Cavity-mediated light–matter coupling can dramatically alter opto-electronic and physico-chemical properties of a molecule. Ab initio theoretical predictions of these systems need to combine non-perturbative, many-body electronic structure theory-based methods with cavity quantum electrodynamics and theories of open-quantum systems. Here, we generalize quantum-electrodynamical density functional theory to account for dissipative dynamics of the cavity and describe coupled cavity–single molecule interactions in the weak-to-strong-coupling regimes. Specifically, to establish this generalized technique, we study excited-state dynamics and spectral responses of benzene and toluene under weak-to-strong light–matter coupling. By tuning the coupling, we achieve cavity-mediated energy transfer between electronically excited states. This generalized ab initio quantum-electrodynamical density functional theory treatment can be naturally extended to describe cavity-mediated interactions in arbitrary electromagnetic environments, accessing correlated light–matter observables and thereby closing the gap between electronic structure theory, quantum optics, and nanophotonics.

Recent experiments have explored regimes of light–matter interaction of molecular systems in nanophotonic cavities, where the interaction of a photon with the excitations of a single molecule can be substantially enhanced. When the loss in the cavity, characterized by a loss rate κ, dominates over the light–matter coupling rate g (g < κ), the system is in the weak-coupling regime. In this regime, the cavity mode enhances light–matter interactions, which can result in decay enhancement of molecular excited states via the Purcell effect.1 This regime has been leveraged to improve spectroscopic detection of molecular species2–6 without qualitatively modifying the electronic character of the molecular excitations.

A qualitative change in the physico-chemical properties of matter emerges when g > κ, which leads to the hybridization of excitations of matter with cavity photons to form polaritonic states.7–14 This coherent non-perturbative regime is denoted as strong light–matter coupling (for more detailed discussions of the definition of light–matter coupling regimes, see Refs. 13 and 15–18), resulting in modifications of molecular chemical reactivity via electronic or vibrational strong coupling,19–26 electrical27 and excitonic28 conductivity, optical properties,29–35 energy transfer,36–39 polariton lasing,40,41 or superconductivity.42–44 

In the intermediate regime, when gκ, the distinction between weak and strong light–matter coupling vanishes. This regime has been shown in optical cavities characterized by a small quality factor Q, where strong confinement of light in plasmonic45 or conventional46 cavities can lead to efficient coupling of light with excitons in many47–49 or a few organic molecules.50 Here, the polaritonic states are no longer well defined long-lived states51 nor can the light–matter interaction be treated as a weak perturbation to molecular excitations. The intermediate regime has been suggested to influence incoherent dissipative rates governing steady-state populations of molecular electronically excited states and can, for example, protect organic molecules from photo-oxidation processes.52 

Theoretical descriptions of coupling of light with molecular excitations often rely on models based on cavity quantum electrodynamics (cQED) and theories of open-quantum systems.53–56 Alternatively, parametrically described emitters can be coupled to an electromagnetic environment quantized within the linear-response theory of dielectric lossy media.57–62 Fully ab initio predictions of the effects of light–matter coupling on the molecular properties require non-perturbative methods combining the power of cQED, theories of open-quantum systems, and many-body electronic structure theory.13 While longitudinal electromagnetic fields of dissipative matter excitations, such as plasmons, can be naturally included in time-dependent density functional theories by including their corresponding charge explicitly in the calculation,63–71 standard theories must be extended to correctly account for transverse electromagnetic fields.

To bridge this critical gap in the field, we build on the formalism of quantum-electrodynamical density functional theory (QEDFT),12–14,72–76 briefly described in Sec. II, to incorporate cavity losses. We apply our generalized formalism to study ab initio vacuum light–matter interaction in both the weak- and strong-coupling regimes. Concretely, we study an isolated electronic transition in a benzene molecule in Sec. III A and a toluene molecule in Sec. III B, where many electronic transitions lead to an effective many-level electronic system. We show that light–matter interactions can induce interactions among the electronic states of the molecules that can result in energy transfer between excited states.

The Hamiltonian H for a non-relativistic system of M electronically excited states interacting with the quantized light field of N photon modes, in the absence of an external classical current and under the dipole approximation in the length gauge, is given by9,14,72

H=He+k=1N12pk2+ωk2qkλkωkR2,
(1)

where He is the electronic Hamiltonian, the kth quantized photon mode is given by the operators for the photon conjugate momentum pk=iωk2(akak), and the photon displacement coordinate is qk=2ωk(ak+ak), where is the reduced Planck constant. Both photon operators are given in terms of the photon annihilation ak and creation ak operators, and ωk is the frequency of mode k. The electronic system couples to the photon modes through the total position operator R=i=1Mri of the electronic system and qk of the photonic system. This coupling is scaled by the cavity strength λk, which is given by

λk=2ωkEke,
(2)

where Ek is the amplitude of the electric field at the center of the electronic charge density and e is the elementary charge. In a lossless cavity, λk represents a set of discrete modes whose excitation energies are well separated, and thus, usually only one or a few modes need to be considered in the description of light–matter coupling. Note that, in principle, molecular vibrational modes may be included in Eq. (1)12,77 in future studies.

To calculate λk, as a representative example, we couple the molecular electronic degrees of freedom with a lossy transverse optical mode. Following the reasoning of Refs. 78 and 79, we replace the cavity strength of the single mode of a lossless 3D cavity λc at energy ℏωc with many photon modes (the cavity mode coupled with the “modes of the universe”) whose spectral density follows a Lorentzian profile Lω, κ, ωkc),

|λk|2=|λc|2L(Δω,κ,ωkc),
(3)

where

L(Δω,κ,ωkc)=Δω12πκ(ωkc)2+(κ/2)2.
(4)

In principle, the appropriate realistic spectral density can be obtained using any method of quantizing lossy electromagnetic environments,56,57,62,78,80 although within QEDFT, longitudinal fields, such as plasmonic fields, should be included in the matter many-body Hamiltonian. The Lorentzian profile distributes the total cavity strength λc over a range of frequency values ωk with a Lorentzian line shape around the central mode frequency (ωkc = ωkωc), where the degree of broadening is controlled by the loss rate κ, which can be related, e.g., to the reflectivity of cavity mirrors. Here, the modes are uniformly spaced, that is, the frequency spacing Δω = ωk+1ωk for all k is constant. In general, transforming to or from another energy spacing Δω(ωk) may offer computational benefits for an arbitrary electromagnetic environment; the detailed mathematical formalism is given in  Appendix B. Equation (3) also implies that the cavity strength obeys the sum rule,

dωD(ω)|λk|2=|λc|2,
(5)

where Δω(ω) → D(ω)dω in the continuum limit, D(ω) is the density of photon modes, and dω → 0.

In the following, we also define the coupling rate gi,k that relates to the cavity strength λk and is a measure of the coupling strength between the cavity mode k and a particular electronic excitation i connecting the electronic ground state |g⟩ with an electronically excited state |ei⟩ of the molecule,

gi,k=eEkg|R|ei=ωk2λkedi.
(6)

Here, di = e⟨g|R|ei⟩ are transition dipole moments that can be, for example, obtained together with their corresponding excitation energies ℏωi from a standard linear-response time-dependent density-functional theory (TDDFT).81 These quantities can then be used as input parameters for cQED models describing light–matter interactions. This effective interaction rate naturally emerges in parametric cQED models, such as the Jaynes–Cummings model.82 We adapt these approaches by including the Lorentzian profile of the cavity strength to formulate a Fano-type cQED model that describes the linear-response of a discrete many-level electronic system coupled to a continuous photonic reservoir, permitting analysis in both the time and frequency domains. The relevant Hamiltonian HcQED, in the rotating-wave approximation, is given by

HcQED=i=1Mωielσiσi+k=1Nωkakak+i,k=1M,N(gi,kσiak+H.c.),
(7)

where σi (σi) and ak (ak) are creation (annihilation) operators for the ith of M excited electronic states and kth of N photon modes, respectively, H.c. is the Hermitian conjugate, ωiel and ℏωk are the mode energies, and gi,k is the coupling defined as in Eq. (6).

The state of the system can be described in the single excitation subspace as

|ψ=c0|g,{0k}+i=1Mciel|ei,{0k}+k=1Nckph|g,{1k},
(8)

where the coefficients c0, ciel, and ckph are time dependent. We plug this ansatz into the Schrödinger equation to obtain the following system of linear differential equations:

ċiel=iωielcielik=1Ngi,kckph,
(9)
ċkph=ii=1Mgi,k*cieliωkck.
(10)

Note that the coefficient c0 drops from the system of equations as the Hamiltonian conserves the number of excitations due to the rotating-wave approximation. This system of linear differential equations can be solved to obtain the eigenvalues, ℏωl (i.e., energies), and eigenvectors, |vl⟩ (i.e., polaritonic states), of the coupled system. With these parameters, we can calculate the electronic absorption spectrum and weight of the original, unmixed electronic and photonic states. In this cQED model, we describe the light–matter coupling via a many-mode Jaynes–Cummings term (adopting the rotating-wave approximation) and neglect the term ∝ R2 appearing in the full light–matter coupling Hamiltonian given in Eq. (1). Several studies9,83–86 have analyzed the effect of this term, which becomes, in particular, relevant in the ultrastrong coupling limit. In  Appendix A, we compare our full QEDFT calculations with R2 contributions included to the cQED model without R2 contributions and comment on their influence.

