Strong light-matter coupling in quantum chemistry and quantum photonics

2.1 Quantum theory of photon-emitter interaction

We start by briefly discussing the light-matter interaction in the nonrelativistic limit [44], [45], [46], [47], [48]. For a system of nonrelativistic charges, the light-matter interaction Hamiltonian can be introduced by the minimal coupling principle in Coulomb gauge [48](p^i→p^i−eA^(ri)), where p^i is the momentum of the particle i and the vector potential of the electromagnetic field is denoted by A^. This gauge is also called the “momentum gauge”. In an alternative representation, the “multipole representation” or “length gauge”, the interaction is formulated in terms of electric displacement and magnetic fields D^(r) and B^(r) coupling to the matter quantities, which are the polarization P^(r) and the magnetization density M^(r), respectively [48].

In the dipole approximation where the wavelength of the photon is much longer than the physical extent of the system of charges, only the lowest-order multipole moment of the polarization P^, i.e. the dipole moment μ^=−e∑i=1Nr^i for N electrons, is considered. Here, the interaction Hamiltonian H^int between light and matter takes the following explicit form [48]:

where the electric displacement field operator D^ is evaluated at the center of charge r₀. In Eq. (1), ε^dip is a dipole self-energy term that appears as a result of the canonical transformation from the momentum gauge to the length gauge and emerges only in a fully quantum analysis of light-matter interactions in which the electromagnetic field is quantized [47], [49]. The dipole self-energy term does not contain photon creation and annihilation operators. Although this term does not play a direct role in emission and absorption processes due to the lack of photon operators, it has important consequences when considering energy shifts and nonperturbative phenomena. We refer the reader to Refs. [32], [36], [49], [50] for more details on this subtle discussion.

To analyze strong light-matter coupling, we consider a single molecule interacting with a low-loss single-mode optical cavity. Physically, this is realized by having a molecular transition being nearly on-resonance with a particular cavity mode. In the case where the optical modes are of low (but nonzero) loss, the electric displacement field operator D^(r) of the single-mode field can be expressed in terms of a normalized cavity mode F(r) and a photon annihilation (creation) operator a^(†) as [45]

where V denotes the mode volume of the cavity, i.e. the effective amount of space of the cavity mode. In general, a cavity mode is broadened by losses as expressed by a density of states, typically taken as Lorentzian function ρ(ω)=1πκ / 2(ω−ωc)2+(κ / 2)2, where κ is the photon dissipation rate and ω_c is the central frequency of the cavity mode. The prefactor ℏω2ε0V in Eq. (2) is the electric field of a single photon. It is equal to the root mean square of the quantized electric field in the electromagnetic vacuum and thus represents the strength of the quantum-fluctuating electric field that couples to an emitter.

Further, in the limit of a very low-loss cavity, much narrower than any relevant emitter linewidths, one usually replaces ρ(ω) with δ(ω – ω_c), writing the field operator in terms of a single mode. By emitter linewidth, we mean the full-width at half-maximum of the density of states of excited state in question. It is this single-mode expression that is used in the famous Rabi model [51], [52]. In the Rabi model, the matter part is described by a two-level system with the electronic ground state |g, the electronic excited state |e, and the electronic excitation energy ω_a. This two-level system is coupled to a single-cavity mode of frequency ω_c. The Rabi model takes the following form including the Rabi frequency Ω:

where σ^x and σ^z are Pauli matrices, respectively. The Rabi model of Eq. (3) does not contain the dipolar self-energy ε^dip of Eq. (1), as this term takes the form of a constant for a two-site model. We note that only recently an analytical solution to the Rabi model was found [53].

By adopting the rotating wave approximation, which neglects nonresonant terms in the interaction Hamiltonian of Eq. (3), we recover the Jaynes-Cummings model [54], [55]. The Jaynes-Cummings model can be analytically solved and thus has heavily contributed to the understanding of the fundamental nature of the light-matter interaction, e.g. see Ref. [55] and the references therein. In the Jaynes-Cummings model, the Rabi splitting, i.e. the splitting between the lower and upper polariton branches, is directly connected to the Rabi frequency and is given by 2Ω. We will discuss the magnitude of the Rabi frequency in the following.

2.2 When is light-matter coupling strong or ultra-strong?

An important quantity in characterizing the coupling between light and matter is the Rabi frequency Ω that enters the Rabi model of Eq. (3) and is defined by [9], [56]

The Rabi frequency contains the dipole moment μ of the particular electronic transition that forms the two-level system in the Rabi model and is proportional to the root-mean-square electric field of a single photon that effectively drives the molecule to interact with light.

In the wide field of light-matter interactions, the term “strong coupling” is usually used for two distinct situations. In one situation, the term refers to the situation where the cavity is of high enough quality such that the two-level system can emit and reabsorb a photon several times before it is irreversibly lost to the environment. Only if this is the case, experiments are able to clearly resolve the Rabi splitting in spectroscopic measurements. In another usage of the phrase, “strong coupling” sometimes refers to situations where Rabi splitting is so strong that the rotating wave approximation used to derive the Jaynes-Cummings model is not applicable anymore. This regime is commonly referred to as the ultra-strong or deep-strong coupling regime depending on the precise magnitude strength of the coupling between light and matter. We start by discussing the first situation and then analyzing the latter meaning.

