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Making two-photon processes dominate one-photon processes using mid-IR phonon polaritons

Making two-photon processes dominate one-photon processes using mid-IR phonon polaritons Making two-photon processes dominate one-photon processes using mid-IR phonon polaritons a,1 a,b a,1 a,c a Nicholas Rivera , Gilles Rosolen , John D. Joannopoulos , Ido Kaminer , and Marin Soljaci ˇ c ´ a b Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; Micro- and Nanophotonic Materials Group, University of Mons, 7000, Mons, Belgium; and Department of Electrical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel Contributed by John D. Joannopoulos, November 7, 2017 (sent for review August 3, 2017; reviewed by Dmitri Basov and Marinko Jablan) Phonon polaritons are guided hybrid modes of photons and opti- taneous parametric down-conversion sources) have become a cal phonons that can propagate on the surface of a polar dielec- staple in quantum information protocols. From a more funda- tric. In this work, we show that the precise combination of con- mental perspective, the frequency spectrum of two-photon spon- finement and bandwidth offered by phonon polaritons allows taneous emission can be broad, with a spectral width of the order for the ability to create highly efficient sources of polariton pairs of the transition frequency itself. This is in sharp contrast to one- in the mid-IR/terahertz frequency ranges. Specifically, these polar photon emission, which is spectrally very sharp in the absence dielectrics can cause emitters to preferentially decay by the emis- of external broadening mechanisms. That implies that if two- sion of pairs of phonon polaritons, instead of the previously dom- photon processes were sufficiently fast, an emitter with discrete inant single-photon emission. We show that such two-photon energy levels could be a source of light at continuous frequen- emission processes can occur on nanosecond time scales and can cies, which is in contrast to one of the first things that one learns be nearly 2 orders of magnitude faster than competing single- about quantum mechanics. photon transitions, as opposed to being as much as 8–10 orders In this work, we propose a design strategy for fast production of magnitude slower in free space. These results are robust to the of polariton pairs with quantum efficiencies potentially >90%. choice of polar dielectric, allowing potentially versatile implemen- In other words, we propose a scheme in which an excited emit- tation in a host of materials such as hexagonal boron nitride, sili- ter prefers to decay via the simultaneous emission of two quanta, con carbide, and others. Our results suggest a design strategy for despite the possibility of allowed single-photon decay pathways. quantum light sources in the mid-IR/terahertz: ones that prefer As a special case, we show how phonon polaritons (PhPs) in to emit a relatively broad spectrum of photon pairs, potentially boron nitride and silicon carbide (SiC) may allow for the design allowing for new sources of both single and multiple photons. of a kind of quantum optics source which emits polariton pairs at rates over an order of magnitude faster than any competing one-photon transitions, corresponding to lifetimes approaching two-photon processes j phonon polaritons j light–matter interactions j 1 ns in atomic-scale two-photon emitters. Our results not only Purcell effect j nanophotonics elucidate a technique for efficient sources of potentially entan- gled polariton pairs but also reveal a class of materials by which fundamental rule in light–matter interaction is that when to realize extreme regimes of light–matter interactions in which an excited electron in an atom has a choice between emit- conventionally unobservable high-order emissions become dom- ting one photon and emitting two photons simultaneously, it inant light–matter interactions. will nearly always decay via the emission of a single photon The general scheme for accessing high-efficiency second-order (1–3). The reason for this is impedance mismatch, or equiva- transitions is illustrated in Fig. 1. In free space (and near most lently, the mismatch in size between an emitter and its emitted radiation. Applying this idea to two-photon emission processes, Significance one realizes that two-photon emission suffers much more from impedance mismatch than one-photon emission, leading to its The recent discovery of nanoscale-confined phonon polaritons relative suppression. More quantitatively, the radiation resistance or impedance of a in polar dielectric materials has generated vigorous interest because it provides a path to low-loss nanoscale photonics at dipole radiator is proportional to (a=) , which up to other technologically important mid-IR and terahertz frequencies. In fundamental constants is proportional to (a=) (4). It turns this work, we show that these polar dielectrics can be used to out that the radiation rate of an atomic dipole is precisely pro- develop a bright and efficient spontaneous emitter of photon portional to (a=) , where a is the atomic size,  is the wave- pairs. The two-photon emission can completely dominate the length of the emitted light, and  1=137 is the fine-structure total emission for realistic electronic systems, even when com- constant. In contrast, the rate of a two-photon process scales as peting single-photon emission channels exist. We believe this 2 4 (a=) (5–7) and suffers much more than the one-photon pro- work acts as a starting point for the development of sources of cess when impedance is mismatched. In atomic systems, (a=)  entangled nano-confined photons at frequency ranges where 1=1000, and thus, two-photon emission in atoms is consistently photon sources are generally considered lacking. Additionally, slower than one-photon emission by more than eight orders of we believe that these results add a dimension to the great magnitude. It is because of this simple scaling argument that two- promise of phonon polaritonics. photon processes are considered insignificant and can thus almost always be ignored for the purposes of analyzing the dynamics of Author contributions: N.R., I.K., and M.S. designed research; N.R., G.R., I.K., and M.S. performed research; G.R., J.D.J., I.K., and M.S. analyzed data; and N.R., G.R., J.D.J., I.K., excited emitters. It is also because of this simple scaling argu- and M.S. wrote the paper. ment that, while conventional (one-photon) spontaneous emis- Reviewers: D.B., Columbia University; and M.J., University of Zagreb. sion engineering is a paradigm in quantum nanophotonics (8– 12), similar engineering has not been nearly as actively pursued The authors declare no conflict of interest. for two-photon spontaneous emission processes (13–18). This open access article is distributed under Creative Commons Attribution- Nevertheless, two-photon spontaneous emission processes NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). have several distinctive features which make them very desirable 1 To whom correspondence may be addressed. Email: [email protected] or nrivera@ to access. For example, the photons emitted in such a process mit.edu. are entangled due to energy and angular momentum conserva- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tion. For this reason, sources of entangled photons (such as spon- 1073/pnas.1713538114/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1713538114 PNAS j December 26, 2017 j vol. 114 j no. 52 j 13607–13612 APPLIED PHYSICAL SCIENCES nanophotonic structures), an emitter given a choice between dif- Maxwell’s equations. The primary difference between them and ferent transition pathways will generally take a single-photon free-space photons is that they are highly confined and cannot dipole (E1) transition over other choices. However, if we have escape to spatial infinity in the direction transverse to the polar highly confined modes (to impedance match) over a sufficiently dielectric slab. narrow frequency band, then we can create a situation in which To translate our intuition to a quantitative theory, we develop the single-photon E1 transition is negligibly enhanced, while the a formalism to compute the rates of two-photon transitions for two-photon transition is highly enhanced (>10 orders of mag- emitters placed near films supporting PhP modes (Supporting nitude relative to the enhancement of the competing E1 transi- Information). Our main assumption is that these PhP support- tion), so much so that the two-photon transition is strongly pre- ing materials are well-described by a local Lorentz oscillator ferred. As we will now demonstrate, one of the recent material model (although we allow for the permittivity to be complex advances that may admit the construction of such an emitter is and anisotropic). Our permittivity data are taken from ref. 32. the