Within QEDFT, on the other hand, the electronic and photonic excitations, in the presence of nuclei, and their interactions are treated on the same quantized footing.72,73,75,76 In particular, in this article, we adopt a linear-response time-dependent QEDFT that generalizes the Casida equation of TDDFT, first introduced in Refs. 14 and 76, where the electron–electron interactions included in TDDFT and the electron–photon interactions are solved simultaneously in an iterative solver, whereas the cQED approach involves adding electron–photon coupling to the TDDFT output before solving. We use the QEDFT implementation in a publicly available modified version of the pseudopotential, real-space DFT code OCTOPUS.87–89 

In Sec. III A, we first demonstrate the present method on a model system by coupling an isolated electronic excitation of benzene to a cavity with varying cavity strength and losses. We note the qualitative similarities between the results calculated from first principles and those expected from the well-known, parametric Jaynes–Cummings model.82 In Sec. III B, we study a more complex system that consists of many electronically states of toluene coupled to a cavity with varying cavity strength and loss, respectively. Importantly, we observe interactions between electronically excited states mediated by the cavity that can result in population transfer between excited states.

To demonstrate our generalized QEDFT approach, we consider an example of a single benzene molecule placed in a lossy 1D optical cavity with a linearly polarized electric field along the x axis. The orientation of the molecule is shown in Figs. 1(a) and 1(b). In the first principles QEDFT calculation, we consider the full electronic structure with all single-particle excitations. Nevertheless, since the target excitation |e1, 0⟩ with energy ω1el=6.93 eV and strong x-polarized transition dipole moment d1,x = 0.96 eÅ tuned in resonance with the cavity mode is well separated in energy from other optically active excited states, benzene can be effectively understood as a two-level electronic system composed of the ground state |g, 0⟩ and an excited state |e1, 0⟩ for the range of parameters we consider. We verify this simplification by calculating an electronic x-polarized absorption spectrum Ax of the molecule in the framework of linear-response QEDFT,14,76

Ax(ω)=Cl=1M+Nδ(ωωl)ωl|i=1MCileldi,x|2,
(11)

where C=2me/(32) (with me being the electron mass) is a frequency-independent pre-factor, ℏωl is the mode energy, di,x is the x-component of the transition dipole moment of an electronic excitation i, and Cilel (Cklph) is the projection of an original, unmixed electronic (photonic) state |ei, 0⟩ (|g, 1k⟩) to a resulting polaritonic state |vl⟩. M excited electronic states of benzene are coupled with N photon modes to produce M + N hybrid electron–photon states called polaritons. For presentation purposes, we broaden the delta function δ(ωωl) with a Lorentzian,

δ(ωωl)12πΓ(ωωl)2+(Γ/2)2.
(12)

The spectral broadening Γ can be interpreted as, for instance, the resolution of the spectrometer. We note that this artificial broadening is not included in our ab initio calculations and does not represent properties of the studied system. In  Appendix D, we show the influence of Γ on the visibility of fine features.

FIG. 1.

The ground state |g, 0⟩ and first excited electronic state |e1, 0⟩ of benzene with energy ω1el = 6.93 eV and transition dipole moment d1,x = 0.96 eÅ comprise an effective two-level electronic system. (a) Coupling |e1, 0⟩ with lossy cavity modes |g, 1k⟩ centered at ωc=ω1el results in unresolved polaritonic states. (b) |e1, 0⟩ couples with a resonantly tuned, low loss optical cavity |g, 1⟩ to form distinct upper and lower polaritonic states. (c) The normalized optical x-polarized absorption spectrum of benzene in the absence of light–matter coupling. (d) The normalized optical x-polarized absorption spectrum of benzene in the presence of light–matter coupling for spectral broadening Γ = 0.001 eV. As the cavity strength λc increases from (i) 0.001 eV12/nm to (ii) 0.002 eV12/nm to (iii) 0.008 eV12/nm with loss ℏκ = 0.001 eV in (i)–(iii), the energy splitting between the lower and upper polaritons increases; they are unable to be resolved after increasing ℏκ from (iii) 0.001 eV to (iv) 0.004 eV to (v) 0.008 eV with λ = 0.008 eV12/nm in (iii)–(v). The color of the curve corresponds to the relative photonic and electronic weight of the polaritonic eigenstate. To retrieve the unnormalized absorption spectra, multiply the absorption intensities by the scaling factor on the right side of the plot.

FIG. 1.

The ground state |g, 0⟩ and first excited electronic state |e1, 0⟩ of benzene with energy ω1el = 6.93 eV and transition dipole moment d1,x = 0.96 eÅ comprise an effective two-level electronic system. (a) Coupling |e1, 0⟩ with lossy cavity modes |g, 1k⟩ centered at ωc=ω1el results in unresolved polaritonic states. (b) |e1, 0⟩ couples with a resonantly tuned, low loss optical cavity |g, 1⟩ to form distinct upper and lower polaritonic states. (c) The normalized optical x-polarized absorption spectrum of benzene in the absence of light–matter coupling. (d) The normalized optical x-polarized absorption spectrum of benzene in the presence of light–matter coupling for spectral broadening Γ = 0.001 eV. As the cavity strength λc increases from (i) 0.001 eV12/nm to (ii) 0.002 eV12/nm to (iii) 0.008 eV12/nm with loss ℏκ = 0.001 eV in (i)–(iii), the energy splitting between the lower and upper polaritons increases; they are unable to be resolved after increasing ℏκ from (iii) 0.001 eV to (iv) 0.004 eV to (v) 0.008 eV with λ = 0.008 eV12/nm in (iii)–(v). The color of the curve corresponds to the relative photonic and electronic weight of the polaritonic eigenstate. To retrieve the unnormalized absorption spectra, multiply the absorption intensities by the scaling factor on the right side of the plot.

Close modal

The absorption spectrum with no light–matter coupling is shown in Fig. 1(c) and features a single dominant absorption peak corresponding to an electronic transition to |e1, 0⟩ from |g, 0⟩, agreeing well with computational predictions in Ref. 76 and with experiments presented in Ref. 90. We verify that the excited eigenstate |e1, 0⟩ is separated in energy at least 3 eV from the nearest eigenstates with a substantial transition dipole moment in the x-direction. For the range of coupling rates considered in this paper, |g, 0⟩ and |e1, 0⟩ thus effectively form a two-level electronic system for the considered range of parameters.

Next, we place the molecule in a cavity whose central mode energy ℏωc is in resonance with this transition (ωc=ω1el). In the single excitation subspace, this system thus corresponds closely to the Jaynes–Cummings model, which describes the interaction of a two-level system with a single resonant bosonic mode. In this scenario, schematically shown in Figs. 1(a) and 1(b), the two-level system can be represented by a ground state |g, 0⟩ and an excited electronic state |e1, 0⟩ of a molecule whose transition dipole moment d1 allows for efficient coupling with light and whose frequency is well separated from other electronic excitations. The bosonic mode is then represented by a single electromagnetic mode of an optical cavity (of quantized electric-field amplitude Ec and experiences losses with decay rate κ) that interacts with the electronic states of the molecule via the Jaynes–Cummings coupling rate g=1d1Ec. If the coupling rate is small compared to the loss rate of the cavity, when g < κ, the system is in the weak-coupling regime in which the cavity enhances the decay rate of the electronically excited state, i.e., the Purcell effect. This enhancement manifests as broadening of the electronic absorption spectrum of the molecule, as illustrated in Fig. 1(a). In contrast, when g > κ, the system is in the strong-coupling regime in which the electronically excited state and the cavity photon hybridize and form new hybrid light–matter polariton states denoted as |U⟩, “upper,” and |L⟩, “lower,” polaritons in Fig. 1(b). The polariton states are manifested in the molecular absorption spectrum as a doublet of peaks whose splitting is ∝g.

By tuning the cavity strength λc (which we assume to be polarized in the x-direction) and loss ℏκ, and therefore changing its spectral profile as shown in Fig. 2(a), we are able to control the regime of light–matter coupling in accordance with the predictions of the Jaynes–Cummings model. We first calculate a series of absorption spectra of the molecule coupled with the cavity field: In the calculations shown in Fig. 1(d), N = 5000 cavity modes whose cavity strength is distributed according to Eq. (3) are used to represent the single leaky cavity mode coupled with a single dominant electronically excited state among the M excited states of benzene, and the absorption spectrum is calculated with spectral broadening Γ = 0.001 eV.