Before going into details, we first mention a useful criterion from Ref. [47] regarding when the coupling of light and matter is strong between a discrete emitter state and a continuum of photonic modes. In Complement C_III of Chapter 3, the authors considered a discrete state coupled to a continuum of photon modes with width w₀, which represents the bandwidth of photon frequencies that interact with the emitter. In that chapter, the authors noted that when the calculated decay rate of the emitter Γ is much smaller than the width w₀, i.e. Γw₀ the dynamics of the combined system are well described by weak coupling exponential decay dynamics. In the opposite limit, Γw₀, the continuum looks like a discrete state to the emitter, and Rabi oscillations occur. We now show that this criterion allows to understand when strong coupling arises.

For “small” values of the Rabi frequency Ω, such that the (enhanced) linewidth of the emitter is much smaller than the cavity linewidth κ, the single-cavity mode appears to the emitter as a broad continuum, and the dynamics of an excited emitter is irreversible emission into cavity modes. Note that the “continuum” of photon frequencies seen by the emitter is represented by the continuous set of frequencies in which the density of states of the cavity mode is not negligible. For a cavity with a Lorentzian density of states, the decay rate κ of the cavity mode represents the bandwidth of the continuum. In this weakly coupled limit, the decay dynamics are well described by a first-order perturbation theory calculation of transition amplitudes from the state |e, 0 to |g, ω, where ω is a frequency in the support of the cavity density of states.

In this case, the rate of decay is given by Fermi’s Golden Rule, as expected when considering the dynamics of a discrete level coupled to a continuum. Performing this calculation, one finds that the rate of emission of a cavity photon by a molecule is given by Γ=4Ω2κ[57]. Expressing Ω in terms of the dipole moment, photon frequency, and photon mode volume, one finds that the rate of decay Γ=F_pΓ₀, where Fp=34π2Qn3V​/​λp3, is the famous Purcell factor [58]. In this formula, Q=ωcκ is the quality factor or Q factor of the cavity mode, whereas λp=2πcω is the photon wavelength, n is the index of refraction occupying the cavity, and V is the volume of the cavity mode. The main assumption in this formula is that the emitter line is concentric with the cavity line and that the emitter line is much narrower. We additionally note that if the free-space emitter linewidth is already much wider than the cavity linewidth (because the cavity is extremely low loss or the emitter is broadened by another mechanism such as Doppler or inhomogeneous broadening), the emitter will also be in a regime of weak coupling, as excitations are lost into the far field much faster than the emitter can interact with the cavity.

We refer to the “strong coupling” regime as the regime where the light-matter interaction is strong enough such that emitter coherently interacts with the cavity and can emit and reabsorb cavity photons to undergo Rabi oscillations between excited and ground states several times before the cavity photon is lost or before emission happens into the far field.

Physically, this regime, provided that ΓΓ₀ (where Γ₀ is the bare emitter decay rate), as is typically the case in strong coupling, should coincide with Γ≥κ, as expected from the criterion presented in Ref. [47]. From the expression Γ~Ω²/κ, one has that the onset of strong coupling should coincide with Ω≥κ. Strictly speaking, it is inconsistent to use a weak coupling result to determine the strong coupling condition. However, as the condition Γ~κ represents the case where the emitter and cavity linewidths match, invalidating irreversibility and weak coupling, one should expect this to be the regime where strong coupling sets in. Thus, to summarize, one is well into the strong coupling when Ωκ and ΩΓ₀. The transition to the strong coupling regime is when the upper and lower polariton lines are clearly spectroscopically resolvable. This occurs when the coupling Ω>12κ2+Γ2, as described in the review by Torma and Barnes [59], models the strong coupling as a coupled oscillator problem.

Historically, this was the first regime of strong coupling theoretically considered and experimentally observed in both few-emitter and collective contexts [21], [60], [61], [62], [63], [64], [65], [66]. Theoretically, the dynamics of the molecule in a cavity in this strong coupling regime is well described by a simple Jaynes-Cummings model provided that the Rabi oscillations are much slower than the cavity frequency and the rotating wave approximation is applicable [54], [55].

The second comparison Ω≥ω_c corresponds to a breakdown of the rotating wave approximation. This regime is also called the ultra-strong or deep-strong coupling regime [67], [68] and has been experimentally accessed in few-emitter systems only recently in the context of circuit QED systems [69], [70], [71]. The term “ultra-strong coupling” has been taken by several authors to mean Ω/ω_c≥0.1, whereas “deep-strong coupling” refers to Ω/ω≥1 [67].

In the above, we have implicitly assumed that only a single emitter is coupled to the cavity mode. If an ensemble of N emitters is coupled to the cavity, the Rabi model can be extended to the so-called Dicke model [72], [73]. In the Dicke model, the effective coupling constant (with dimensions of frequency) scales as Ωeff=NΩ. This square-root scaling with the number of emitters can lead to a very large enhancement of the coupling constant for a large number emitters [6], thus allowing experimentalists to avoid creating either very low-loss or very low mode-volume cavities.