discovery of highly confined PhPs in polar dielectrics. Agreement with a local model has been seen in hBN films as thin Polar dielectrics like hexagonal boron nitride (hBN) and SiC as 1 nm, although only the long-wavelength part of the disper- have been the subject of significant attention over the past few sion was measured (20). Nevertheless, we will see that the results years due to their ability to confine electromagnetic energy in we arrive at should be achievable for thicknesses between 5 and small volumes (length scales <5 nm have been theoretically sug- 10 nm, and for polariton wavelengths of a few tens of nanome- gested) (19–32). Moreover, the transversely guided PhPs of these ters, where a local model should certainly suffice. materials have extremely strong spatial confinement only over a narrow spectral band of a few terahertz. Combined with the fact Extreme LDOS Near Polar Dielectrics that these PhPs can have substantially lower losses than other The main operating principle behind our results is that the surface excitations like plasmons (33) (albeit with lower veloci- LDOS of the electromagnetic field in the vicinity of PhP sup- ties and similar propagation lengths), they may provide a poten- porting materials is extremely high. This LDOS can be so high tially exciting platform for nanoscale optics in the technologi- that the emission rate of a dipole within nanometer distances cally interesting IR/terahertz regime. Going beyond hBN and from the surface of the material can be 6 or 7 orders of mag- SiC, it should also be possible to exploit PhPs at mid-IR/far- nitude faster than in free space (i.e., the Purcell factor can be IR frequencies in materials like cubic boron nitride (cBN), gal- 6 7 of order 10 to 10 ). Because the rate of emission of a pair of lium phosphide, and many others (32). Much effort has been polaritons depends on two factors of the LDOS, the emission devoted to demonstrating the coupling and propagation of these of pairs naively can be maximally enhanced by 12 or 13 orders modes. In this work, we focus on the potential of using the of magnitude. Our detailed theory of two-photon emission into extreme local density of states (LDOS) in the vicinity of polar PhPs corroborates this intuition. dielectrics to efficiently access a wide range of light–matter inter- Although Purcell factors for one-photon emission have been actions. Before moving on to a quantitative description of the predicted in ref. 31 in the context of SiC, and in ref. 9 in the two PhP emission process considered here, we note that through- context of hBN, we clarify the basic physics of why the LDOS out the text, we use the terms “two-polariton emission” and is so high in the first place with transparent physical formulae, “two-photon emission” interchangeably. This is because the PhP and show that Purcell factors >10 are to be expected. As is well modes are described fully in terms of propagating modes of BC Fig. 1. General scheme for accessing two-photon transitions at high efficiency. (A, Left) Typical situ- ation for an emitter: An emitter may have many choices for a transition, but the relatively high- frequency single-photon dipole transition is chosen. (A, Right) When coupling that same excited elec- tron to PhPs in a polar dielectric, the electron can be made to prefer a two-polariton spontaneous emis- sion. (B) Color plot of the imaginary part of the p-polarized reflectivity of 5-nm-thick hBN evalu- ated at the upper RS band. The center of the sharp peaks signify the dispersion relation. (C) Same for SiC in its sole RS band. 13608 j www.pnas.org/cgi/doi/10.1073/pnas.1713538114 Rivera et al. known, the spontaneous emission rate of an emitter in free space ings and predicts a potentially even greater degree of confine- at frequency ! (wavelength  ) goes like: ment for thinner slabs. That this effect is more visible for thinner 0 0 films of polar dielectrics is because making the slab thinner pushes the dispersion toward higher wavevectors. A semiinfinite slab of = !  (k a ) ; [1] 0 0 0 such materials would have very weak confinement, as pointed e 2 out, for example, in ref. 25. All this said, an impediment toward where = is the fine-structure constant, k = = , 4 ~c c 0 0 experimentally observing even higher confinements comes from and a is the emitter size. The dimensionless factor (k a ) the limited resolution of scattering-scanning near-field opti- 7 9 is of order 10 10 for typical atoms and molecules and cal microscopy techniques. Perhaps complementary approaches is indicative of the weak coupling between light and matter in can make these extremely confined modes more visible. free space. On the other hand, when the emitter is placed at a distance z from a polar dielectric supporting PhP modes with 0 Two-Photon Emission Enhancement Theory polariton wavelength  and group velocity v , the emission PhP g Moving to two-photon emission, while there is no closed-form rate will scale like expression for the two-photon emission rate of a general emitter 4 4z (because it depends on the detailed level diagram of the emit- 2 0 !  (k a ) exp PhP 0 PhP PhP ter), the emission rate in the absence of radiative cascades will PhP generically scale as: c  4z 0 0 2 4 exp ; [2] 0  (k a ) !  ; [3] 0;2Ph 0 0 g PhP PhP where ! is the bandwidth of the two-photon emission spec- 2 e where k = ,  , and we have omitted PhP PhP trum. Note that in free space, it is generally of the order of half 4 ~v PhP 0 g the transition frequency, but in the situations we consider, it is dimensionless factors that are O (1). The exponential arises typically much less, leading to some suppression relative to what from the evanescent tails of these PhP modes, and the sharp- one might expect. The emission is into a continuum of frequency ness of those tails requires the emitter to be within 10 nm pairs (!; ! !) satisfying energy conservation. Such a result is from the surface of the polar dielectric. Assuming that indeed 0 corroborated by more detailed second-order perturbation theory the emitter is in close enough proximity, the Purcell enhance- calculations. ment is governed by the factor (k a ) , which is an impedance PhP In the presence of a surface supporting PhPs, the rate of two- matching factor, and the factor which is a “slow-light” or photon/two-polariton emission per unit frequency goes as (ignor- ing for now evanescent tails): density-of-states-enhancement factor. Since both of these prop- erties can be easily gleaned from the dispersion relation of the 2PhP 2 4 (k a ) ; [4] PhPs, we now estimate the possible LDOS enhancements using PhP PhP d! the dispersion relations of SiC and hBN plotted in Fig. 1 B and which, according to the estimates from the previous section C. As can be seen from the dispersion relations, the wavelength would lead to enhancement of the emission spectrum of 10 of the polariton can be >100 times shorter than the wavelength for an emitter sufficiently close to the surface. By virtue of Eq. 4, of light in free space. When the confinement factor is 100, the it follows that for PhPs with a bandwidth of 1/10 of the transi- group velocity is well over 100 times slower than the velocity of tion frequency, the total rate enhancement is 10 . Going into light (due to the highly dispersive nature of these materials), and a more rigorous level of detail, we show in Supporting Informa- the corresponding Purcell factors for a sufficiently close emit- tion that for an s ! s transition, the spectral enhancement fac- ter tend toward 10 . We emphasize that, although our result in tor, defined as the ratio of the PhP emission spectrum, , to the Eq. 2 is just an estimate, we have calculated the Purcell factors d! using three approaches: a mode quantization approach assuming free-space emission spectrum is given by: d! lossless polar dielectrics; a macroscopic quantum electrodynam- 2PhP ics approach using dyadic Green’s functions, taking into account 1 d! Spectral Enhancement = = F (!)F (! !); [5] p p 0 medium losses; and numerically in a finite-difference frequency- 0;2Ph d! domain approach by placing a dipole emitter near polar dielec- tric surfaces. All three approaches are in agreement and corrob- where F (!) is the Purcell factor for a dipole perpendicular to orate the estimates provided here, lending support to all of the the surface. As is well known, the expression for the Purcell fac- subsequent results. tor for a z-polarized dipole in the geometry of Fig. 1A is simply: To conclude this section, we briefly comment on the experimen- 2 2qz tal state of the art. For substantially thicker films (of order 100-nm F (!) = dq q e Im r (q; !) [6] p p 2k thickness), confinement factors >80 have already been observed in hBN associated with third-order modes, corresponding to r is the p -polarized reflectivity of the air-slab-substrate system polariton wavelengths of 80 nm (34). Even larger wavelength (plotted in Fig. 1 B and C), and k = is the free-space pho- confinements (up to 200) have been claimed in nanometric- ton wavevector (9, 35). We note that the two-photon spectrum scale SiC nanoresonators (31). In these SiC nanoresonators, the is always symmetric with respect to reflection about half of the authors predicted on the basis of the measured quality factor and transition energy. This is because emission at (!; ! !) is indis- simulated mode volume a Purcell enhancement of about 10 . tinguishable from emission at (! !; !). The theory we provide here is fully consistent with these find- Before turning to a discussion of our results, we discuss the valid- ity of perturbation theory, which we use to arrive at the results above. The main result of this work is that two-phonon-polariton emission beats one-photon emission. Were it also the case that In ref. 33, it was suggested that perhaps the dimensionless factor could be much less third- and higher-order emission processes were more impor- than 1 as a result of the strong phononic character of the excitations, thus limiting tant than second-order processes, it would be the case that our the degree of enhancement. Being quantitative, one finds that this is not the case. Although most of the electromagnetic energy (incorporating the mechanical energy calculations based on second-order perturbation theory would of vibrations) resides in the polariton sustaining slab, it is not much different from not be representative of the full dynamics of the electron. We what happens in plasmonics, where most of the energy is inside the metal. Additionally, claim that this is not the case in our work and that two-photon because of the dispersiveness of the polar dielectric, the group velocity becomes much emission dominates the perturbation series. An explicit estimate less than the phase velocity, leading to a larger density of states enhancement than one is provided in the final section of Supporting Information. To would get in a comparable plasmonic system. This compensates for the strong phononic component of the excitations. summarize those estimates, the only reason that two-polariton Rivera et al. PNAS j December 26, 2017 j vol. 114 j no. 52 j 13609 APPLIED PHYSICAL SCIENCES Fig. 2 presents a system where the two-photon emission is AB made dominant over all other emission pathways. We consider (for concreteness) the example of a hydrogen atom above a z = 5 nm cBN slab and present the two-photon spectral enhancement. We z = 10 nm choose cBN, a material different from the ones discussed thus z = 25 nm far, because in cBN, the Reststrahlen (RS) band coincides with a potential two-photon process in the hydrogen atom. Never- theless, the basic physical arguments presented in the previous section still apply—just at a different frequency range. cBN, in CD contrast to hBN, is not hyperbolic, and thus its dispersion is more similar to that of SiC. The hydrogen transition we con- sider is the 5s to 4s two-photon transition at a transition energy of 2,468 cm (4.05 m), so that the energies of the emitted polari- tons (8 m) fit in the RS band where the Purcell factor is very high (> 10 ) (Fig. 2A). In Fig. 2A, we compute the p -polarized Purcell spectrum for a first-order dipole transition for atom- surface separations of 5, 10, and 25 nm to get an order of magni- tude estimate for the rates of these competing dipole transitions. Fig. 2. Making two-photon emission dominant. (A) Purcell spectra for a At 5 nm, the fastest competing transition occurs with a lifetime z-polarized dipole above 10-nm-thick cBN at atom-surface separations of 5, of order 100 ns. At 10 and 25 nm, the order of magnitude rates 10, and 25 nm. (B–D) Two-photon Purcell spectra for an s ! s transition as of the competing E1 transitions are 1 per s. Interestingly, one a function of photon frequency ! for the same set of atom-surface separa- also sees that just outside the RS band, there can be strong off- tions. Blue denotes the Purcell spectra with losses accounted for, and orange resonant enhancement due to the tail of the resonance lineshape denotes the Purcell spectra for cBN with 100weaker losses. In B–D, we note of the Purcell spectrum. Although we do not make use of this both the overall two-photon transition rate between the 5s and 4s states of fact in the paper, one may also consider using these high Purcell hydrogen and the corresponding radiative ratio ( =). The permittivity of factors to realize the results of this work. the substrate is taken to be 2, and the damping constant for cBN is taken to 1 12 1 be 5 cm (10 s ). In Fig. 2 B–D, we compute the enhancement of the spectrum of two-photon emission (relative to free-space) from 5s to 4s (due to cBN PhPs), the lifetime of the two-photon transition, emission is important is that first-order emission is only negli- and the r -values, the last of which we define as the ratio of gibly enhanced by polaritons for the transitions we consider. If the decay rate computed assuming no losses ( ) to the decay first-order emission were enhanced, we would find that it would rate computed with losses taken into account (). The lossless completely dominate two-polariton emission by >3 orders of decay rates are computed in Supporting Information based on magnitude. Following this logic, even if third-order emission is a mode expansion formalism and are fully consistent with our enhanced by polaritons, it will be much weaker than second- calculation taking losses into account in the limit of vanishing order emission (by 3 orders of magnitude) since the second- losses. The r -value introduced here is a measure of the extent order process is already greatly enhanced by polaritons. That to which quenching dominates the decay dynamics insofar as a said, it would be very interesting to use the idea proposed in this low r value suggests loss-dominated decay. An r value of nearly work to make three- or higher-order emission dominant. 1 suggests that losses have little impact on the decay rate. (That said, the r -value can be >1 because losses can affect the den- Extreme Enhancement of Polariton-Pair Emission Rates sity of states of polariton modes, as is known from the Purcell Having provided the general theory of two-photon emission effect in resonant cavities.) In Fig. 2 B–D, we see that the life- enhancements and estimates for the degree of the enhance- times of two-photon spontaneous emission for an emitter 5, 10, ments, we now show by direct calculation that not only is the and 25 nm away from the surface of 10 nm thick cBN are 1.7 ns, enhancement of two-photon emission very high, but it can be 68 ns, and 28 s, respectively. Their r -values are 0.51, 1.02, made higher than the rate of competing one-photon processes and 1.12, respectively. Thus, two-photon spontaneous emission as a result of the narrow spectral window in which the LDOS of into PhPs can be over an order of magnitude faster than single- the material is so high. photon dipole transitions, computed by taking the one-photon Fig. 3. Hyperbolicity and two photon emitters in multiple bands. Two-polariton spectral enhancement defined as in Eq. 6 for a spherical emitter as a function of transition frequency [1,600 (Left), 2,300 (Center), and 3,000 (Right) cm ], plotted on log scale. The spectral enhancement is plotted with respect to emission frequency and atom-surface separation between 5 and 10 nm. Hyperbolicity allows for enhancement over a large range of frequencies compared with isotropic systems. Moreover, distance can be used to tune the width of the spectrum. 