FIG. 2.

The cavity strength λk with constant frequency spacing Δω = 0.0001 eV of the cavity modes for curves (i)–(v) for (a) benzene in Fig. 1(d) and (b) toluene in Fig. 3(d). For toluene in Figs. 4(a) and 4(b), the cavity mode energy range is truncated to 6.35 eV–6.95 eV with Δω = 0.001 eV.

FIG. 2.

The cavity strength λk with constant frequency spacing Δω = 0.0001 eV of the cavity modes for curves (i)–(v) for (a) benzene in Fig. 1(d) and (b) toluene in Fig. 3(d). For toluene in Figs. 4(a) and 4(b), the cavity mode energy range is truncated to 6.35 eV–6.95 eV with Δω = 0.001 eV.

Close modal

We first assume that the cavity is characterized by a constant cavity loss ℏκ = 0.001 eV and vary the value of cavity strength λc. The calculated absorption spectra are shown in Figs. 1(di) for λc = (0.001, 0.002, 0.008) eV12/nm or g/κ = (0.18, 0.36, 1.43), respectively. The largest ℏg = 0.23 meV 0.1ω1el, so the coupling strength remains far below the ultrastrong regime.91,92 As we increase the coupling strength, we observe the transition from the weak-coupling regime in Fig. 1(di) where the molecular absorption spectrum features a single broadened Lorentzian peak to the strong-coupling regime in Fig. 1(diii) featuring the polaritonic peak doublet.

As the regime of light–matter coupling is a function of the ratio g/κ, we show in Figs. 1(diii) that the strong light–matter coupling regime can transition back into the weak-coupling regime when cavity losses are increased for a given cavity strength λc = 0.008 eV12/nm. Concretely, we increase the cavity loss rate from ℏκ = 0.001 eV in (iii) to ℏκ = 0.004 eV in (iv) to ℏκ = 0.008 eV in (v). We observe that with increasing loss rate, the doublet merges back into a single broadened Lorentzian peak as we reenter the weak-coupling regime. The peak in Fig. 1(dv) is broader than the peak in Fig. 1(di) as the Purcell decay ∝g2/κ is larger in the former case.

To develop further physical intuition on how the N photonic states hybridize with the M electronic states to form M + N polaritonic states, we calculate the weight Wilel=|Cilel|2 (Wklph=|Cklph|2) of an original, unmixed electronic (photonic) state |ei, 0⟩ (|g, 1k⟩) in a resulting lth polaritonic state |vl⟩. We plot the total weight from the electronic states wlel=i=1MWilel and the total weight from the photonic states wlph=k=1NWklph=1wlel as the overlaid color of the curve in Fig. 1(d). Notably, we observe concentrations of the electronic character near the absorption peaks, even when separated in energy as in Fig. 1(diii), which proves the hybrid polaritonic character of the states composing the peaks.

Finally, we remark that in Fig. 5(a) in  Appendix A, we show that the results obtained from QEDFT in this parameter regime can be reproduced well using the cQED model, as expected in a model system.

First principles-based techniques enable quantitative predictions a priori of complex systems. Here, we systematically describe the effect of weak-to-strong light–matter coupling to an effective many-level electronic system, toluene. Although we again include all M electronically excited states from the ab initio calculation, here we discuss three x-polarized optically active excited states clustered in energy: |e1, 0⟩ with energy ω1el=6.64 eV and x-polarized transition dipole moment d1,x = 0.76 eÅ, |e2, 0⟩ with ω2el=6.71 eV and d2,x = 0.11 eÅ, and |e3, 0⟩ with ω3el=6.78 eV and d3,x = 0.08 eÅ. The x-polarized absorption spectrum of toluene in free space is plotted in Fig. 3(c) where we observe absorption peaks that scale as di,x2 corresponding to |e1, 0⟩, |e2, 0⟩, and |e3, 0⟩. There are additional eigenstates that are further off-resonant or have smaller transition dipole moments. We neglect them in our discussion in the main text because they do not interact strongly with the cavity. Nonetheless, we still include their effects in our calculation. As we discuss in  Appendix D, for small enough spectral broadening Γ, these excitations appear in the spectra as additional Fano resonances.

When the molecule is placed in a 1D lossy cavity with central mode frequency ωc=ω1el and M = 36 000 photon modes, we observe the transition from the weak-to-strong coupling regime in the electronic absorption spectra in Fig. 3(d) with spectral broadening Γ = 0.001 eV upon varying cavity strength λc and loss ℏκ, as shown in Fig. 2(b). At suitable cavity strength λc and loss ℏκ, we also observe Fano resonances resulting from photon-mediated interactions between electronic states.

FIG. 3.

The ground state |g, 0⟩ and three excited electronic states—|e1, 0⟩, |e2, 0⟩, and |e3, 0⟩, in the order of increasing energy—of toluene comprise a many-level electronic system. (a) For coupling between |e1, 0⟩ and a resonantly tuned, low loss optical cavity |g, 1⟩ where energy splitting ωpω12el (ω12el=ω2elω1el), we observe splitting into upper and lower polaritons with little interaction with |e2, 0⟩ and |e3, 0⟩. (b) When the coupling strength is tuned such that ωpω12el, the upper polariton can be tuned in resonance and mixed with |e2, 0⟩. (c) Normalized x-polarized optical absorption spectrum of toluene in the absence of light–matter coupling. (d) Normalized optical x-polarized absorption spectrum in the presence of light–matter coupling for spectral broadening Γ = 0.001 eV. The coupling strength λc is set to (i) 0.01, (ii) 0.10, and (iii)–(v) 0.43 eV12/nm. The loss ℏκ is set to (i)–(iii) 0.01 eV, (iv) 0.10 eV, and (v) 0.32 eV. The color of the curves corresponds to the relative photonic and electronic character of the polaritonic eigenstate. To retrieve the unnormalized absorption spectra, multiply the absorption intensities by the scaling factor on the right side of the plot.

FIG. 3.

The ground state |g, 0⟩ and three excited electronic states—|e1, 0⟩, |e2, 0⟩, and |e3, 0⟩, in the order of increasing energy—of toluene comprise a many-level electronic system. (a) For coupling between |e1, 0⟩ and a resonantly tuned, low loss optical cavity |g, 1⟩ where energy splitting ωpω12el (ω12el=ω2elω1el), we observe splitting into upper and lower polaritons with little interaction with |e2, 0⟩ and |e3, 0⟩. (b) When the coupling strength is tuned such that ωpω12el, the upper polariton can be tuned in resonance and mixed with |e2, 0⟩. (c) Normalized x-polarized optical absorption spectrum of toluene in the absence of light–matter coupling. (d) Normalized optical x-polarized absorption spectrum in the presence of light–matter coupling for spectral broadening Γ = 0.001 eV. The coupling strength λc is set to (i) 0.01, (ii) 0.10, and (iii)–(v) 0.43 eV12/nm. The loss ℏκ is set to (i)–(iii) 0.01 eV, (iv) 0.10 eV, and (v) 0.32 eV. The color of the curves corresponds to the relative photonic and electronic character of the polaritonic eigenstate. To retrieve the unnormalized absorption spectra, multiply the absorption intensities by the scaling factor on the right side of the plot.

Close modal

We first assume that the cavity is characterized by cavity strength λc = 0.01eV12/nm and loss ℏκ = 0.01 eV or g/κ = 0.14. As we increase the coupling strength, we observe the transition from the weak-coupling regime in Fig. 3(di) where the molecular absorption spectrum features a broadened Lorentzian peak at ω1el to the strong-coupling regime in Fig. 3(diii) where λc = 0.10 eV12/nm or g/κ = 1.4 features the polaritonic peak doublet. This doublet corresponds to the lower |L⟩ and upper |U⟩ polaritons in Fig. 3(a). The relative electronic character wel of the doublet is lower than those of the peaks at ω2el and ω3el, which do not mix substantially with the photon modes.

In a many-level electronic system, electronic states may interact through the photon modes. We demonstrate the effects of this interaction by increasing the cavity strength λc to 0.43 eV12/nm and maintaining loss ℏκ = 0.01 eV (g/κ = 6.0) in Fig. 3(diii) from 0.10 eV12/nm in Fig. 3(dii). At this cavity strength, ωpω12el=ω2elω1el, so the upper polariton becomes nearly degenerate |e2, 0⟩ with energy ω2el, resulting in the upper polariton mixing with |e2, 0⟩ to form an upper–lower polariton |UL⟩ and an upper–upper polariton |UU⟩. These conditions are schematically illustrated in Fig. 3(b).