We now discuss one more interesting possibility for strong coupling: strong coupling of an emitter to a continuum of photonic modes. On the one hand, as strong coupling appears to be related to reversibility, and a continuum of modes appears to imply irreversible coupling, the answer would appear to be that it is not possible to have strong coupling to a continuum of photonic modes. On the other hand, the strong coupling condition Ωκ (or Γκ) says that the condition for strong coupling is that the enhanced linewidth of the emitter is large compared to the frequency scale at which the photonic density of states varies. Thus, it should be possible, and in fact, this possibility has been considered theoretically in earlier works via a Laplace transform formalism [74]. Moreover, there are now recent experiments that study the coupling of superconducting qubits to a continuum of transmission line modes in the regime where the emission rate exceeds the transition frequency [29]. In general, it is currently understood that a useful criterion for the possibility of continuum ultra-strong coupling is that the spontaneous emission rate calculated from Fermi’s Golden Rule is comparable to the transition frequency, as this is when the rotating wave approximation breaks down [47].

2.3 Quantifying strong coupling

To conclude this section, we provide some estimates for dimensionless coupling constants of the form for different classes of systems. In general, recent developments in the field we review have focused on ultra-strong coupling as opposed to strong coupling. As a result, we focus on ultra-strong coupling constants. In estimates 1–3, the emitter we considered is coupled to a continuum of modes, so we must compare the enhanced decay rate either to the bandwidth of the continuum, if there is one, or the emitter frequency itself, the latter of which signals a breakdown of the rotating wave approximation. The decay rate is the natural choice for the numerator, as a Rabi frequency is typically not considered when coupling to a continuum. In estimate 4, we look at a single emitter in a cavity and estimate the strong and ultra-strong coupling constants to show that the condition for ultra-strong coupling is much more stringent even when one is well into the strong coupling regime. In estimates 5 and 6, we look at ensembles of emitters coupled to a common cavity mode and look at the ultra-strong coupling constant, comparing the Rabi frequency to the cavity frequency, to gauge the breakdown of the rotating wave approximation.

1. Hydrogen 2p to 1s transition: The rate of this transition is 6.25×10⁸ s⁻¹ and the natural frequency is given by 1.55×10¹⁶ rad/s. As a consequence, the effective coupling constant in this case would be 6.25×1081.55×1016≈2×10−4. This transition is well into the regime of weak coupling [75].

2. A dye molecule with a radiative transition frequency of 600 nm with a free-space emission lifetime of 10 ns experiencing a Purcell enhancement of 1000 due a plasmonic cavity, as recently shown in Ref. [76]. In that case, the coupling constant is approximately 0.006, still far into the weak coupling regime.

3. IR Rydberg-like 7500 nm transition between the 6p and 5s levels of a hydrogen atom coupled to highly confined phonon polaritons in a material such that the Purcell factor is on the order of 10⁶[77]. Further assume that the initial decay rate is Γ=10⁶ s⁻¹. In that case, the coupling constant defined by Γ/ω₀ is on the order of 0.06. However, in the specific case of phonon polaritons, the spectral band at which surface phonon polariton modes exist is narrow, about 1/10 of the transition frequency. In this case, the denominator of the coupling constant, which should represent the frequency bandwidth of the photonic density of states, should be about a factor of 10 smaller, making the coupling constant more like 0.2. In this case, it could be possible to observe substantial corrections to the weak coupling dynamics. Interestingly, this suggests that large dipole moment transitions in the IR, where high optical confinement and low losses are possible, are an almost ideal parameter range for strong coupling physics.

4. GaAs quantum dot with a 744 nm transition coupled to a photonic microcavity with a quality factor of 12.000 and a mode volume of 0.07 μm³: The coupling constant for conventional cavity strong coupling of Ω/κ is about 2. On the contrary, the coupling constant defined by Ω/ω₀ is on the order of 0.02. This shows that the conditions for cavity strong coupling are much less stringent than the conditions for ultra-strong coupling and the breakdown of the rotating wave approximation [64].

5. For molecular liquids that are strongly coupled to multiple IR cavity modes, the collective coupling constant Ω/ω=0.24 has been reached in Ref. [32].

6. Consider a stack of quantum wells (QWs) collectively coupled to a metallic cavity as in Ref. [36]. For the case of a stack of 25 QWs of width 32 nm with a first excited-state energy of 12 meV, the collective coupling constant Ω/ω is 0.48.

3.1 Theoretical tools for strong plasmon-enhanced light-matter interactions

We now move to a theoretical description regarding the interactions between emitters and plasmonic systems, such as nanoplasmonic cavities, thin films, or much more complicated geometries. To understand the coarse features of the experiments discussed above, one can largely follow the approach of Section 2: quantize the field of the plasmons by expanding the electric field in properly normalized plasmon modes F(r) and normalizing them such that the energy in the plasmonic mode is equal to ħω. However, such an approach is only strictly valid for very low-loss modes, although reasonable agreement with a more complex theory incorporating losses can be achieved even when the modes have a quality factor as low as 5 [83]. Incorporating losses introduces complications because, in the extreme near field of a lossy material, large deviations can arise from the single-mode picture depicted in Eq. (3) arising effectively from interactions in which an emitters dipole induces dipoles in a highly polarizable (conductive or lossy) material and transfers energy to the material dipoles. It turns out that the analysis of light-matter interactions for highly lossy materials is possible and falls under the so-called “macroscopic QED formalism”. Although we do not derive the basic formalism here, leaving that to dedicated reviews on the formalism such as Refs. [84], [85], we can quickly make the basic results plausible.