13610 j www.pnas.org/cgi/doi/10.1073/pnas.1713538114 Rivera et al. Table 1. A summary of the dependence of the angular spectrum in the lower RS band, two photons in the upper RS band, or one of two-photon radiation as a function of initial and final electronic in the upper RS band and one in the lower RS band. A hypothet- states for a few selected initial electronic states ical material having three separate RS bands would offer six fre- Transition Angular spectrum quency ranges for two-photon emission enhancement. We also found that increasing the emitter separation causes the emission s ! s sin spectrum not only to be weaker, but also narrower. This allows 2 0 d ! s sin ( +  ) xy one to tune not only the emission rate, but also the emission spec- 0 2 d ! s (cos  + cos  ) xz trum with atom-surface separation for emitters whose location 0 2 d ! s (sin  + sin  ) yz can potentially be precisely controlled. Before moving on to a discussion of the angular properties of the emitted PhP pairs, we quickly comment on the fact that hyperbolic hBN and its nonhy- perbolic relative cBN give similar degrees of enhancement. This free space rates and applying the one-photon Purcell factor. In a is interesting because the character of PhP modes in hBN is very system where there is only one competing dipole transition (at a different from in cBN: In hBN, optical rays can propagate in the rate of 1 per s), the branching ratio/probability of second-order slab, giving a mode that looks like a slab-waveguide mode in a 1:4710 decay at 10 nm would be  94%. This is in 7 7 dielectric waveguide. Meanwhile, in cBN, the mode is more like 1:4710 +0:110 a plasmonic mode in shape. Nevertheless, they both give similar sharp contrast to the situation in free space, where two-photon amounts of emission enhancement because, at the end of the day, spontaneous emission is 8–10 orders of magnitude slower. We the degree of enhancement depends on the modal volume of the thus conclude that, by using PhPs, it is possible to create a source PhPs, whose order of magnitude is set by the polariton in-plane of a pair of polaritons pairs with very high efficiency. We now wavelength. move on to describing the spectral properties of this quantum light source, both in frequency and angle. Finally, we consider the angular spectrum of emitted pho- tons. In Supporting Information, we derive the general result In Fig. 3, we consider the second-order spectral enhancement of Eq. 5 (plotted on log scale) for an emitter now near hBN that the angle and frequency spectrum of two-photon emission, as a function of the transition frequency of the emitter (! = S (!; ;  ), is proportional to: 1600; 2300; 3000 cm ), the emission frequencies (!), and the emitter-surface separation (z = 5 10 nm). We chose a dif- 0 0   0 S (!; ;  )  e ^ ()e ^ ( )Tij (!) ; [7] i j ferent material than that of Fig. 2 to show explicitly that two- ij photon spectral enhancements similar to those in thin cBN are achievable in other materials (the spectral enhancement is of where the same order of magnitude as that in cBN (10 ) and also gn gn ne ne d d d d j j to show a large number of frequency bands where a two-photon i T (!) = + ; [8] ij emitter can be designed. What we found is that hBN offers very E E ~! E E ~(! !) e n e n 0 high spectral enhancement in three different frequency bands, as ab opposed to one in isotropic polar dielectrics, allowing for com- in which d denotes a dipole matrix element between states a patibility with many more atomic, molecular, or quantum well and b , n denotes an intermediate atomic state, g denotes the ground state, e denotes the excited state, and E is the energy of systems. The reason for this is intercombination: A two-photon i emission through near-field polaritons can occur via two photons the i th state. The e ^ () are the PhP polarizations in the vicinity Fig. 4. Using the shape of the wavefunction to control the angle spectrum of emitted polariton pairs. Plots of the angular spectrum S(! =2; ;  ) of two-photon emission as a function of the initial state of the electron for initial states s; d ; d ; d . xy xz yz Rivera et al. PNAS j December 26, 2017 j vol. 114 j no. 52 j 13611 APPLIED PHYSICAL SCIENCES 1 of the emitter, given by (cos ; sin ; i ) in the high-confine- designing all other one-photon transitions to fall out of the band. Yet another advantage of these artificial atomic systems is their ment limit. relatively large sizes, allowing for impedance matching to be real- The angular dependence of the spectrum in Eq. 2 will lead to ized which much less requisite polariton confinement. very different angular spectra for different transitions. In Table In this work, we focused our attention on quantum optics at 1, we show the angular spectrum as a function of different tran- mid-IR/terahertz frequencies, a relatively undeveloped field, but sitions (at ! = ! =2). Strictly speaking, the angular spectrum is nonetheless one which would be rather exciting to develop. Mov- frequency-dependent. However, due to the narrowness of the ing beyond the mid-IR to the near-IR and eventually visible, it may RS band(s), this can be neglected, and thus we only consider the be possible to realize the effects we describe here using “shaped spectrum at half the transition frequency. Remarkably, just by polaritonic media” (such as nanoresonators of graphene or plas- changing the initial state of the system, one can change whether monic crystals of graphene and other 2D plasmonic materials) the polariton pairs are preferentially emitted in the same direc- supporting (respectively) narrow spectral response or photonic tion (as in Fig. 4, Upper Right and Lower Left) or in opposite bandgaps. We note that in all of these examples, the emitted directions (as in Fig. 4, Upper Left). There are a number of ways quanta are confined modes and do not escape into the far-field to preferentially populate a particular initial state. One is by unless outcoupled. Outcoupling efficiencies of a few percent were exciting the atoms with light of a fixed polarization. Another, demonstrated in highly confined graphene plasmons, and even appropriate in systems with less extreme degeneracy than hydro- higher efficiencies should be demonstrable through optimization. gen, is to simply excite the atoms with the appropriate frequency. We believe that the results presented in this work may have direct implications for spectroscopy (to infer electronic transitions Discussion which cannot be determined with conventional photons), sensors Although we have considered the hydrogen atom in our calcu- based on forbidden transitions, quantum radiation sources (on- lations, this was just for concreteness, and the physical mecha- demand generation of single photons and potentially entangled nism for efficient two-photon emitters can readily be extended pairs of photons), new platforms for quantum nonlinear optics, to many atomic and molecular systems. Here, we propose some the possibility of realizing nonlinearities at the single-photon emitter platforms for testing the predictions of our theory. For level, the ability to turn narrow-band emitters into broadband atoms, there are many which have a level structure conducive emitters, the ability to turn narrow-band absorbers into broad- to the situation described throughout this paper. For example, band absorbers, and, most generally, the ability to completely in the lithium atom there is a d ! s transition at 1,611 cm , reshape the ostensibly fixed optical properties of materials. which can potentially occur via emission of a pair of PhPs. All other competing transitions fall well outside of the RS bands of ACKNOWLEDGMENTS. The authors thank Prof. E. Ippen, J. J. Lopez, and hBN, just as in the example we discussed in Fig. 2. More gener- Prof. B. Zhen for fruitful discussions. Research was supported as part of ally, as all atoms have electronic transitions in the mid-IR, our the Army Research Office through the Institute for Soldier Nanotechnolo- results should apply to a large portion of the periodic table. It gies under Contract W911NF-13-D-0001 [photon management for devel- may also be possible that vibrational transitions may be used oping nuclear-TPV (thermophotovoltaics) and fuel-TPV mm-scale systems]. to observe these effects, although one obstacle toward realizing Research was also supported as part of the S3TEC, an Energy Frontier Research Center funded by the US Department of Energy under Grant DE- these effects with molecular vibrations is the generally low mul- SC0001299 [for fundamental photon transport related to solar TPVs and tipole moments associated with vibrational transitions. Another solar TEs (thermoelectrics)]. I.K. was supported in part by Marie Curie Grant exciting possibility is using emitters whose level structure is des- 328853-MC-BSiCS. N.R. was supported by Department of Energy Fellowship ignable, such as quantum dots or wells. There, one can envision DE-FG02-97ER25308. G.R. was supported by a fellowship of The Belgian designing a two-photon transition to fall in the RS bands while American Educational Foundation and Wallonie-Bruxelles International. 1. Craig DP, Thirunamachandran T (1984) Molecular Quantum Electrodynamics: An 19. Hillenbrand R, Taubner T, Keilmann F (2002) Phonon-enhanced light–matter interac- Introduction to Radiation-Molecule Interactions (Academic, London). tion at the nanometre scale. Nature 418:159–162. 2. Berestetskii VB, Lifshitz EM, Pitaevskii LP (1982) Quantum Electrodynamics (Butter- 20. Dai S, et al. (2014) Tunable phonon polaritons in atomically thin van der Waals crystals worth-Heinemann, Oxford), Vol 4. of boron nitride. Science 343:1125–1129. 3. 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Nat Photon 2:238–241. hyperbolic material hexagonal boron nitride. Nat Commun 5:5221. 14. Nevet A, et al. (2010) Plasmonic nanoantennas for broad-band enhancement of two- 31. Caldwell JD et al. (2013) Low-loss, extreme subdiffraction photon confinement via sil- photon emission from semiconductors. Nano Lett 10:1848–1852. icon carbide localized surface phonon polariton resonators. Nano Lett 13:3690–3697. 15. Hayat A, Nevet A, Ginzburg P, Orenstein M (2011) Applications of two-photon 32. Caldwell JD, et al. (2015) Low-loss, infrared and terahertz nanophotonics using sur- processes in semiconductor photonic devices: Invited review. Semicond Sci Technol face phonon polaritons. Nanophotonics, https://doi.org/10.1515/nanoph-2014-0003. 26:083001. 33. Khurgin JB (2015) How to deal with the loss in plasmonics and metamaterials. Nat 16. Ota Y, Iwamoto S, Kumagai N, Arakawa Y (2011) Spontaneous two-photon emission Nanotechnol 10:2–6. from a single quantum dot. Phys Rev Lett 107:233602. 34. Giles AJ, et al. (2017) Ultra-low-loss polaritons in isotopically pure materials: A new 17. Munoz ˜ CS, et al. (2014) Emitters of n-photon bundles. Nat Photon 8:550–555. approach. arXiv:1705.05971. 18. Rivera N, Kaminer I, Zhen B, Joannopoulos JD, Soljaci ˇ c ´ M (2016) Shrinking light to 35. Koppens FH, Chang DE, Garcia de Abajo FJ (2011) Graphene plasmonics: A platform allow forbidden transitions on the atomic scale. Science 353:263–269. for strong light–matter interactions. Nano Lett 11:3370–3377. 13612 j www.pnas.org/cgi/doi/10.1073/pnas.1713538114 Rivera et al. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Proceedings of the National Academy of Sciences of the United States of America Pubmed Central