Increasing loss ℏκ for the toluene-lossy cavity system such that |e1, 0⟩ and |e2, 0⟩ interact through the photon modes results in Fano resonances in the absorption spectrum. To capture this effect, from the strong-coupling regime in Fig. 3(diii) where the cavity strength λc = 0.43 eV12/nm and loss ℏκ = 0.01 eV, we increase loss ℏκ to 0.08 eV and maintain λc = 0.43 eV12/nm, or g/κ = 0.74, as shown in Fig. 3(div). Further increasing loss ℏκ enables resonant electronic eigenstates to interact with other electronic eigenstates even further in energy: in Fig. 3(dv), where λc = 0.43 eV12/nm and loss ℏκ is increased to 0.32 eV, the peak at ω3el broadens. In addition, the individual polaritonic peaks merge into a single unresolved one, a characteristic of the weak-coupling regime.

In Fig. 5(b) in  Appendix A, we discuss how the results obtained from QEDFT can be interpreted using the cQED model and compare the absorption spectra obtained from both models. We show that up to small discrepancies arising from the rotating-wave approximation applied in cQED and the mean-field approximation applied in QEDFT, as well as the exclusion of the R2 term arising from expansion of Eq. (1) in the cQED model, the two methods are in good quantitative agreement.

To elucidate the physical implications of the absorption spectra, we plot the weights Wilel from QEDFT calculations in Figs. 4(ai). The conditions correspond to the same cavity conditions as in Figs. 3(di), respectively, except that the input cavity mode frequency range is restricted from 6.35 eV to 6.95 eV in Figs. 4(a) and 4(b) for computational tractability as opposed to 4.85 eV–8.45 eV in Fig. 3(d). Absorption spectra are nearly identical for both energy ranges. Since the absorption spectra calculated from first principles in Fig. 3(d) agree with those of the cQED model in Fig. 5 parameterized with first principles calculations of the electronic structure and cavity mode profiles, we can use the cQED model to calculate the explicit time dependence of population transfer in Fig. 4(b) for a toluene-lossy cavity system initially prepared in |e1, 0⟩.

FIG. 4.

(a) Normalized weights of original, unmixed electronic states in the polaritonic states calculated with the QEDFT framework. All electronic states are decoupled from each other with low light–matter coupling in (i). The upper polariton resulting from mixing between |e1, 0⟩ and |g, 1⟩ mixes with |e2, 0⟩ in (iii). λ and κ for (i)–(v) in Fig. 4(a) equal those in (i)–(v) in Fig. 3(d). The unnormalized weights are recovered by scaling the plotted curves by the listed scale factors. (b) Population transfer in the time domain via a cQED model with inputs from first principles calculations of the excited electronic eigenstates and photonic spectral profile. Toluene is initially prepared in |e1, 0⟩. Rabi oscillations in (ii) are a clear indication of strong coupling between |e1, 0⟩ and |g, 1⟩. Population transfer between |e1, 0⟩ and |e2, 0⟩ occurs when ωpω12el, as shown in (iii) and schematically illustrated in Fig. 3(b). Increasing κ in (iv) and (v) decreases excited electronic state lifetimes and enables slight population transfer to |e3, 0⟩, as observed in (v). λ and κ for (i)–(v) in Fig. 4(b) equal those in (i)–(v) in Fig. 3(d).

FIG. 4.

(a) Normalized weights of original, unmixed electronic states in the polaritonic states calculated with the QEDFT framework. All electronic states are decoupled from each other with low light–matter coupling in (i). The upper polariton resulting from mixing between |e1, 0⟩ and |g, 1⟩ mixes with |e2, 0⟩ in (iii). λ and κ for (i)–(v) in Fig. 4(a) equal those in (i)–(v) in Fig. 3(d). The unnormalized weights are recovered by scaling the plotted curves by the listed scale factors. (b) Population transfer in the time domain via a cQED model with inputs from first principles calculations of the excited electronic eigenstates and photonic spectral profile. Toluene is initially prepared in |e1, 0⟩. Rabi oscillations in (ii) are a clear indication of strong coupling between |e1, 0⟩ and |g, 1⟩. Population transfer between |e1, 0⟩ and |e2, 0⟩ occurs when ωpω12el, as shown in (iii) and schematically illustrated in Fig. 3(b). Increasing κ in (iv) and (v) decreases excited electronic state lifetimes and enables slight population transfer to |e3, 0⟩, as observed in (v). λ and κ for (i)–(v) in Fig. 4(b) equal those in (i)–(v) in Fig. 3(d).

Close modal
FIG. 5.

Comparison of normalized x-polarized absorption spectra for (a) benzene and (b) toluene between QEDFT and a cQED model parameterized with the eigenenergies and transition dipole moments of the electronically excited states and the same respective cavity mode profiles as the first principles, self-consistent QEDFT calculations. QEDFT curves of Figs. 5(a) and 5(b) are re-plotted from Figs. 1(d) and 3(d), respectively.

FIG. 5.

Comparison of normalized x-polarized absorption spectra for (a) benzene and (b) toluene between QEDFT and a cQED model parameterized with the eigenenergies and transition dipole moments of the electronically excited states and the same respective cavity mode profiles as the first principles, self-consistent QEDFT calculations. QEDFT curves of Figs. 5(a) and 5(b) are re-plotted from Figs. 1(d) and 3(d), respectively.

Close modal

In the weak-coupling regime in Fig. 4(ai), we see that all electronic characters from |e1, 0⟩, |e2, 0⟩, and |e3, 0⟩ are localized in one peak each. To demonstrate weak-coupling behavior in the time domain, the time decay of the toluene-lossy cavity system initialized in |e1, 0⟩ is plotted in Fig. 4(bi). As expected for an electronic system weakly coupled to photon modes, we observe exponential decay of the population of |e1, 0⟩.

In the strong-coupling regime where the upper |U⟩ and lower |L⟩ polaritons are well-resolved in Fig. 3(dii), in Fig. 4(aii), the electronic character of |e1, 0⟩ is concentrated within the two peaks corresponding to |U⟩ and |L⟩. We calculate the time decay of the toluene-lossy cavity system initialized in |e1, 0⟩ in Fig. 4(bii) for the same cavity parameters as in Fig. 3(dii) and observe a signature of strong-coupling behavior: vacuum Rabi oscillations. In this regime, the difference in energy ℏωp between the polaritons and |e1, 0⟩ is much lower than the difference in energy ω12el between |e1, 0⟩ and |e2, 0⟩, so there is negligible interaction between |e1, 0⟩ and |e2, 0⟩. This lack of interaction is apparent in both Figs. 4(aii) and 4(bii). In Fig. 4(aii), the two polaritonic peaks have no contribution from |e2, 0⟩, and in Fig. 4(bii), we observe no population transfer to |e2, 0⟩.

In Fig. 3(diii), λc is further increased such that electronic states |e1, 0⟩ and |e2, 0⟩ interact. In Fig. 4(aiii), the peaks corresponding to |UU⟩ and |UL⟩ have contributions from both |e1, 0⟩ and |e2, 0⟩. The consequence in the time domain is population transfer between |e1, 0⟩ and |e2, 0⟩, as well as Rabi oscillations with the photon modes, suggesting that this toluene-lossy cavity system experiences both strong light–matter and strong matter–matter coupling via the cavity modes. As loss ℏκ increases to 0.10 eV in Fig. 3(div), although population still transfers between |e1, 0⟩ and |e2, 0⟩, the excitation lifetime commensurately decreases compared to when ℏκ = 0.01 eV. Finally, as κ is increased further to 0.32 eV in Fig. 3(dv), where |e3, 0⟩ interacts non-negligibly with the cavity modes, we observe slight population transfer to |e3, 0⟩ in Fig. 4(bv).

Overall, we demonstrate that tuning the cavity strength λc and loss ℏκ can generate polaritonic states with contributions from several excited electronic states and can control population transfer to higher-lying states in energy and excitation lifetimes.

To summarize, we study ab initio correlated optical interactions in matter ranging from the weak-coupling to strong-coupling regime. As an example, we calculate excited-state dynamics and spectral responses of benzene and toluene as effective two-level and many-level electronic systems, respectively, under variable light–matter coupling controlled by the cavity strength λc and loss ℏκ. By tuning the cavity parameters, we notice transitions between the weak-coupling and the strong-coupling regimes where polaritonic states can be resolved. In the many-level electronic system, we observe Fano resonances in the electronic absorption spectrum resulting from interactions between electronically excited states mediated by the cavity. These interactions enable cavity-mediated population transfer between electronically excited states where the lifetimes and the degree of population transfer are controlled by the cavity parameters. We reproduce the first principles results using a cQED model parameterized with the QEDFT data and generally note excellent agreement.