One reason for the failure of a mode expansion approach when losses are appreciable is that the completeness and orthonormality of the eigenmodes no longer necessarily hold. This is a known issue not just in light-matter interactions but simply classical electromagnetism, where convenient approaches based on modes fail when losses are present [86]. In that domain of classical electrodynamics, one can still understand the dynamics of fields and electromagnetic resonances through the dyadic Green’s function of the system [87], defined formally via G(r, r′, ω)≡(∇×∇−ε(r, ω)ω2c2)−1δ(r−r′)I3. From the dyadic Green’s function of the system, one can calculate the fields induced by an arbitrary arrangement of electric currents (equivalently, an arbitrary arrangement of electric dipoles). The real-valued electric field associated with an arbitrary arrangement of dipoles with spectrum j(r, ω) is given by 12∫dωω∫dr′iωμ0G(r, r′​, ω)j(r′, ω)e−iωt−c.c. It stands to reason that, if the current density in a material at a given point in space, at a given frequency, and in a given direction can be considered as an elementary excitation of a field, which in this case is a polarization field, then the electromagnetic field can be quantized in terms of these current operators and the classical dyadic Green’s function. This assumption is the essence of the macroscopic QED method. Following canonical commutation relations for the currents based on the linear response theory and the fluctuation-dissipation theorem (see Ref. [85]), the Schrödinger picture (t=0) quantized electric displacement field operator can be written as

In this expression, the f^ operators are rescaled dipole operators such that they satisfy canonical commutation relations: [f^i(r, ω), f^j†(r′, ω′)]=δ(r−r′)δ(ω−ω′)δij and ε(r, ω) is the dielectric function, which relates the macroscopic polarization to macroscopic electric fields. As in the “low-loss” case, the coupling Hamiltonian between an emitter and a lossy material in the long-wavelength approximation is the dipole Hamiltonian of Eq. (1). A key result from this formalism is that, if one computes the rate of spontaneous emission of material dipoles (anywhere in the material and in any direction) by an emitter of natural frequency ω₀, the spontaneous emission rate follows as [88]

This expression leads to a Purcell factor for an emitter with an n^-directed dipole moment of magnitude 6πcωn^ ⋅ Im G(r, r, ω0)⋅n^. Because, as discussed in Section 2, the emission rate, compared to a photon dissipation rate, can be used as a proxy for a coupling constant, in experimental work, this expression for Γ is commonly used to assess whether or not the strong coupling regime is reached. This was done in the experiment on strong coupling to a plasmonic mode. Thus, the calculation of the Green’s function of an optical structure allows to quantitatively assess plasmon-enhanced decay rates and evaluate whether the strong coupling regime was reached.

The Green’s function is determined by the dielectric function ε(r, ω) (currently assumed to be spatially local); thus, finding the Green’s function G(r, r, ω) boils down to finding the electric field at position r created by a dipole at position r and frequency ω, which is a classical electromagnetic problem. In the domain where the emitter is more than roughly 20 nm from the surface of the plasmonic cavity, and in the domain where the plasmonic cavity can be described by a local dielectric function, the relevant Green’s function can be calculated using standard numerical algorithms for solving Maxwell’s equations such as finite-difference time-domain (FDTD) methods [89] or boundary element methods [90]. These methods are applied in recent experiments such as those discussed above, both assuming a spatially local dielectric function for the NPoM [28], [76].

It is now well known experimentally [78], [91], [92], [93], [94] that for metal nanoparticles with sizes below 10 nm, nanoscopic metallic gaps, and very high wavevector plasmons such as in plasmonic film geometries, the effects of spatial dispersion become relevant and the electromagnetism must now be described by the nonlocal permittivity ε(r, r′, ω). In general, these effects are relevant once the spatial scale of the electromagnetic field variation is comparable to the Fermi wavelength of the underlying metal. Phrased differently, it matters when the momentum components of the electromagnetic field are comparable to the momentum of electrons in the underlying metal. It is in this regime that plasmons can, for example, induce strong nonvertical interband transitions in the underlying metal, allowing for an increased phase space of plasmon damping and lower plasmon lifetimes [95], [96], [97], [98]. Although this review is not dedicated to the treatment of nonlocality, leaving that for more comprehensive reviews such as Refs. [99], [100], [101], [102], [103], we note that there are many ways to treat nonlocality in metals, from hydrodynamic models [100] to Feibelman d-parameter approaches [104], [105] with input from time-dependent DFT (TDDFT) [103]. In any of these, once the dielectric function is known and the Green’s function is computed, it turns out that Eq. (6) still holds, but with the Green’s function that accounts for nonlocality. By and large, the effect of nonlocality on spontaneous emission enhancements is to reduce them due to increased damping that invariably comes with spatial nonlocality.

