to a state with more (slowly decaying) entangled antisymmetric character and thus greater concurrence. However, the dynamics is relatively fast, with the maximum in concurrence occurring after about 100 fs (Figure 4c). In comparison to the hybridized system dynamics, the cavity case (Figure 4d−f) displays longer decays and a lower concurrence maximum.

Figure 3. (a) Comparison of the QE decay dynamics for different QE−plasmon coupling schemes. Quantum simulations for an initially excited single QE placed at the positions used for point dipole emitter excitation as shown in Figure 1b,c for a WGM antenna (B, red solid line) and gaptenna (C, black solid line) and in Figure 2a for the hybridized system with 8 nm gap width (blue solid line). In addition, the QE decay for the hybridized system with reduced gap width (E, magenta solid line) and the gap plasmon coupling case reported in ref 16 (F, green solid line) are shown. The dashed lines for cases E and F represent linear fits for determining FQD in the intermediate and strong coupling case. (b) Dynamical Purcell factor FQD as a function of the plasmon decay constant γs and QE emission rate γq normalized to the QE−plasmon coupling constant g. The QE dephasing rate has been fixed at γd/g = 0.1. The points labeled A−F refer to the QE coupling schemes listed in Table 1. The solid black contour lines correspond to values FQD = 102, 103, ..., 106. The dashed straight lines correspond to the weak coupling limit Purcell factor determined using eq 4.

and in this limit the results do not depend on ω0. From Figure 3a we see that systems B and C do indeed exhibit exponential decay. However, systems D−F all exhibit some degree of oscillatory behavior superimposed on an underlying exponential decay indicative of larger QE−plasmon coupling. Actually, the systems A−F can be represented as isolated points in a much wider range of possibilities. It turns out that a universal plot can be made if FQD is plotted as a function of the dimensionless decay constants γs/g and γq/g, as in Figure 3b. Strong coupling manifests itself by leading to the bent, Λshaped contours. For the depicted ranges, the strong coupling limit is the region to the left of the “apex” of the Λ-contours, i.e., the region for log10(γs/g) < 0.5. The deviations of the solid lines from the dashed ones, more noticeable in the left side of Figure 3b, also suggest that the use of simple formulas such as eq 4 and possibly also the approach of Hümmer et al.12 may not be appropriate, although a detailed analysis of better approaches to obtaining g must be deferred to future work. 244


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Figure 4. Time evolution of two QE embedded in the hybridized cavity−gaptenna system (a−c) and the elliptic SPP cavity (d−f). The hybridized system is a result of all components of the system involved, i.e., the cavity, the WGM antenna, and the gaptenna of dgap = 8 nm, while on the right the dynamics of the QEs in the “bare” cavity are plotted. The insets in (a) and (d) depict the corresponding coupling schemes. (a) and (d) show the time evolution of the populations of both QE, and (b) and (e) show the evolution of the populations of the symmetric and antisymmetric states for the two configurations. (c) and (f) show the evolution of the concurrence, which is a measure of the entanglement between the two QEs.

We once again would like to stress that the currently discussed onset of strong coupling was reached with a rather large gap size of 8 nm. Although the emitter properties of quantum systems are expected to be quenched close to metallic surfaces,36 Baumberg and co-workers found extraordinary high Purcell factors16 (see Table 1) for a gap size of about 1 nm. Therefore, we finally demonstrate that our proposed cavity design explicitly allows strong coupling of widely separated QEs by reducing the gap size from 8 to 4 nm. Without loss of generality, and to show the flexibility of our system, we adhere to the identical WGM antenna geometry for this scenario. Because of the reduced gap size, the antenna resonance shifts to slightly lower frequencies. To account for this resonance shift, the cavity’s major axis is increased by 222 nm, moving the cavity mode again into resonance with the gaptenna mode. The spectral resonance of both systems is then centered at λ0 = 860 nm (photon energy 1.44 eV or ν0 = 348.6 THz). The spectral Purcell function f(ω), retrieved from FDTD simulations in the weak coupling limit, is shown in Figure 5a: Again, three clearly separated eigenmodes are observed, spanning a frequency range of 37 THz, i.e., more than 10% of the center frequency. By applying the abovementioned fitting procedure (red solid line in Figure 5a) we retrieve a Purcell factor of 44900 for the red-shifted eigenmode at 331.6 THz. Hence, a reduction of gap size from 8 nm to 4 nm leads to a 3.5 times higher Purcell enhancement, and the coupling constant increases to ℏg = 16.7 meV, while the plasmon resonance line-width remains rather constant at ℏγs = 33.1 meV. Maintaining plasmon dissipation properties while increasing the coupling strength of the QE−plasmon system facilitates in finally passing the threshold toward strong coupling: Figure 5b shows the population dynamics of two QEs having the same properties as before (γq = 1 GHz at ν0 = 348.6 THz) but positioned now in narrower gaps. The QEs undergo a partial oscillation in their respective excited state populations; however, dissipation and decoherence still prevent additional oscillations. In contrast to previous works in which plasmonmediated quantum entanglement was realized via cooperative decay effects,18,19 i.e., subradiance, we clearly see modulations in the state populations (Figure 5b,c) and also in the