This generalized QEDFT formalism is especially useful for predictions where interactions of single molecules with arbitrary electromagnetic environments dominate over vibrationally mediated losses. Looking forward, in electronic structure theory, we anticipate that the development of improved TDDFT methods and integration of light–matter interactions of molecular vibrations from first principles12 will improve prediction accuracy for higher-lying excited states and enable studies of correlated cavity-electron–nuclei interactions,93 respectively. In cQED, further understanding the effects of the correct inclusion of the R2 term in the QEDFT formalism vs a conventional cQED model invoked in the present study will enable the extension of this method to the ultrastrong coupling regime. Finally, a natural extension of this work is to study the interactions of many molecules in lossy cavities from first principles, as formation of collective dark states leads to modifications of the polariton dynamics.19,25,30,39,41,52,94–96

This work demonstrates a predictive technique for single-molecule experiments involving the weak- and strong-coupling light–matter regimes, including modifications of the excited-state potential energy surfaces in the field of polaritonic chemistry. This extension to QEDFT moves toward closing the loop between first principles calculations in electronic structure theory and parametric models of the quantum optics community.

This work was supported by the DOE “Photonics at Thermodynamic Limits” Energy Frontier Research Center under Grant No. DE-SC0019140. D.S.W. is an NSF Graduate Research Fellow. The Flatiron Institute is a division of the Simons Foundation. P.N. is a Moore Inventor Fellow.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Normalized x-polarized absorption spectra calculated with the cQED model for benzene and toluene with the same cavity conditions in Figs. 2(a) and 2(b) are plotted in Figs. 5(a) and 5(b), respectively. We note that the QEDFT and cQED agree quantitatively well for both benzene and toluene. There are more substantial differences between the spectra, specifically a blue shift of the cQED spectra, for the toluene study, which used higher coupling strengths than the benzene study. This trend agrees with a comparison between QEDFT and the Rabi model in Ref. 76. To understand the effect of the lack of the R2 term in the cQED model that arises from expanding the last term in Eq. (1), in Fig. 5(b), we also plot the absorption spectra calculated with the present QEDFT formalism without the R2 term. The effect of the R2 is small for the coupling strengths here, so the differences in spectra can be more directly attributed to differences in the rotating-wave approximation made in the cQED model and the mean-field approximation made in QEDFT. Finally, Eqs. (8) and (9) can also be explicitly propagated in time given a set of initial conditions to plot population transfer, as shown in Fig. 4(b).

Here, we demonstrate how to transform the frequency spacing of a photon spectral profile from an exact calculation of the electromagnetic environment of a cavity to an arbitrary mode frequency spacing. Such an equivalency has clear numerical advantages whereby frequencies of interest can be more densely sampled at the cost of lowered frequency density at less relevant ranges for the same computational cost.

For specificity, we study the Hamiltonian HcQED,cont in the cQED model, Eq. (7), expressed in terms of a continuously, constantly spaced frequency variable ω for photons and discrete frequencies ωiel for electronically excited states,

HcQED,cont=i=1Mωielσiσi+dωωa(ω)a(ω)+dωi=1M(gi(ω)σia(ω)+H.c.).
(B1)

We desire to change the integration variable ω to a new one Ω such that ω(Ω) has a more favorable spacing once the Hamiltonian is necessarily discretized for numerical calculations. Assuming a general functional dependence connecting the two frequency variables, note that the transformation of variables leads to modification of all integrals,

dωdωdΩD(ωΩ)dΩ,
(B2)

where we have defined the density of states D(Ω). This transformation is accompanied by re-definition of creation and annihilation operators of the continuum,

ã(ω(Ω))=D(Ω)a(ω),
(B3)

which follows from the requirement that the commutation relation [ã(Ω),ã(Ω)]=δ(ΩΩ) holds.

We can rewrite HcQED,cont as

HcQED,cont=i=1Mωielσiσi+dΩω(Ω)ã(Ω)ã(Ω)+dΩi=1MD(Ω)Gi[ω(Ω)]G̃i(Ω)σiã(Ω)+H.c.,
(B4)

where we have defined a new coupling constant G̃i(Ω). Equation (B4) can be discretized by assuming a constant spacing in variable Ω,

dΩk=1NΔΩk,
(B5)

where ΔΩ = Ωi+1 − Ωi is the step of the discretization. The annihilation and creation operators of the continuum must be transformed back to the operators of the discrete set of modes as

Ã(Ωi)ÃkΔΩã(Ωk),
(B6)

which follows from the requirement that [Ãi,Ãj]=δij. We write the discrete version of HcQED,cont, H̃cQED, with new mode frequency spacing,

H̃Fano=i=1Mωielσiσi+k=1Nω(Ωk)ÃkÃk+i,k=1M,NΔΩG̃(Ωk)σiÃk+H.c..
(B7)

We use a publicly available modified version of the pseudopotential, real-space DFT code OCTOPUS.87–89 The calculation of electronically excited states via the Casida TDDFT method first requires a converged ground state electronic structure on optimized molecular geometries. Molecular geometries are optimized with norm-conserving, plane-wave Hamann, Schlüter, Chiang, and Vanderbilt (HSCV) pseudopotentials97 and a local density approximation (LDA) functional.98 The real-space simulation box is parameterized with a mesh spacing of 0.16 Å and consists of spheres with 6 Å radius around each atom.

We calculate the converged ground state electron densities with a standard LDA functional for benzene and a long-range adapted LDA functional for toluene;99 the long-range LDA (LRLDA) functional requires the ionization potential from a ΔSCF calculation. Ground state self-consistent field (SCF) energies converged within ∼1 meV/atom with 0.08 Å and 0.14 Å real-space mesh spacing for benzene and toluene, respectively, and a simulation box consisting of a 6 Å radius around each atom.

The energies of the excited states of interest converged within ∼1 meV/atom with 500 extra states in the Casida method. We further confirm the reliability of the excited electronic eigenstates that we couple to the cavity modes by noting that all excited states studied lie below the ionization potentials determined with a ΔSCF calculation.

Molecules can have excitations densely spaced in energy that all interact through the cavity photons. However, because excitations with low transition dipole moments or that are sufficiently off-resonant from the line width of the cavity modes do not interact strongly with the cavity, the effects of their interactions are small. Nonetheless, it is possible to observe their effects by increasing the measurement quality.

For instance, for toluene in the energy range shown in Fig. 3(c), there are several excitations beyond |e1, 0⟩, |e2, 0⟩, and |e3, 0⟩ that do not have an appreciable effect on the absorption spectra when coupled to photons, as shown in Fig. 3(d). The only observable impact is a slight dip on the side of the lower polariton in Figs. 3(diii) caused by the eigenstate at 6.58 eV with transition dipole moment dx = 0.01 eÅ. We study the impact of varying the spectral broadening Γ on the absorption spectra for the conditions in Fig. 3(c), as demonstrated in Fig. 6. Physically, the spectral broadening can be lowered by decreasing the temperature. As the spectral broadening Γ decreases from Γ = (0.005, 0.002, 0.001, 0.0005, 0.0002) eV in Figs. 6(a)6(e), respectively, we observe the appearance of an additional Fano resonance at the eigenenergy at 6.58 eV, a signature of the presence of an electronic eigenstate interacting with others through the photon modes. Given sufficient measurement resolution, we expect that further decreasing the spectral broadening Γ would enable the observation of the remaining eigenstates in Fig. 3(c).

FIG. 6.

Normalized x-polarized absorption spectra for toluene in a lossy cavity with λc = 0.43 eV12/nm and ℏκ = 0.01 eV with spectral broadening Γ = (0.005, 0.002, 0.001, 0.0005, 0.0002) eV for (a)–(e), respectively. Curve (c) corresponds to Fig. 3(diii).

FIG. 6.

Normalized x-polarized absorption spectra for toluene in a lossy cavity with λc = 0.43 eV12/nm and ℏκ = 0.01 eV with spectral broadening Γ = (0.005, 0.002, 0.001, 0.0005, 0.0002) eV for (a)–(e), respectively. Curve (c) corresponds to Fig. 3(diii).