3.2 Novel light-matter coupling phenomena at the extreme nanoscale

In the rest of this section, we discuss systems that fall outside common treatments of emitter-plasmon interactions. In particular, we discuss methods that become relevant when the electromagnetic energy is confined to scales comparable to the length scale of an emitter. One reason to expect the physics of emission to change when electromagnetic fields are highly confined is hinted by the common treatment of emitters as dipoles. In principle, the emitter has to be described by a complicated transition current density determined by the spatial variation of the wave functions participating in the transition. Thus, in principle, it can (and does) have many multipole moments beyond the dipole moment. However, when the emitter is very small compared to the wavelength of light, the dipole component is strongly dominant. The validity of the long-wavelength approximation is given by the parameter [1]

where k is the wavevector of the optical field, λ is its corresponding wavelength, and a is the “emitter size”, which can be roughly parameterized, for example, by the transition multipole moments. In the regime where ka≪1, the rates of various electric multipolar transitions scale as Γ(En)~α(ka)2ⁿ⁺¹, where the terminology “En” denotes a 2ⁿ-pole transition (n=1 is dipole, n=2 is quadrupole, etc.). Taking a=1 nm and a typical free-space wavelength of 1 μm, (ka)²≈4×10⁻⁵, meaning that higher-order transitions become successively slower by this factor, rendering them irrelevant to the vast majority of optical experiments involving emitters.

Taking a=1 nm, the optical wavevector k corresponding to a complete breakdown of the dipole approximation is k=1 nm⁻¹ or λ=6 nm. These kinds of spatial scales are readily available in recent experiments involving plasmonic Purcell enhancement [76], in single-molecule plasmonic strong coupling [28], and in strong coupling experiments involving “picocavities” [20]. Even for substantially longer wavelengths such as 20 or 30 nm, higher-order multipolar effects can potentially be detected, as discussed in Ref. [83], where it was suggested that graphene plasmons of these wavelengths could already lead to relatively short lifetimes (microseconds) of transitions, which have thus far been spectroscopically unobserved in any experiment such as E4 and E5 transitions, which have typical free-space lifetimes of hundreds and billions of years, respectively. These kinds of effects with higher-order transitions have been considered theoretically in a number of other studies such as [106], [106], [107], [108], [109], [110], [111], [112], [113], typically focusing on electric quadrupole and magnetic dipole transitions. Nondipolar effects have also been reported experimentally with larger emitters such as carbon nanotubes [114] and larger quantum dots coupled to plasmonic nanowires with plasmon wavelengths on the order of 300 nm [115].

That the shrinking of plasmonic wavelengths can lead to normally “forbidden” transitions has been long known. A related fact that is much less appreciated, however, is that the shrinking of the plasmonic wavelength may also lead to extremely high probability of multiplasmon spontaneous emission processes in emitters. Two-photon spontaneous emission by an emitter is a second-order process in quantum electrodynamics that has been known since the 1930s. It was predicted by Goppert-Mayer [116] and subsequently estimated for the hydrogen 2s to 1s transition by Breit and Teller [117]. Being a second-order process in the coupling constant, it scales as α²(ka)⁴ω, in the case where the two-photon transition takes place via two virtual dipole transitions. From this estimation, it follows that two-photon emission is anywhere between 8 and 10 orders of magnitude slower than one-photon emission for atomic emitters in free space. For semiconductor QW-based emitters, the scaling is more favorable, with it having been experimentally determined that the two-photon emission is only five orders of magnitude than one-photon emission [118], [119].

In the presence of a plasmonic environment, analogous to the Purcell effect, the two-photon emission can be greatly enhanced. The theory of plasmon- or cavity-enhanced two-photon emission effects has been worked out in Refs. [77], [83], [120], [121] and has been experimentally supported with the observation of two plasmon-emission rate enhancements of more than 1000 for QW electrons in the vicinity of a bowtie plasmonic antenna. This same theory was used in Ref. [83] to show that, due to the extreme confinement of plasmons in graphene in the IR, atoms could experience two-plasmon Purcell enhancements of more than 10¹⁰, leading to emission rates in the microsecond to nanosecond regime. It was also shown that, by combining plasmon confinement with density-of-states engineering, it is also possible to make these two-polariton decays dominant over one-polariton decays by simultaneously having high Purcell enhancement of two-polariton emission frequencies while having only negligible Purcell enhancement at the one-polariton emission frequency [77]. Ultimately, as the spatial confinement of plasmonic modes becomes larger, even without using density-of-states engineering, the multipolariton spontaneous emission processes will have comparable amplitude to the first-order processes, representing the breakdown of the perturbation series and the ultra-strong coupling regime. It is of critical importance to emphasize that the kinds of multiphoton spontaneous emission effects described here are out of the scope of Rabi models that consider either single- or few-photonic modes and also two-level systems. This is for a few reasons: (1) a pair of emitter levels that have a dipole moment is incompatible with two-photon emission as, in the dipole approximation, two-photon transitions preserve parity and (2) two-photon emission rates require knowledge of intermediate emitter states, which is beyond a two-level description. As a result of multiphoton phenomena not being part of the typical theoretical framework, it is interesting to see what role they should play in a description of ultra-strong light-matter coupling.