Figure 5. Onset of strong long-range QE coupling in a hybridized cavity with a 4 nm gaptenna gap width. (a) Spectral Purcell function f(ω) for the hybridized modes in the elliptical cavity with embedded gaptennas (dotted line) together with the best fit of a sum of Lorentzian line-shape functions (red solid line) and their individual contributions (solid black lines). The low-frequency resonance (grayshaded peak) is used in the following to mediate efficient coupling between QEs placed in the gaps of both gaptennas. Here the QE dephasing parameter ℏγs = 1.7 meV. (b) Population and concurrence dynamics for the hybrid system with reduced gap width (E). The arrangement of panels (b)−(d) is the same as in Figure 4a−c.

concurrence, C (Figure 5d). While a concurrence of unity is the optimal value which indicates a maximally entangled state, the transient values achieved over a short period (λ0) from the simulation volume boundaries. We employed absorbing boundary conditions using a minimum number of 32 perfectly matched layers at each boundary in order to prevent reflections of propagating fields. The adaptive mesh accuracy of the simulation volume was set to 22 sampling points per effective wavelength λ inside the corresponding material, and we omitted a conformal mesh refinement. Instead, we encased the WGM antenna, particle, and the gap in between with additional meshes providing a resolution of 2 nm in each direction, except the axial gap direction, which was resolved by 1 nm. In the case of the system featuring the 4 nm gap size the structures are resolved by 1 and 0.5 nm, respectively. Note that these monitors are active in all simulations, irrespective of the presence of antennas or nanoparticles, in order to guarantee similar mesh conditions. The excitation source is a pulsed dipole (4.3 fs and λ0 = 800 nm), which exhibits a minimum distance of four mesh cells to interfaces in all simulations. Purcell factors were calculated by encasing the dipole source with a built-in transmission monitor: 2 nm × 2 nm × 1 nm in the case of a 4 nm gap and 2 nm × 2 nm × 1 nm in the case of an 8 nm gap size. The monitor integrates the Poynting vector

H = ℏωsb†b + ℏωq

∑ k = 1,2

σk+σk− + ℏg

∑

(σk+b + σk−b†)

k = 1,2

(7) +

where b and b are the harmonic oscillator raising and lowering operators, σk− = |0k ⟩⟨1k |, σk+ = |1k ⟩⟨0k |, and we will take the plasmon and QE transition frequencies to be equal, ωs = ωq = ω0. Each QE is assumed to be coupled to the plasmon mode with the same dipolar coupling constant g. See ref 32 for the explicit form of L, and we note that it is parametrized by the QE’s spontaneous emission rate γq, the environmentally induced dephasing rate γd, and the plasmon decay constant γs. Insertion of eq 5 into eq 6 with eq 7 leads to a set of linear differential equations for the 16(smax + 1)2 density matrix elements ρq1,q2,s,q1′,q2′,s′ that can be solved numerically to yield, via appropriate traces of the density matrix, populations and other quantities of interest as a function of time. We focus on an initial condition corresponding to the QE 1 being in its excited state with the rest of the system cold, and for such calculations it suffices to include just two plasmon states (s = 0 and 1) and simply solve the density matrix equations numerically with a standard ordinary differential equations numerical solver. In addition to the excited state populations of the two QEs, we calculate the populations in the symmetric and antisymmetric entangled states, |S⟩ = 1/ 2 (|10⟩ + |01⟩) and |A⟩ = 1/ 2 (|10⟩ − |01⟩), where |q1q2⟩ = |q1⟩|q2⟩ denotes a product of the two QE states. We also compute the bipartite concurrence, 246


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C(t), which is a measure of the degree of entanglement between the two QEs35 and thus the potential relevance of the systems to quantum information and sensing applications. The concurrence is between 0 and 1, with 0 indicating a separable (unentangled) state, while any C > 0 indicates the state is an entangled state, i.e., a state that cannot be written as a separable state. Quantum information applications generally use “maximally entangled” states with C = 1.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Matthias Hensen: 0000-0002-5578-0118 Author Contributions

The manuscript was written through contributions of all authors. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has received funding from the German Research Foundation (DFG) within the priority program SPP 1391 “Ultrafast Nanooptics” under Grant Agreement PF 317/5. This work was performed, in part, at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility under Contract No. DEAC02-06CH11357.

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REFERENCES

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