Close modal
1.
E. M.
Purcell
,
H. C.
Torrey
, and
R. V.
Pound
, “
Resonance absorption by nuclear magnetic moments in a solid
,”
Phys. Rev.
69
,
37
38
(
1946
).
2.
G.
Wrigge
,
I.
Gerhardt
,
J.
Hwang
,
G.
Zumofen
, and
V.
Sandoghdar
, “
Efficient coupling of photons to a single molecule and the observation of its resonance fluorescence
,”
Nat. Phys.
4
,
60
66
(
2008
).
3.
A.
Kinkhabwala
,
Z.
Yu
,
S.
Fan
,
Y.
Avlasevich
,
K.
Müllen
, and
W. E.
Moerner
, “
Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna
,”
Nat. Photonics
3
,
654
657
(
2009
).
4.
R. F.
Aroca
, “
Plasmon enhanced spectroscopy
,”
Phys. Chem. Chem. Phys.
15
,
5355
5363
(
2013
).
5.
M.
Pelton
, “
Modified spontaneous emission in nanophotonic structures
,”
Nat. Photonics
9
,
427
435
(
2015
).
6.
T.
Itoh
,
Y. S.
Yamamoto
, and
Y.
Ozaki
, “
Plasmon-enhanced spectroscopy of absorption and spontaneous emissions explained using cavity quantum optics
,”
Chem. Soc. Rev.
46
,
3904
3921
(
2017
).
7.
J. A.
Ćwik
,
P.
Kirton
,
S.
De Liberato
, and
J.
Keeling
, “
Excitonic spectral features in strongly coupled organic polaritons
,”
Phys. Rev. A
93
,
033840
(
2016
).
8.
T. W.
Ebbesen
, “
Hybrid light–matter states in a molecular and material science perspective
,”
Acc. Chem. Res.
49
,
2403
2412
(
2016
).
9.
J.
Flick
,
M.
Ruggenthaler
,
H.
Appel
, and
A.
Rubio
, “
Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry
,”
Proc. Natl. Acad. Sci. U. S. A.
114
,
3026
3034
(
2017
).
10.
F.
Herrera
and
J.
Owrutsky
, “
Molecular polaritons for controlling chemistry with quantum optics
,”
J. Chem. Phys.
152
,
100902
(
2020
).
11.
N.
Rivera
,
J.
Flick
, and
P.
Narang
, “
Variational theory of nonrelativistic quantum electrodynamics
,”
Phys. Rev. Lett.
122
,
193603
(
2019
).
12.
J.
Flick
and
P.
Narang
, “
Cavity-correlated electron-nuclear dynamics from first principles
,”
Phys. Rev. Lett.
121
,
113002
(
2018
).
13.
J.
Flick
,
N.
Rivera
, and
P.
Narang
, “
Strong light-matter coupling in quantum chemistry and quantum photonics
,”
Nanophotonics
7
,
1479
1501
(
2018
).
14.
J.
Flick
and
P.
Narang
, “
Excited-state nanophotonic and polaritonic chemistry with ab initio potential-energy surfaces
,”
J. Chem. Phys.
153
,
094116
(
2020
).
15.
P.
Törmä
and
W. L.
Barnes
, “
Strong coupling between surface plasmon polaritons and emitters: A review
,”
Rep. Prog. Phys.
78
,
013901
(
2014
).
16.
D.
Melnikau
,
R.
Esteban
,
D.
Savateeva
,
A.
Sánchez-Iglesias
,
M.
Grzelczak
,
M. K.
Schmidt
,
L. M.
Liz-Marzán
,
J.
Aizpurua
, and
Y. P.
Rakovich
, “
Rabi splitting in photoluminescence spectra of hybrid systems of gold nanorods and J-aggregates
,”
J. Phys. Chem. Lett.
7
,
354
362
(
2016
).
17.
G.
Zengin
,
T.
Gschneidtner
,
R.
Verre
,
L.
Shao
,
T. J.
Antosiewicz
,
K.
Moth-Poulsen
,
M.
Käll
, and
T.
Shegai
, “
Evaluating conditions for strong coupling between nanoparticle plasmons and organic dyes using scattering and absorption spectroscopy
,”
J. Phys. Chem. C
120
,
20588
20596
(
2016
).
18.
M.
Autore
,
P.
Li
,
I.
Dolado
,
F. J.
Alfaro-Mozaz
,
R.
Esteban
,
A.
Atxabal
,
F.
Casanova
,
L. E.
Hueso
,
P.
Alonso-González
,
J.
Aizpurua
 et al, “
Boron nitride nanoresonators for phonon-enhanced molecular vibrational spectroscopy at the strong coupling limit
,”
Light: Sci. Appl.
7
,
17172
(
2018
).
19.
J.
Galego
,
F. J.
Garcia-Vidal
, and
J.
Feist
, “
Cavity-induced modifications of molecular structure in the strong-coupling regime
,”
Phys. Rev. X
5
,
041022
(
2015
).
20.
J.
Galego
,
F. J.
Garcia-Vidal
, and
J.
Feist
, “
Suppressing photochemical reactions with quantized light fields
,”
Nat. Commun.
7
,
13841
(
2016
).
21.
A.
Thomas
,
J.
George
,
A.
Shalabney
,
M.
Dryzhakov
,
S. J.
Varma
,
J.
Moran
,
T.
Chervy
,
X.
Zhong
,
E.
Devaux
,
C.
Genet
,
J. A.
Hutchison
, and
T. W.
Ebbesen
, “
Ground-state chemical reactivity under vibrational coupling to the vacuum electromagnetic field
,”
Angew. Chem., Int. Ed.
55
,
11462
11466
(
2016
).
22.
F.
Herrera
and
F. C.
Spano
, “
Cavity-controlled chemistry in molecular ensembles
,”
Phys. Rev. Lett.
116
,
238301
(
2016
).
23.
J.
Flick
,
C.
Schäfer
,
M.
Ruggenthaler
,
H.
Appel
, and
A.
Rubio
, “
Ab initio optimized effective potentials for real molecules in optical cavities: Photon contributions to the molecular ground state
,”
ACS Photonics
5
,
992
1005
(
2017
).
24.
J.
Galego
,
C.
Climent
,
F. J.
Garcia-Vidal
, and
J.
Feist
, “
Cavity Casimir-Polder forces and their effects in ground-state chemical reactivity
,”
Phys. Rev. X
9
,
021057
(
2019
).
25.
G.
Groenhof
,
C.
Climent
,
J.
Feist
,
D.
Morozov
, and
J. J.
Toppari
, “
Tracking polariton relaxation with multiscale molecular dynamics simulations
,”
J. Phys. Chem. Lett.
10
,
5476
5483
(
2019
).
26.
A.
Thomas
,
L.
Lethuillier-Karl
,
K.
Nagarajan
,
R. M. A.
Vergauwe
,
J.
George
,
T.
Chervy
,
A.
Shalabney
,
E.
Devaux
,
C.
Genet
,
J.
Moran
, and
T. W.
Ebbesen
, “
Tilting a ground-state reactivity landscape by vibrational strong coupling
,”
Science
363
,
615
619
(
2019
).
27.
E.
Orgiu
,
J.
George
,
J. A.
Hutchison
,
E.
Devaux
,
J. F.
Dayen
,
B.
Doudin
,
F.
Stellacci
,
C.
Genet
,
J.
Schachenmayer
,
C.
Genes
,
G.
Pupillo
,
P.
Samorì
, and
T. W.
Ebbesen
, “
Conductivity in organic semiconductors hybridized with the vacuum field
,”
Nat. Mater.
14
,
1123
1129
(
2015
).
28.
J.
Feist
and
F. J.
Garcia-Vidal
, “
Extraordinary exciton conductance induced by strong coupling
,”
Phys. Rev. Lett.
114
,
196402
(
2015
).
29.
D. G.
Lidzey
,
D. D. C.
Bradley
,
T.
Virgili
,
A.
Armitage
,
M. S.
Skolnick
, and
S.
Walker
, “
Room temperature polariton emission from strongly coupled organic semiconductor microcavities
,”
Phys. Rev. Lett.
82
,
3316
3319
(
1999
).
30.
J.
del Pino
,
J.
Feist
, and
F. J.
Garcia-Vidal
, “
Quantum theory of collective strong coupling of molecular vibrations with a microcavity mode
,”
New J. Phys.
17
,
053040
(
2015
).
31.
J.
George
,
S.
Wang
,
T.
Chervy
,
A.
Canaguier-Durand
,
G.
Schaeffer
,
J.-M.
Lehn
,
J. A.
Hutchison
,
C.
Genet
, and
T. W.
Ebbesen
, “
Ultra-strong coupling of molecular materials: Spectroscopy and dynamics
,”
Faraday Discuss.
178
,
281
294
(
2015
).
32.
F.
Herrera
and
F. C.
Spano
, “
Theory of nanoscale organic cavities: The essential role of vibration-photon dressed states
,”
ACS Photonics
5
,
65
79
(
2018
).
33.
M. A.
Zeb
,
P. G.
Kirton
, and
J.
Keeling