We conclude this section with an outlook to the future of plasmon-emitter coupling experiments in the extreme-small-wavelength regime. In addition to the experiments by Baumberg et al. demonstrating few-molecule strong coupling to plasmons in nanocavities and picocavities, recent experiments in graphene plasmonics have shown that the mode volume of a plasmon on a graphene-hBN-gold structure can potentially be confined to volumes that are a billion times smaller than that of a photon in free space [3]. A naive estimate of emitter-plasmon coupling suggests that an emitter coupled to such plasmons can be in the ultra-strong coupling regime. This suggests that the regime of ultra-strong coupling of an emitter to a continuum of plasmonic modes is within reach. This regime features a confluence of rich physics such as nondipolar effects, multiphoton effects, quantum nonlocality, and the change of emitter wave functions due to electronic interactions with the nearby plasmonic material. To our knowledge, no theoretical study has considered the confluence of all these effects, and no experimental study has been able to dissect all of these classes of effects. To include all of these effects in a single theoretical model is a formidable challenge due to the need to take into account the following:

1. The detailed shape of the emitter wave function as it becomes very relevant to nondipolar effects;

2. The shape of the emitter wave function as it is affected by the proximity to the plasmonic metal;

3. Nonlocal effects as well as tunneling effects;

4. Potentially large material dissipation, leading to the breakdown of few-mode pictures; and

5. A Hilbert space that has not only many modes (or quasi-modes) but also many potential excitations of these modes as a result of multiphoton interactions.

An interesting testing ground for such a comprehensive theory would naturally be the recent experiments probing the strong coupling of molecule vibrations to “picocavities” [20], formed by the presence of an adsorbed molecule between two metallic facets not even 1 nm apart. In this regime, the molecular system can behave as optomechanical systems [122], as shown in Figure 3A, and the picocavity setup is shown in Figure 3B. Benz et al. [20] have demonstrated that the optomechanical coupling of single molecules in picocavities reveals the changes of the surface-enhanced Raman spectroscopy (SERS) selection rules, as shown in Figure 3C.

Figure 4:

Experiments demonstrating collective strong light-matter coupling.

(A) Adapted with permission from Ref. [123] (© 2015 Springer Nature). (B and C) Adapted with permission from Ref. [20] (© 2016 The American Association for the Advancement of Science). (A) General setup [5] when organic molecules are placed between two Ag mirrors that form an optical cavity to create dressed polaritonic states. (B) Transmission spectrum of such a system. By increasing the density of the Fe(CO)₅ compound in the optical cavity, the system shows a transition from the strong (blue) to the ultra-strong (red) coupling limit [32]. (C) Top: Schematic illustration of a strong coupling experiment [8]. The first excitation of an Si-C vibrational mode is strongly coupled to a cavity mode of the same frequency, leading to a Rabi splitting. Bottom: Deprotection reaction of PTA with TBAF. (D) Reaction rate depending on the temperature for reactions inside the cavity (red) and outside the cavity (blue) [8]. (E) Experimental setup [11] of the organic and inorganic molecules that are strongly coupled in a microcavity. (F) Strong light-matter coupling leads to a Frenkel-Wannier-Mott hybridization of excitons [11]. UP, MP, and LP denote the upper, middle, and lower polaritons, respectively. (A) Adapted with permission from Ref. [5] (© 2016 American Chemical Society). (B) Adapted with permission from Ref. [32] (© 2016 American Physical Society). (C and D) Adapted with permission from Ref. [8] (© 2016 John Wiley and Sons). (E and F) Adapted with permission from Ref. [11] (© 2014 American Physical Society).

We note that one emerging theoretical technique for addressing electromagnetic response in subnanometer gaps is that of applying a quantum corrected model (QCM) [101], [124]. The QCM is a phenomenological model in which the gap is modeled as an effectively local material, typically assigned to take the form of a Drude model with a dissipation parameter that is adjustable. Physically, this dissipation rate is meant to reflect the tunneling current between the two sides of the gap. Accordingly, this rate parameter is very large for large gaps, making this effective material very resistive so that tunneling current cannot go through. Similarly, for very small gaps, the dissipation goes to the intrinsic dissipation of the metal region surrounding the gap. Thus, in the limit of zero gap, the model maps to a homogeneous Drude metal model. Notably, this very simple model has had some success in qualitatively explaining the electromagnetic response of gaps at these scales, where the success is determined by direct comparison to TDDFT coupled to Maxwell equations [125].

4.1 QEDFT

As a first step toward a unified ab initio description of light-matter coupled systems, recently, different novel and general concepts have been developed. Density-functional theories (DFTs) have played an important role in explaining plasmonic effects, such as ground-state DFT [155] to study the electronic structure of plasmonic resonators or TDDFT [156] to study the dynamical behavior of plasmonic particles [157]. In the original formulation of TDDFT, the electromagnetic field enters via the time-dependent external potential v_ext(r, t) and thus can, for example, describe the effect of a classical pumping laser field on the system. Typically, such external fields assume the dipole coupling of the electric field to matter, i.e. vext(r, t)=E(r, t)⋅μ^. To go beyond the dipole approximation, time-dependent current DFT [158] can be used that is based on the electron current J and an external vector field A_ext as conjugated variables. To overcome the classical approximation of the electromagnetic field, QEDFT [159], [160], [161] can be employed. QEDFT generalizes TDDFT to the realm of correlated electron-photon interactions treating electrons and photons on the same quantized footing. Furthermore, being formulated as an exact reformulation of the combined electron-photon Schrödinger equation, QEDFT is not restricted to few-level approximations as commonly used in quantum optics but allows to study correlated electron-photon dynamics in full real-space. Thus far, QEDFT has been applied to study effective quantum optics systems [161], [162], coherent excitations [163], polaritonic chemistry and multimode effects [163], static chemical systems in optical cavities and correlated spectroscopies [154], lifetimes and excited states [164], and even extended to correlated electron-nuclear motion in optical cavities [165].