Making two-photon processes dominate one-photon processes using mid-IR phonon polaritons

Rivera, Nicholas; Rosolen, Gilles; Joannopoulos, John D.; Kaminer, Ido; Soljačić, Marin
Proceedings of the National Academy of Sciences of the United States of America , Volume 114 (52) – Dec 12, 2017
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Abstract

Making two-photon processes dominate one-photon processes using mid-IR phonon polaritons a,1 a,b a,1 a,c a Nicholas Rivera , Gilles Rosolen , John D. Joannopoulos , Ido Kaminer , and Marin Soljaci ˇ c ´ a b Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; Micro- and Nanophotonic Materials Group, University of Mons, 7000, Mons, Belgium; and Department of Electrical Engineering, Technion–Israel Institute of Technology, Haifa 32000, Israel Contributed by John D. Joannopoulos, November 7, 2017 (sent for review August 3, 2017; reviewed by Dmitri Basov and Marinko Jablan) Phonon polaritons are guided hybrid modes of photons and opti- taneous parametric down-conversion sources) have become a cal phonons that can propagate on the surface of a polar dielec- staple in quantum information protocols. From a more funda- tric. In this work, we show that the precise combination of con- mental perspective, the frequency spectrum of two-photon spon- finement and bandwidth offered by phonon polaritons allows taneous emission can be broad, with a spectral width of the order for the ability to create highly efficient sources of polariton pairs of the transition frequency itself. This is in sharp contrast to one- in the mid-IR/terahertz frequency ranges. Specifically, these polar photon emission, which is spectrally very sharp in the absence dielectrics can cause emitters to preferentially decay by the emis- of external broadening mechanisms. That implies that if two- sion of pairs of phonon polaritons, instead of the previously dom- photon processes were sufficiently fast, an emitter with discrete inant single-photon emission. We show that such two-photon energy levels could be a source of light at continuous frequen- emission processes can occur on nanosecond time scales and can cies, which is in contrast to one of the first things that one learns be nearly 2 orders of magnitude faster than competing single- about quantum mechanics. photon transitions, as opposed to being as much as 8–10 orders In this work, we propose a design strategy for fast production of magnitude slower in free space. These results are robust to the of polariton pairs with quantum efficiencies potentially >90%. choice of polar dielectric, allowing potentially versatile implemen- In other words, we propose a scheme in which an excited emit- tation in a host of materials such as hexagonal boron nitride, sili- ter prefers to decay via the simultaneous emission of two quanta, con carbide, and others. Our results suggest a design strategy for despite the possibility of allowed single-photon decay pathways. quantum light sources in the mid-IR/terahertz: ones that prefer As a special case, we show how phonon polaritons (PhPs) in to emit a relatively broad spectrum of photon pairs, potentially boron nitride and silicon carbide (SiC) may allow for the design allowing for new sources of both single and multiple photons. of a kind of quantum optics source which emits polariton pairs at rates over an order of magnitude faster than any competing one-photon transitions, corresponding to lifetimes approaching two-photon processes j phonon polaritons j light–matter interactions j 1 ns in atomic-scale two-photon emitters. Our results not only Purcell effect j nanophotonics elucidate a technique for efficient sources of potentially entan- gled polariton pairs but also reveal a class of materials by which fundamental rule in light–matter interaction is that when to realize extreme regimes of light–matter interactions in which an excited electron in an atom has a choice between emit- conventionally unobservable high-order emissions become dom- ting one photon and emitting two photons simultaneously, it inant light–matter interactions. will nearly always decay via the emission of a single photon The general scheme for accessing high-efficiency second-order (1–3). The reason for this is impedance mismatch, or equiva- transitions is illustrated in Fig. 1. In free space (and near most lently, the mismatch in size between an emitter and its emitted radiation. Applying this idea to two-photon emission processes, Significance one realizes that two-photon emission suffers much more from impedance mismatch than one-photon emission, leading to its The recent discovery of nanoscale-confined phonon polaritons relative suppression. More quantitatively, the radiation resistance or impedance of a in polar dielectric materials has generated vigorous interest because it provides a path to low-loss nanoscale photonics at dipole radiator is proportional to (a=) , which up to other technologically important mid-IR and terahertz frequencies. In fundamental constants is proportional to (a=) (4). It turns this work, we show that these polar dielectrics can be used to out that the radiation rate of an atomic dipole is precisely pro- develop a bright and efficient spontaneous emitter of photon portional to (a=) , where a is the atomic size,  is the wave- pairs. The two-photon emission can completely dominate the length of the emitted light, and  1=137 is the fine-structure total emission for realistic electronic systems, even when com- constant. In contrast, the rate of a two-photon process scales as peting single-photon emission channels exist. We believe this 2 4 (a=) (5–7) and suffers much more than the one-photon pro- work acts as a starting point for the development of sources of cess when impedance is mismatched. In atomic systems, (a=)  entangled nano-confined photons at frequency ranges where 1=1000, and thus, two-photon emission in atoms is consistently photon sources are generally considered lacking. Additionally, slower than one-photon emission by more than eight orders of we believe that these results add a dimension to the great magnitude. It is because of this simple scaling argument that two- promise of phonon polaritonics. photon processes are considered insignificant and can thus almost always be ignored for the purposes of analyzing the dynamics of Author contributions: N.R., I.K., and M.S. designed research; N.R., G.R., I.K., and M.S. performed research; G.R., J.D.J., I.K., and M.S. analyzed data; and N.R., G.R., J.D.J., I.K., excited emitters. It is also because of this simple scaling argu- and M.S. wrote the paper. ment that, while conventional (one-photon) spontaneous emis- Reviewers: D.B., Columbia University; and M.J., University of Zagreb. sion engineering is a paradigm in quantum nanophotonics (8– 12), similar engineering has not been nearly as actively pursued The authors declare no conflict of interest. for two-photon spontaneous emission processes (13–18). This open access article is distributed under Creative Commons Attribution- Nevertheless, two-photon spontaneous emission processes NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). have several distinctive features which make them very desirable 1 To whom correspondence may be addressed. Email: [email protected] or nrivera@ to access. For example, the photons emitted in such a process mit.edu. are entangled due to energy and angular momentum conserva- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tion. For this reason, sources of entangled photons (such as spon- 1073/pnas.1713538114/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1713538114 PNAS j December 26, 2017 j vol. 114 j no. 52 j 