, “
Exact states and spectra of vibrationally dressed polaritons
,”
ACS Photonics
5
,
249
257
(
2018
).
34.
F.
Herrera
and
F. C.
Spano
, “
Dark vibronic polaritons and the spectroscopy of organic microcavities
,”
Phys. Rev. Lett.
118
,
223601
(
2017
).
35.
B.
Xiang
,
R. F.
Ribeiro
,
A. D.
Dunkelberger
,
J.
Wang
,
Y.
Li
,
B. S.
Simpkins
,
J. C.
Owrutsky
,
J.
Yuen-Zhou
, and
W.
Xiong
, “
Two-dimensional infrared spectroscopy of vibrational polaritons
,”
Proc. Natl. Acad. Sci. U. S. A.
115
,
4845
4850
(
2018
).
36.
X.
Zhong
,
T.
Chervy
,
L.
Zhang
,
A.
Thomas
,
J.
George
,
C.
Genet
,
J. A.
Hutchison
, and
T. W.
Ebbesen
, “
Energy transfer between spatially separated entangled molecules
,”
Angew. Chem., Int. Ed.
56
,
9034
9038
(
2017
).
37.
D. M.
Coles
,
N.
Somaschi
,
P.
Michetti
,
C.
Clark
,
P. G.
Lagoudakis
,
P. G.
Savvidis
, and
D. G.
Lidzey
, “
Polariton-mediated energy transfer between organic dyes in a strongly coupled optical microcavity
,”
Nat. Mater.
13
,
712
(
2014
).
38.
D. M.
Juraschek
,
T.
Neuman
,
J.
Flick
, and
P.
Narang
, “
Cavity control of nonlinear phononics
,” arXiv:1912.00122 (
2019
).
39.
M.
Du
,
L. A.
Martínez-Martínez
,
R. F.
Ribeiro
,
Z.
Hu
,
V. M.
Menon
, and
J.
Yuen-Zhou
, “
Theory for polariton-assisted remote energy transfer
,”
Chem. Sci.
9
,
6659
(
2018
).
40.
S.
Kéna-Cohen
and
S. R.
Forrest
, “
Room-temperature polariton lasing in an organic single-crystal microcavity
,”
Nat. Photonics
4
,
371
375
(
2010
).
41.
L.
Mazza
,
S.
Kéna-Cohen
,
P.
Michetti
, and
G. C. L.
Rocca
, “
Microscopic theory of polariton lasing via vibronically assisted scattering
,”
Phys. Rev. B
88
,
075321
(
2013
).
42.
F. P.
Laussy
,
A. V.
Kavokin
, and
I. A.
Shelykh
, “
Exciton-polariton mediated superconductivity
,”
Phys. Rev. Lett.
104
,
106402
(
2010
).
43.
F.
Schlawin
,
A.
Cavalleri
, and
D.
Jaksch
, “
Cavity-mediated electron-photon superconductivity
,”
Phys. Rev. Lett.
122
,
133602
(
2019
).
44.
J. B.
Curtis
,
Z. M.
Raines
,
A. A.
Allocca
,
M.
Hafezi
, and
V. M.
Galitski
, “
Cavity quantum Eliashberg enhancement of superconductivity
,”
Phys. Rev. Lett.
122
,
167002
(
2019
).
45.
T.
Neuman
,
R.
Esteban
,
D.
Casanova
,
F. J.
García-Vidal
, and
J.
Aizpurua
, “
Coupling of molecular emitters and plasmonic cavities beyond the point-dipole approximation
,”
Nano Lett.
18
,
2358
2364
(
2018
).
46.
D.
Hagenmüller
,
S.
Schütz
,
G.
Pupillo
, and
J.
Schachenmayer
, “
Adiabatic elimination for ensembles of emitters in cavities with dissipative couplings
,”
Phys. Rev. A
102
,
013714
(
2020
).
47.
G.
Zengin
,
G.
Johansson
,
P.
Johansson
,
T. J.
Antosiewicz
,
M.
Käll
, and
T.
Shegai
, “
Approaching the strong coupling limit in single plasmonic nanorods interacting with J-aggregates
,”
Sci. Rep.
3
,
3074
(
2013
).
48.
G.
Zengin
,
M.
Wersäll
,
S.
Nilsson
,
T. J.
Antosiewicz
,
M.
Käll
, and
T.
Shegai
, “
Realizing strong light-matter interactions between single-nanoparticle plasmons and molecular excitons at ambient conditions
,”
Phys. Rev. Lett.
114
,
157401
(
2015
).
49.
M.
Wersäll
,
J.
Cuadra
,
T. J.
Antosiewicz
,
S.
Balci
, and
T.
Shegai
, “
Observation of mode splitting in photoluminescence of individual plasmonic nanoparticles strongly coupled to molecular excitons
,”
Nano Lett.
17
,
551
558
(
2017
).
50.
R.
Chikkaraddy
,
B.
de Nijs
,
F.
Benz
,
S. J.
Barrow
,
O. A.
Scherman
,
E.
Rosta
,
A.
Demetriadou
,
P.
Fox
,
O.
Hess
, and
J. J.
Baumberg
, “
Single-molecule strong coupling at room temperature in plasmonic nanocavities
,”
Nature
535
,
127
130
(
2016
).
51.
R. E.
Silva
,
J.
del Pino
,
F. J.
García-Vidal
, and
J.
Feist
, “
Polaritonic molecular clock for all-optical ultrafast imaging of wavepacket dynamics without probe pulses
,”
Nat. Commun.
11
,
1423
(
2020
).
52.
B.
Munkhbat
,
M.
Wersäll
,
D. G.
Baranov
,
T. J.
Antosiewicz
, and
T.
Shegai
, “
Suppression of photo-oxidation of organic chromophores by strong coupling to plasmonic nanoantennas
,”
Sci. Adv.
4
,
eaas9552
(
2018
).
53.
H.-P.
Breuer
and
F.
Petruccione
,
The Theory of Open Quantum Systems
(
Oxford University Press
,
2003
).
54.
K.
Blum
,
Density Matrix Theory and Applications
(
Springer Science & Business Media
,
2013
).
55.
C.
Sauvan
,
J. P.
Hugonin
,
I. S.
Maksymov
, and
P.
Lalanne
, “
Theory of the spontaneous optical emission of nanosize photonic and plasmon resonators
,”
Phys. Rev. Lett.
110
,
237401
(
2013
).
56.
S.
Franke
,
S.
Hughes
,
M. K.
Dezfouli
,
P. T.
Kristensen
,
K.
Busch
,
A.
Knorr
, and
M.
Richter
, “
Quantization of quasinormal modes for open cavities and plasmonic cavity quantum electrodynamics
,”
Phys. Rev. Lett.
122
,
213901
(
2019
).
57.
B.
Huttner
and
S. M.
Barnett
, “
Dispersion and loss in a Hopfield dielectric
,”
Europhys. Lett.
18
,
487
492
(
1992
).
58.
T.
Gruner
and
D.-G.
Welsch
, “
Correlation of radiation-field ground-state fluctuations in a dispersive and lossy dielectric
,”
Phys. Rev. A
51
,
3246
3256
(
1995
).
59.
T.
Gruner
and
D.-G.
Welsch
, “
Green-function approach to the radiation-field quantization for homogeneous and inhomogeneous Kramers-Kronig dielectrics
,”
Phys. Rev. A
53
,
1818
1829
(
1996
).
60.
S.
Scheel
,
L.
Knöll
, and
D.-G.
Welsch
, “
QED commutation relations for inhomogeneous Kramers-Kronig dielectrics
,”
Phys. Rev. A
58
,
700
706
(
1998
).
61.
H. T.
Dung
,
L.
Knöll
, and
D.-G.
Welsch
, “
Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics
,”
Phys. Rev. A
57
,
3931
(
1998
).
62.
S.
Scheel
and
S. Y.
Buhmann
,
Acta Phys. Slova
58
,
675
(
2008
).
63.
E.
Townsend
and
G. W.
Bryant
, “
Plasmonic properties of metallic nanoparticles: The effects of size quantization
,”
Nano Lett.
12
,
429
434
(
2012
).
64.
D. C.
Marinica
,
A. K.
Kazansky
,
P.
Nordlander
,
J.
Aizpurua
, and
A. G.
Borisov
, “
Quantum plasmonics: Nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer
,”
Nano Lett.
12
,
1333
1339
(
2012
).
65.
K.
Iida
,
M.
Noda
,
K.
Ishimura
, and
K.
Nobusada
, “
First-principles computational visualization of localized surface plasmon resonance in gold nanoclusters
,”
J. Phys. Chem. A
118
,
11317
11322
(
2014
).
66.
J.
Ma
,
Z.
Wang
, and
L.-W.
Wang
, “
Interplay between plasmon and single-particle excitations in a metal nanocluster
,”
Nat. Commun.
6
,
10107
(
2015
).
67.
M.
Barbry
,
P.
Koval
,
F.
Marchesin
,
R.
Esteban
,
A. G.
Borisov
,
J.
Aizpurua
, and
D.
Sánchez-Portal