4.2 General theory

In the following, we will give a brief review on the field of QEDFT and also refer the reader to the review in Ref. [40].

Although the (semi)classical approximation of the light-matter interaction is justified in the limit of strong laser pulses, in which a high number of photons occurs, this approximation breaks down in the single-photon limit [161], [163] and therefore necessitates the development of a coherent ab initio QED theory. As exact solutions to quantum many-body theories are usually impractical due to their unfavorable exponential scaling [155], alternative routes have to be pursued. Solutions to problems in quantum mechanics are further usually identified as finding the corresponding (ground-state) wave function of the specific problem. However, as the wave function depends on all coordinates of the system, it is a high-dimensional object that cannot even be stored on a computer hard drive for more than a few particles [155].

One alternative route is the idea of “density functionalizing” [166]. Here, a reduced object (or a set of them), i.e. a specific observable, is chosen that does not depend on all coordinates of the system. Then, a conjugated variable is found that is usually an external control. If one now can demonstrate a so-called one-to-one correspondence (bijective mapping) between these internal and external variables, then we can completely circumvent the necessity to calculate the wave function and all solutions of the problem can be interpreted in terms of the internal variable. Thus, any observable becomes a function of this internal variable. The solution of the quantum problem is then found once the corresponding internal variable is determined.

One such method is DFT [155], where the internal variable is the electron density and the external variable is the external potential. For light-matter coupling, this specific method can be employed to any level of theory. As shown in Refs. [159], [161], in principle, such a formulation can also be formulated for the Dirac’s equation to describe relativistic (anti)particles. In the nonrelativistic limit, length gauge, and dipole approximation, QEDFT is based on the one-to-one correspondence between sets of internal and external variables (for a fixed initial state Ψ₀) that can be written as follows [160]:

Here, the external variables are the electronic external potential v_ext(r, t) and the time derivative of a classical current jext(α)(t) and allow to control the electronic and photonic subsystems, respectively. The internal variables in this limit are given by the electron density n(r, t) and the mode resolved displacement coordinate of photon mode α, q_α proportional to the mode-resolved displacement field D_α(r, t). Both internal variables enter in Eq. (1); further, the external Hamiltonian is defined as

where the internal variables couple to the external variables.

4.3 Kohn-Sham system

DFTs became only popular, as the one-to-one correspondence can be exploited by replacing the problem of solving an interacting many-body problem by the problem of solving a noninteracting problem. The noninteracting system is called the Kohn-Sham system [155]. In the Kohn-Sham system of QEDFT [160], [161], the interaction of Eq. (1) vanishes and the electronic degrees of freedom become decoupled from the photonic degrees of freedom. This comes with a price; however, in Eq. (10), the external potential become Kohn-Sham potentials, which have to be chosen such that they reproduce the correct many-body dynamics. Therefore, typically, we write formally

where the mean-field exchange correlation (Mxc) part now contains the effects of the electron-electron interaction as well as the electron-photon interaction. Although the mean-field part is explicitly known, for the exchange-correlation part, we have to employ approximations.

4.4 Route toward approximations

The applicability of any theory depends on its quality to predict the observable of interest. The same is true in the case of DFTs. Here, the quality of the calculation is closely intertwined with the underlying approximation for the famous exchange-correlation (xc) functional. In QEDFT, we can take advantage of existing approximation to describe the electronic structure of the system. Here, a wide range of different approximation schemes for the xc functional is available [167]. In contrast, still in its infancy, QEDFT currently lacks the rich variety of electronic DFT to choose suitable approximations. Some work has been done to close this gap, most prominently, to connect the optimized effective potential (OEP) approach to QEDFT [154], [162]. The OEP photon exchange energy can be formulated in two ways: (1) the expression of Ref. [162] depends on all occupied and unoccupied states and therefore is computationally demanding for larger systems. (2) Alternatively, the photon OEP equation and the OEP photon exchange energy can also be formulated in terms of occupied orbitals and so-called orbital shifts. These orbital shifts (Φiσ, α(1), Φiσ, α(2)) are electronic wave functions and can be calculated using Sternheimer equations [154]. The photon exchange energy in terms of these orbital shifts is then given by

where we now have introduced so-called orbital shifts that are shifts in electronic wave functions (Φiσ, α(1), Φiσ, α(2)) with spin index Φ. Whereas Φiσ,  α(1) contains effects due to the one-photon processes, Φiσ,  α(2) originates from the dipole self-energy term. The spin-resolved potential then follows from

By formulating the OEP equation in terms of Sternheimer equations, an efficient numerical algorithm has been developed, now allowing to calculate changes in ground-state properties for coupled electron-photon systems [154], as depicted in Figure 6D, and new correlated observables [40] can be calculated as shown in Figure 6F.