13607–13612 APPLIED PHYSICAL SCIENCES nanophotonic structures), an emitter given a choice between dif- Maxwell’s equations. The primary difference between them and ferent transition pathways will generally take a single-photon free-space photons is that they are highly confined and cannot dipole (E1) transition over other choices. However, if we have escape to spatial infinity in the direction transverse to the polar highly confined modes (to impedance match) over a sufficiently dielectric slab. narrow frequency band, then we can create a situation in which To translate our intuition to a quantitative theory, we develop the single-photon E1 transition is negligibly enhanced, while the a formalism to compute the rates of two-photon transitions for two-photon transition is highly enhanced (>10 orders of mag- emitters placed near films supporting PhP modes (Supporting nitude relative to the enhancement of the competing E1 transi- Information). Our main assumption is that these PhP support- tion), so much so that the two-photon transition is strongly pre- ing materials are well-described by a local Lorentz oscillator ferred. As we will now demonstrate, one of the recent material model (although we allow for the permittivity to be complex advances that may admit the construction of such an emitter is and anisotropic). Our permittivity data are taken from ref. 32. the discovery of highly confined PhPs in polar dielectrics. Agreement with a local model has been seen in hBN films as thin Polar dielectrics like hexagonal boron nitride (hBN) and SiC as 1 nm, although only the long-wavelength part of the disper- have been the subject of significant attention over the past few sion was measured (20). Nevertheless, we will see that the results years due to their ability to confine electromagnetic energy in we arrive at should be achievable for thicknesses between 5 and small volumes (length scales <5 nm have been theoretically sug- 10 nm, and for polariton wavelengths of a few tens of nanome- gested) (19–32). Moreover, the transversely guided PhPs of these ters, where a local model should certainly suffice. materials have extremely strong spatial confinement only over a narrow spectral band of a few terahertz. Combined with the fact Extreme LDOS Near Polar Dielectrics that these PhPs can have substantially lower losses than other The main operating principle behind our results is that the surface excitations like plasmons (33) (albeit with lower veloci- LDOS of the electromagnetic field in the vicinity of PhP sup- ties and similar propagation lengths), they may provide a poten- porting materials is extremely high. This LDOS can be so high tially exciting platform for nanoscale optics in the technologi- that the emission rate of a dipole within nanometer distances cally interesting IR/terahertz regime. Going beyond hBN and from the surface of the material can be 6 or 7 orders of mag- SiC, it should also be possible to exploit PhPs at mid-IR/far- nitude faster than in free space (i.e., the Purcell factor can be IR frequencies in materials like cubic boron nitride (cBN), gal- 6 7 of order 10 to 10 ). Because the rate of emission of a pair of lium phosphide, and many others (32). Much effort has been polaritons depends on two factors of the LDOS, the emission devoted to demonstrating the coupling and propagation of these of pairs naively can be maximally enhanced by 12 or 13 orders modes. In this work, we focus on the potential of using the of magnitude. Our detailed theory of two-photon emission into extreme local density of states (LDOS) in the vicinity of polar PhPs corroborates this intuition. dielectrics to efficiently access a wide range of light–matter inter- Although Purcell factors for one-photon emission have been actions. Before moving on to a quantitative description of the predicted in ref. 31 in the context of SiC, and in ref. 9 in the two PhP emission process considered here, we note that through- context of hBN, we clarify the basic physics of why the LDOS out the text, we use the terms “two-polariton emission” and is so high in the first place with transparent physical formulae, “two-photon emission” interchangeably. This is because the PhP and show that Purcell factors >10 are to be expected. As is well modes are described fully in terms of propagating modes of BC Fig. 1. General scheme for accessing two-photon transitions at high efficiency. (A, Left) Typical situ- ation for an emitter: An emitter may have many choices for a transition, but the relatively high- frequency single-photon dipole transition is chosen. (A, Right) When coupling that same excited elec- tron to PhPs in a polar dielectric, the electron can be made to prefer a two-polariton spontaneous emis- sion. (B) Color plot of the imaginary part of the p-polarized reflectivity of 5-nm-thick hBN evalu- ated at the upper RS band. The center of the sharp peaks signify the dispersion relation. (C) Same for SiC in its sole RS band. 13608 j www.pnas.org/cgi/doi/10.1073/pnas.1713538114 Rivera et al. known, the spontaneous emission rate of an emitter in free space ings and predicts a potentially even greater degree of confine- at frequency ! (wavelength  ) goes like: ment for thinner slabs. That this effect is more visible for thinner 0 0 films of polar dielectrics is because making the slab thinner pushes the dispersion toward higher wavevectors. A semiinfinite slab of = !  (k a ) ; [1] 0 0 0 such materials would have very weak confinement, as pointed e 2 out, for example, in ref. 25. All this said, an impediment toward where = is the fine-structure constant, k = = , 4 ~c c 0 0 experimentally observing even higher confinements comes from and a is the emitter size. The dimensionless factor (k a ) the limited resolution of scattering-scanning near-field opti- 7 9 is of order 10 10 for typical atoms and molecules and cal microscopy techniques. Perhaps complementary approaches is indicative of the weak coupling between light and matter in can make these extremely confined modes more visible. free space. On the other hand, when the emitter is placed at a distance z from a polar dielectric supporting PhP modes with 0 Two-Photon Emission Enhancement Theory polariton wavelength  and group velocity v , the emission PhP g Moving to two-photon emission, while there is no closed-form rate will scale like expression for the two-photon emission rate of a general emitter 4 4z (because it depends on the detailed level diagram of the emit- 2 0 !  (k a ) exp PhP 0 PhP PhP ter), the emission rate in the absence of radiative cascades will PhP generically scale as: c  4z 0 0 2 4 exp ; [2] 0  (k a ) !  ; [3] 0;2Ph 0 0 g PhP PhP where ! is the bandwidth of the two-photon emission spec- 2 e where k = ,  , and we have omitted PhP PhP trum. Note that in free space, it is generally of the order of half 4 ~v PhP 0 g the transition frequency, but in the situations we consider, it is dimensionless factors that are O (1). The exponential arises typically much less, leading to some suppression relative to what from the evanescent tails of these PhP modes, and the sharp- one might expect. The emission is into a continuum of frequency ness of those tails requires the emitter to be within 10 nm pairs (!; ! !) satisfying energy conservation. Such a result is from the surface of the polar dielectric. Assuming that indeed 0 corroborated by more detailed second-order perturbation theory the emitter is in close enough proximity, the Purcell enhance- calculations. ment is governed by the factor (k a ) , which is an impedance PhP In the presence of a surface supporting PhPs, the rate of two- matching factor, and the factor which is a “slow-light” or photon/two-polariton emission per unit frequency goes as (ignor- ing for now evanescent tails): density-of-states-enhancement factor. Since both of these prop- erties can be easily gleaned from the dispersion relation of the 2PhP 2 4 (k a ) ; [4] PhPs, we now estimate the possible LDOS enhancements using PhP PhP d! the dispersion relations of SiC and hBN plotted in Fig. 1 B and which, according to the estimates from the previous section C. As can be seen from the dispersion relations, the wavelength would lead to enhancement of the emission spectrum of 10 of the polariton can be >100 times shorter than the wavelength for an emitter sufficiently close to the surface. By virtue of Eq. 4, of light in free space. When the confinement factor is 100, the it follows that for PhPs with a bandwidth of 1/10 of the transi- group velocity is well over 100 times slower than the velocity of tion frequency, the total rate enhancement is 10 . Going into light (due to the highly dispersive nature of these materials), and a more rigorous level of detail, we show in Supporting Informa- the corresponding Purcell factors for a sufficiently close emit- tion that for an s ! s transition, the spectral enhancement fac- ter tend toward 