, “
Atomistic near-field nanoplasmonics: Reaching atomic-scale resolution in nanooptics
,”
Nano Lett.
15
,
3410
3419
(
2015
).
68.
F.
Marchesin
,
P.
Koval
,
M.
Barbry
,
J.
Aizpurua
, and
D.
Sánchez-Portal
, “
Plasmonic response of metallic nanojunctions driven by single atom motion: Quantum transport revealed in optics
,”
ACS Photonics
3
,
269
277
(
2016
).
69.
R.
Sinha-Roy
,
P.
García-González
,
H.-C.
Weissker
,
F.
Rabilloud
, and
A. I.
Fernández-Domínguez
, “
Classical and ab initio plasmonics meet at sub-nanometric noble metal rods
,”
ACS Photonics
4
,
1484
1493
(
2017
).
70.
D. C.
Marinica
,
V. M.
Silkin
,
A. K.
Kazansky
, and
A. G.
Borisov
, “
Controlling gap plasmons with quantum resonances
,”
Phys. Rev. B
98
,
155426
(
2018
).
71.
T. P.
Rossi
,
T.
Shegai
,
P.
Erhart
, and
T. J.
Antosiewicz
, “
Strong plasmon-molecule coupling at the nanoscale revealed by first-principles modeling
,”
Nat. Commun.
10
,
3336
(
2019
).
72.
I. V.
Tokatly
, “
Time-dependent density functional theory for many-electron systems interacting with cavity photons
,”
Phys. Rev. Lett.
110
,
233001
(
2013
).
73.
M.
Ruggenthaler
,
J.
Flick
,
C.
Pellegrini
,
H.
Appel
,
I. V.
Tokatly
, and
A.
Rubio
, “
Quantum-electrodynamical density-functional theory: Bridging quantum optics and electronic-structure theory
,”
Phys. Rev. A
90
,
012508
(
2014
).
74.
T.
Dimitrov
,
J.
Flick
,
M.
Ruggenthaler
, and
A.
Rubio
, “
Exact functionals for correlated electron–photon systems
,”
New J. Phys.
19
,
113036
(
2017
).
75.
M.
Ruggenthaler
,
N.
Tancogne-Dejean
,
J.
Flick
,
H.
Appel
, and
A.
Rubio
, “
From a quantum-electrodynamical light–matter description to novel spectroscopies
,”
Nat. Rev. Chem.
2
,
0118
(
2018
).
76.
J.
Flick
,
D. M.
Welakuh
,
M.
Ruggenthaler
,
H.
Appel
, and
A.
Rubio
, “
Light–matter response in nonrelativistic quantum electrodynamics
,”
ACS Photonics
6
,
2757
2778
(
2019
).
77.
J.
del Pino
,
F. A.
Schröder
,
A. W.
Chin
,
J.
Feist
, and
F. J.
Garcia-Vidal
, “
Tensor network simulation of non-Markovian dynamics in organic polaritons
,”
Phys. Rev. Lett.
121
,
227401
(
2018
).
78.
M. O.
Scully
and
M. S.
Zubairy
,
Quantum Optics
(
Cambridge University Press
,
1997
).
79.
S.
Esfandiarpour
,
H.
Safari
,
R.
Bennett
, and
S. Y.
Buhmann
, “
Cavity-QED interactions of two correlated atoms
,”
J. Phys. B: At., Mol. Opt. Phys.
51
,
094004
(
2018
).
80.
M.
Sánchez-Barquilla
,
R. E.
Silva
, and
J.
Feist
, “
Cumulant expansion for the treatment of light-matter interactions in arbitrary material structures
,”
J. Chem. Phys.
152
,
034108
(
2020
).
81.
M. E.
Casida
, “
Time-dependent density functional response theory for molecules
,” in
Recent Advances in Density Functional Methods: Part I
(
World Scientific
,
1995
), pp.
155
192
.
82.
B. W.
Shore
and
P. L.
Knight
, “
The Jaynes–Cummings model
,”
J. Mod. Opt.
40
,
1195
1238
(
1993
).
83.
C.
Schäfer
,
M.
Ruggenthaler
,
V.
Rokaj
, and
A.
Rubio
, “
Relevance of the quadratic diamagnetic and self-polarization terms in cavity quantum electrodynamics
,”
ACS Photonics
7
,
975
(
2020
).
84.
V.
Rokaj
,
D. M.
Welakuh
,
M.
Ruggenthaler
, and
A.
Rubio
, “
Light–matter interaction in the long-wavelength limit: No ground-state without dipole self-energy
,”
J. Phys. B: At., Mol. Opt. Phys.
51
,
034005
(
2018
).
85.
D.
De Bernardis
,
P.
Pilar
,
T.
Jaako
,
S.
De Liberato
, and
P.
Rabl
, “
Breakdown of gauge invariance in ultrastrong-coupling cavity QED
,”
Phys. Rev. A
98
,
053819
(
2018
); arXiv:1805.05339.
86.
J.
Li
,
D.
Golez
,
G.
Mazza
,
A. J.
Millis
,
A.
Georges
, and
M.
Eckstein
, “
Electromagnetic coupling in tight-binding models for strongly correlated light and matter
,”
Phys. Rev. B
101
,
205140
(
2020
).
87.
M. A. L.
Marques
,
A.
Castro
,
G. F.
Bertsch
, and
A.
Rubio
, “
octopus: A first-principles tool for excited electron-ion dynamics
,”
Comput. Phys. Commun.
151
,
60
78
(
2003
).
88.
X.
Andrade
,
D.
Strubbe
,
U.
De Giovannini
,
A. H.
Larsen
,
M. J. T.
Oliveira
,
J.
Alberdi-Rodriguez
,
A.
Varas
,
I.
Theophilou
,
N.
Helbig
,
M. J.
Verstraete
,
L.
Stella
,
F.
Nogueira
,
A.
Aspuru-Guzik
,
A.
Castro
,
M. A. L.
Marques
, and
A.
Rubio
, “
Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems
,”
Phys. Chem. Chem. Phys.
17
,
31371
(
2015
).
89.
N.
Tancogne-Dejean
,
M. J. T.
Oliveira
,
X.
Andrade
,
H.
Appel
,
C. H.
Borca
,
G.
Le Breton
,
F.
Buchholz
,
A.
Castro
,
S.
Corni
,
A. A.
Correa
,
U.
De Giovannini
,
A.
Delgado
,
F. G.
Eich
,
J.
Flick
,
G.
Gil
,
A.
Gomez
,
N.
Helbig
,
H.
Hübener
,
R.
Jestädt
,
J.
Jornet-Somoza
,
A. H.
Larsen
,
I. V.
Lebedeva
,
M.
Lüders
,
M. A. L.
Marques
,
S. T.
Ohlmann
,
S.
Pipolo
,
M.
Rampp
,
C. A.
Rozzi
,
D. A.
Strubbe
,
S. A.
Sato
,
C.
Schäfer
,
I.
Theophilou
,
A.
Welden
, and
A.
Rubio
, “
Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems
,”
J. Chem. Phys.
152
,
124119
(
2020
).
90.
E.
Koch
and
A.
Otto
, “
Optical absorption of benzene vapour for photon energies from 6 eV to 35 eV
,”
Chem. Phys. Lett.
12
,
476
(
1971
).
91.
A.
Frisk Kockum
,
A.
Miranowicz
,
S.
De Liberato
,
S.
Savasta
, and
F.
Nori
, “
Ultrastrong coupling between light and matter
,”
Nat. Rev. Phys.
1
,
19
40
(
2019
).
92.
P.
Forn-Díaz
,
L.
Lamata
,
E.
Rico
,
J.
Kono
, and
E.
Solano
, “
Ultrastrong coupling regimes of light-matter interaction
,”
Rev. Mod. Phys.
91
,
025005
(
2019
).
93.
D.
Wang
,
H.
Kelkar
,
D.
Martin-Cano
,
D.
Rattenbacher
,
A.
Shkarin
,
T.
Utikal
,
S.
Götzinger
, and
V.
Sandoghdar
, “
Turning a molecule into a coherent two-level quantum system
,”
Nat. Phys.
15
,
483
489
(
2019
).
94.
T.
Neuman
and
J.
Aizpurua
, “
Origin of the asymmetric light emission from molecular exciton-polaritons
,”
Optica
5
,
1247
1255
(
2018
).
95.
P.
Michetti
and
G. C.
La Rocca
, “
Simulation of J-aggregate microcavity photoluminescence
,”
Phys. Rev. B
77
,
195301
(
2008
).
96.
F.
Herrera
and
F. C.
Spano
, “
Absorption and photoluminescence in organic cavity QED
,”
Phys. Rev. A
95
,
053867
(
2017
).
97.
M.
van Setten
,
M.
Giantomassi
,
E.
Bousquet
,
M.
Verstraete
,
D.
Hamann
,
X.
Gonze
, and
G.-M.
Rignanese
,
Comput. Phys. Commun.
26
,
39
(
2019
).
98.
J. P.
Perdew
and
Y.
Wang
, “
Accurate and simple analytic representation of the electron-gas correlation energy
,”
Phys. Rev. B
45
,
13244
(
1992
).
99.
M. E.
Casida
and
D. R.
Salahub
,
J. Chem. Phys.
113
,
8918
(
2000
).