4.5 Excited states

TDDFT is widely used due to its capabilities to calculate excited-state behavior of electronic systems [156]. One prominent example is the ability to calculate absorption spectra that naturally can be calculated using a linear response formalism [168]. In linear response, the system is initially in the ground state of an time-independent Hamiltonian H^0. Then, a weak laser pulse that is localized in time (delta pulse) is applied to the system and its time evolution is recorded. The Fourier transform of the time-dependent dipole moment then yields the corresponding absorption spectrum [169].

When light is involved in an interaction, naturally processes occur that involve electronic excited states. In strong coupling situations, we find Rabi splitting also in spectroscopic quantities, such as the absorption spectra, as depicted in Figure 6G. Recently, a simple and numerically manageable linear response formalism has been worked out for QEDFT [164], which allows to calculate the Rabi splitting and intrinsic lifetimes from first principles.

5.1 Summary

In Figure 7, we classify some of the current computational methods by their treatment of light and matter. In total, we classify the methods into four categories ranging from a more analytical treatment to a more computationally heavy treatment of the matter and light field, respectively. State-of-the-art methods of the electronic structure theory, such as DFT [155], many-body perturbation theory (GW) [170], or coupled cluster [171], treat the electrons quantum mechanically, and the light field on a classical level. Such methods are depicted in Figure 7 (bottom right), as they are computationally heavy on the matter part, whereas the light field is usually either not considered or taken into account only implicitly by assuming a semiclassical approximation.

Figure 7:

Theoretical methods that treat the light-matter problem can be classified by the level of theory they treat the individual subsystems and from more analytical to more computationally heavy.

More computational methods such as FDTD, FEM, photonic crystals, and the transfer matrix approach (top left) treat the matter part of the problem using classical equations of motion and the light field in great detail, mostly on the level of Maxwell’s equations. In bottom right, we denote methods from the electronic structure theory such as DFT, GW, and coupled cluster [CCSD(T)] that solve the electronic Schrödinger equation and often ignore the transverse electromagnetic field effects. In the diagonal, we show methods that treat matter and light on a coherent quantized footing, such as the more analytical models heavily used in quantum optics, such as lattice models, Rabi/Dicke models, and the Jaynes-Cummings model. In top right, we depict more recent methods such as NEGF and QEDFT that treat the light-matter problem on an equally quantized level.

Methods from nonlinear optics, the field of nanoplasmonics [172], or photonic crystals [27] are depicted in Figure 7 (top left). These methods, such as FDTD [89] and finite-element method (FEM) [173], focus on the computational aspect of the light field, and an effective treatment of the matter is employed.

In the diagonal in Figure 7, we depict methods that treat the electromagnetic field and the electronic structure on the same quantized footing. Whereas analytical solutions are often employed in quantum optics models, such as the Jaynes-Cummings model [55] (bottom left), generalized methods of the electronic structure theory such as nonequilibrium Green’s function (NEGF) [174] and QEDFT [160], [161] (top right) are heavy computational methods that require massive computing power with the promise of an ab initio description of the combined matter-photon system.

5.2 Perspectives/outlook

Thus far, most of the works in ab initio many-body QED have focused on the changes to “matter observables” such as wave functions, electron energies, and chemical properties. As strong coupling by definition implies that both the matter and the photon are strongly modified, it will be of great value and interest to develop ab initio techniques that assess the modification of “field observables” due to light-matter coupling. Examples of interesting observables that could be tracked are the mode functions for the photons modified by matter, the new dispersion relation of the electromagnetic modes, and the amount of photons that get bound to the ground state of the correlated light-matter system [154] but also correlated light-matter correlation functions [40], [154]. What would such observables tell us? For example, if we could find how photon modes are modified by the presence of a single emitter ultra-strongly coupled to the electromagnetic field, then one could answer a very interesting fundamental question: How does a single atom modify electromagnetic modes? Can it rearrange a significant fraction of ħω field energy in the vicinity of an atom? This would be analogous to how electromagnetic field modes can be altered by a dielectric inclusion. The major difference is that whereas a dielectric function description is conceptually simple for macroscopic objects, such a treatment for a single atom strongly coupled to a field is not known. Once one answers these questions, one is able to take a fundamentally new approach to quantum photonics: building waveguides and cavities with judiciously placed single emitters.

Controlling and directing reactions in single molecule-optical cavity hybrids will provide mechanistic knowledge as well as guidelines for similar quantum optical control of molecular catalytic systems. For example, chemical reactions could be performed in optical high-quality factor cavities without the need to explicitly drive the system. Conversely, these optical cavities could be used to monitor the kinetic and thermodynamic properties of chemical reactions [175], creating a new method of “quantum chemical spectroscopy”.

Beyond catalysis, we envision these single molecule-optical cavity hybrids as building blocks to create novel platforms for implementing specific quantum algorithms and simulations, thereby integrating the field of quantum information with quantum chemistry [176]. Chemical systems are ultimately governed by the laws of quantum mechanics. Therefore, a quantum information perspective would provide new tools for theoretical and computational chemistry. An understanding of chemical phenomena within the framework of quantum information will shift the paradigm in theoretical chemistry by approaching problems with a different toolset.