10 . We emphasize that, although our result in tor, defined as the ratio of the PhP emission spectrum, , to the Eq. 2 is just an estimate, we have calculated the Purcell factors d! using three approaches: a mode quantization approach assuming free-space emission spectrum is given by: d! lossless polar dielectrics; a macroscopic quantum electrodynam- 2PhP ics approach using dyadic Green’s functions, taking into account 1 d! Spectral Enhancement = = F (!)F (! !); [5] p p 0 medium losses; and numerically in a finite-difference frequency- 0;2Ph d! domain approach by placing a dipole emitter near polar dielec- tric surfaces. All three approaches are in agreement and corrob- where F (!) is the Purcell factor for a dipole perpendicular to orate the estimates provided here, lending support to all of the the surface. As is well known, the expression for the Purcell fac- subsequent results. tor for a z-polarized dipole in the geometry of Fig. 1A is simply: To conclude this section, we briefly comment on the experimen- 2 2qz tal state of the art. For substantially thicker films (of order 100-nm F (!) = dq q e Im r (q; !) [6] p p 2k thickness), confinement factors >80 have already been observed in hBN associated with third-order modes, corresponding to r is the p -polarized reflectivity of the air-slab-substrate system polariton wavelengths of 80 nm (34). Even larger wavelength (plotted in Fig. 1 B and C), and k = is the free-space pho- confinements (up to 200) have been claimed in nanometric- ton wavevector (9, 35). We note that the two-photon spectrum scale SiC nanoresonators (31). In these SiC nanoresonators, the is always symmetric with respect to reflection about half of the authors predicted on the basis of the measured quality factor and transition energy. This is because emission at (!; ! !) is indis- simulated mode volume a Purcell enhancement of about 10 . tinguishable from emission at (! !; !). The theory we provide here is fully consistent with these find- Before turning to a discussion of our results, we discuss the valid- ity of perturbation theory, which we use to arrive at the results above. The main result of this work is that two-phonon-polariton emission beats one-photon emission. Were it also the case that In ref. 33, it was suggested that perhaps the dimensionless factor could be much less third- and higher-order emission processes were more impor- than 1 as a result of the strong phononic character of the excitations, thus limiting tant than second-order processes, it would be the case that our the degree of enhancement. Being quantitative, one finds that this is not the case. Although most of the electromagnetic energy (incorporating the mechanical energy calculations based on second-order perturbation theory would of vibrations) resides in the polariton sustaining slab, it is not much different from not be representative of the full dynamics of the electron. We what happens in plasmonics, where most of the energy is inside the metal. Additionally, claim that this is not the case in our work and that two-photon because of the dispersiveness of the polar dielectric, the group velocity becomes much emission dominates the perturbation series. An explicit estimate less than the phase velocity, leading to a larger density of states enhancement than one is provided in the final section of Supporting Information. To would get in a comparable plasmonic system. This compensates for the strong phononic component of the excitations. summarize those estimates, the only reason that two-polariton Rivera et al. PNAS j December 26, 2017 j vol. 114 j no. 52 j 13609 APPLIED PHYSICAL SCIENCES Fig. 2 presents a system where the two-photon emission is AB made dominant over all other emission pathways. We consider (for concreteness) the example of a hydrogen atom above a z = 5 nm cBN slab and present the two-photon spectral enhancement. We z = 10 nm choose cBN, a material different from the ones discussed thus z = 25 nm far, because in cBN, the Reststrahlen (RS) band coincides with a potential two-photon process in the hydrogen atom. Never- theless, the basic physical arguments presented in the previous section still apply—just at a different frequency range. cBN, in CD contrast to hBN, is not hyperbolic, and thus its dispersion is more similar to that of SiC. The hydrogen transition we con- sider is the 5s to 4s two-photon transition at a transition energy of 2,468 cm (4.05 m), so that the energies of the emitted polari- tons (8 m) fit in the RS band where the Purcell factor is very high (> 10 ) (Fig. 2A). In Fig. 2A, we compute the p -polarized Purcell spectrum for a first-order dipole transition for atom- surface separations of 5, 10, and 25 nm to get an order of magni- tude estimate for the rates of these competing dipole transitions. Fig. 2. Making two-photon emission dominant. (A) Purcell spectra for a At 5 nm, the fastest competing transition occurs with a lifetime z-polarized dipole above 10-nm-thick cBN at atom-surface separations of 5, of order 100 ns. At 10 and 25 nm, the order of magnitude rates 10, and 25 nm. (B–D) Two-photon Purcell spectra for an s ! s transition as of the competing E1 transitions are 1 per s. Interestingly, one a function of photon frequency ! for the same set of atom-surface separa- also sees that just outside the RS band, there can be strong off- tions. Blue denotes the Purcell spectra with losses accounted for, and orange resonant enhancement due to the tail of the resonance lineshape denotes the Purcell spectra for cBN with 100weaker losses. In B–D, we note of the Purcell spectrum. Although we do not make use of this both the overall two-photon transition rate between the 5s and 4s states of fact in the paper, one may also consider using these high Purcell hydrogen and the corresponding radiative ratio ( =). The permittivity of factors to realize the results of this work. the substrate is taken to be 2, and the damping constant for cBN is taken to 1 12 1 be 5 cm (10 s ). In Fig. 2 B–D, we compute the enhancement of the spectrum of two-photon emission (relative to free-space) from 5s to 4s (due to cBN PhPs), the lifetime of the two-photon transition, emission is important is that first-order emission is only negli- and the r -values, the last of which we define as the ratio of gibly enhanced by polaritons for the transitions we consider. If the decay rate computed assuming no losses ( ) to the decay first-order emission were enhanced, we would find that it would rate computed with losses taken into account (). The lossless completely dominate two-polariton emission by >3 orders of decay rates are computed in Supporting Information based on magnitude. Following this logic, even if third-order emission is a mode expansion formalism and are fully consistent with our enhanced by polaritons, it will be much weaker than second- calculation taking losses into account in the limit of vanishing order emission (by 3 orders of magnitude) since the second- losses. The r -value introduced here is a measure of the extent order process is already greatly enhanced by polaritons. That to which quenching dominates the decay dynamics insofar as a said, it would be very interesting to use the idea proposed in this low r value suggests loss-dominated decay. An r value of nearly work to make three- or higher-order emission dominant. 1 suggests that losses have little impact on the decay rate. (That said, the r -value can be >1 because losses can affect the den- Extreme Enhancement of Polariton-Pair Emission Rates sity of states of polariton modes, as is known from the Purcell Having provided the general theory of two-photon emission effect in resonant cavities.) In Fig. 2 B–D, we see that the life- enhancements and estimates for the degree of the enhance- times of two-photon spontaneous emission for an emitter 5, 10, ments, we now show by direct calculation that not only is the and 25 nm away from the surface of 10 nm thick cBN are 1.7 ns, enhancement of two-photon emission very high, but it can be 68 ns, and 28 s, respectively. Their r -values are 0.51, 1.02, made higher than the rate of competing one-photon processes and 1.12, respectively. Thus, two-photon spontaneous emission as a result of the narrow spectral window in which the LDOS of into PhPs can be over an order of magnitude faster than single- the material is so high. photon dipole transitions, computed by taking the one-photon Fig. 3. Hyperbolicity and two photon emitters in multiple bands. Two-polariton spectral enhancement defined as in Eq. 6 for a spherical emitter as a function of transition frequency [1,600 (Left), 2,300 (Center), and 3,000 (Right) cm ], plotted on log scale. The spectral enhancement is plotted with respect to emission frequency and atom-surface separation between 5 and 10 nm. Hyperbolicity allows for enhancement over a large range of frequencies compared with isotropic systems. Moreover, distance can be used to tune the width of the spectrum. 13610 j www.pnas.org/cgi/doi/10.1073/pnas.1713538114 Rivera et al. Table 1. A summary of the dependence of the angular spectrum in the lower RS band, two photons in the upper RS band, or one of two-photon radiation as a function of initial and final electronic in the upper RS band and one in the lower RS band. A hypothet- states for a few selected initial electronic states ical material having three separate RS bands would offer six fre- Transition Angular spectrum quency ranges for two-photon emission enhancement. We also found that increasing the emitter separation causes the emission s ! s sin spectrum not only to be weaker, but also narrower. This allows 2 0 d ! s sin ( +  ) xy one to tune not only the emission rate, but also the emission spec- 0 2 d ! s (cos  + cos  ) xz trum with atom-surface separation for emitters whose location 0 2 d ! s (sin  + sin  ) yz can potentially be precisely controlled. Before moving on to a discussion of the angular properties of the emitted PhP pairs, we quickly comment on the fact that hyperbolic hBN and its nonhy- perbolic relative cBN give similar degrees of enhancement. This free space rates and applying the one-photon Purcell factor. In a is interesting because the character of PhP modes in hBN is very system where there is only one competing dipole transition (at a different from in cBN: In hBN, optical rays can propagate in the rate of 1 per s), the branching ratio/probability of second-order slab, giving a mode that looks like a slab-waveguide mode in a 1:4710 decay at 10 nm would be  94%. This is in 7 7 dielectric waveguide. Meanwhile, in cBN, the mode is more like 1:4710 +0:110 a plasmonic mode in shape. Nevertheless, they both give similar sharp contrast to the situation in free space, where two-photon amounts of emission enhancement because, at the end of the day, spontaneous emission is 8–10 orders of magnitude slower. We the degree of enhancement depends on the modal volume of the thus conclude that, by using PhPs, it is possible to create a source PhPs, whose order of magnitude is set by the polariton in-plane of a pair of polaritons pairs with very high efficiency. We now wavelength. move on to describing the spectral properties of this quantum light source, both in frequency and angle. Finally, we consider the angular spectrum of emitted pho- tons. In Supporting Information, we derive the general result In Fig. 3, we consider the second-order spectral enhancement of Eq. 5 (plotted on log scale) for an emitter now near hBN that the angle and frequency spectrum of two-photon emission, as a function of the transition frequency of the emitter (! = S (!; ;  ), is proportional to: 1600; 2300; 3000 cm ), the emission frequencies (!), and the emitter-surface separation (z = 5 10 nm). We chose a dif- 0 0   0 S (!; ;  )  e ^ ()e ^ ( )Tij (!) ; [7] i j ferent material than that of Fig. 2 to show explicitly that two- ij photon spectral enhancements similar to those in thin cBN are achievable in other materials (the spectral enhancement is of where the same order of magnitude as that in cBN (10 ) and also gn gn ne ne d d d d j j to show a large number of frequency bands where a two-photon i T (!) = + ; [8] ij emitter can be designed. What we found is that hBN offers very E E ~! E E ~(! !) e n e n 0 high spectral enhancement in three different frequency bands, as ab opposed to one in isotropic polar dielectrics, allowing for com- in which d denotes a dipole matrix element between states a patibility with many more atomic, molecular, or quantum well and b , n denotes an intermediate atomic state, g denotes the ground state, e denotes the excited state, and E is the energy of systems. The reason for this is intercombination: A two-photon i emission through near-field polaritons can occur via two photons the i th state. The e ^ () are the PhP polarizations in the vicinity Fig. 4. Using the shape of the wavefunction to control the angle spectrum of emitted polariton pairs. Plots of the angular spectrum S(! =2; ;  ) of two-photon emission as a function of the initial state of the electron for initial states s; d ; d ; d . xy xz yz Rivera et al. PNAS j December 26, 2017 j vol. 114 j no. 52 j 13611 APPLIED PHYSICAL SCIENCES 1 of the emitter, given by (cos ; sin ; i ) in the high-confine- designing all other one-photon transitions to fall out of the band. Yet another advantage of these artificial atomic systems is their ment limit. relatively large sizes, allowing for impedance matching to be real- The angular dependence of the spectrum in Eq. 2 will lead to ized which much less requisite polariton confinement. very different angular spectra for different transitions. In Table In this work, we focused our attention on quantum optics at 1, we show the angular spectrum as a function of different tran- mid-IR/terahertz frequencies, a relatively undeveloped field, but sitions (at ! = ! =2). Strictly speaking, the angular spectrum is nonetheless one which would be rather exciting to develop. Mov- frequency-dependent. However, due to the narrowness of the ing beyond the mid-IR to the near-IR and eventually visible, it may RS band(s), this can be neglected, and thus we only consider the be possible to realize the effects we describe here using “shaped spectrum at half the transition frequency. Remarkably, just by polaritonic media” (such as nanoresonators of graphene or plas- changing the initial state of the system, one can change whether monic crystals of graphene and other 2D plasmonic materials) the polariton pairs are preferentially emitted in the same direc- supporting (respectively) narrow spectral response or photonic tion (as in Fig. 4, Upper Right and Lower Left) or in opposite bandgaps. We note that in all of these examples, the emitted directions (as in Fig. 4, Upper Left). There are a number of ways quanta are confined modes and do not escape into the far-field to preferentially populate a particular initial state. One is by unless outcoupled. Outcoupling efficiencies of a few percent were exciting the atoms with light of a fixed polarization. Another, demonstrated in highly confined graphene plasmons, and even appropriate in systems with less extreme degeneracy than hydro- higher efficiencies should be demonstrable through optimization. gen, is to simply excite the atoms with the appropriate frequency. We believe that the results presented in this work may have direct implications for spectroscopy (to infer electronic transitions Discussion which cannot be determined with conventional photons), sensors Although we have considered the hydrogen atom in our calcu- based on forbidden transitions, quantum radiation sources (on- lations, this was just for concreteness, and the physical mecha- demand generation of single photons and potentially entangled nism for efficient two-photon emitters can readily be extended pairs of photons), new platforms for quantum nonlinear optics, to many atomic and molecular systems. Here, we propose some the possibility of realizing nonlinearities at the single-photon emitter platforms for testing the predictions of our theory. For level, the ability to turn narrow-band emitters into broadband atoms, there are many which have a level structure conducive emitters, the ability to turn narrow-band absorbers into broad- to the situation described throughout this paper. For example, band absorbers, and, most generally, the ability to completely in the lithium atom there is a d ! s transition at 1,611 cm , reshape the ostensibly fixed optical properties of materials. which can potentially occur via emission of a pair of PhPs. All other competing transitions fall well outside of the RS bands of ACKNOWLEDGMENTS. The authors thank Prof. E. Ippen, J. J. Lopez, and hBN, just as in the example we discussed in Fig. 2. More gener- Prof. B. Zhen for fruitful discussions. Research was supported as part of ally, as all atoms have electronic transitions in the mid-IR, our the Army Research Office through the Institute for Soldier Nanotechnolo- results should apply to a large portion of the periodic table. It gies under Contract W911NF-13-D-0001 [photon management for devel- may also be possible that vibrational transitions may be used oping nuclear-TPV (thermophotovoltaics) and fuel-TPV mm-scale systems]. to observe these effects, although one obstacle toward realizing Research was also supported as part of the S3TEC, an Energy Frontier Research Center funded by the US Department of Energy under Grant DE- these effects with molecular vibrations is the generally low mul- SC0001299 [for fundamental photon transport related to solar TPVs and tipole moments associated with vibrational transitions. Another solar TEs (thermoelectrics)]. 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Published: Dec 12, 2017

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