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From a quantum-electrodynamical light–matter description to novel spectroscopies

Michael Ruggenthaler1*, Nicolas Tancogne-Dejean1*, Johannes Flick1,3*, Heiko Appel1* and Angel Rubio1,2* Abstract | Insights from spectroscopic experiments led to the development of quantum mechanics as the common theoretical framework for describing the physical and chemical properties of atoms, molecules and materials. Later, a full quantum description of charged particles, electromagnetic radiation and special relativity was developed, leading to quantum electrodynamics (QED). This is, to our current understanding, the most complete theory describing photon–matter interactions in correlated many-body systems. In the low-energy regime, simplified models of QED have been developed to describe and analyse spectra over a wide spatiotemporal range as well as physical systems. In this Review, we highlight the interrelations and limitations of such theoretical models, thereby showing that they arise from low-energy simplifications of the full QED formalism, in which antiparticles and the internal structure of the nuclei are neglected. Taking molecular systems as an example, we discuss how the breakdown of some simplifications of low-energy QED challenges our conventional understanding of light–matter interactions. In addition to high-precision atomic measurements and simulations of particle physics problems in solid-state systems, new theoretical features that account for collective QED effects in complex interacting many-particle systems could become a material-based route to further advance our current understanding of light–matter interactions.

Spectroscopy (from the Latin speciō ‘look at or view’ and now reaching sub-angstroms and atto­seconds. The dif-

1Max Planck Institute for the Greek skopéō ‘to see’) investigates the various properties ferent static and time-resolved spectro­scopic techniques Structure and Dynamics of of physical systems over different space, time and energy encompass optical (vibrational, rotational and electronic), Matter and Center for Free- scales by looking at their response to different pertur- magnetic, magnetic-resonance (nuclear and electron Electron Laser Science, bations. The different particles (electrons, protons and spin-resonance), energy-loss, mechanical (atomic force), Hamburg, Germany. 2Center for Computational neutrons) taking part in a physico-chemical process, transport (electronic, spin or heat), and imaging (diffrac- Quantum Physics (CCQ), electro­magnetic and thermal radiations or mechanical tion, scanning-tunnelling microscopy and holography) The Flatiron Institute, New deformations can be investigated and simultaneously spectroscopies. Therefore, it is no longer possible to refer York, NY, USA. used as perturbations. In atomic, molecular and solid- to ‘spectroscopy’ without specifying the technique or the 3 Present address: John state physics, spectroscopic methods are specifically spatial and temporal regime. A. Paulson School of Engineering and Applied used to probe and control various aspects of coupled Diverse spectroscopic techniques are applied in differ- 11 Sciences, Harvard University, photon–matter systems, such as the quantum nature of ent fields — from atomic and molecular physics to solid- Cambridge, MA, USA. light1, matter–matter interactions2, electronic3 and nuclear state physics12 and biochemistry13 — and the analysis *e-mail: michael. degrees of freedom4 and collective effects in molecular and and interpretation of the complex experimental data in [email protected]; solid-state systems5. The study of these photon–matter each of these fields are routinely performed by means nicolas.tancogne-dejean@ interactions has informed us, for example, on unknown of purpose-built theoretical tools. These theoretical mpsd.mpg.de; 6,7 [email protected]; states of matter , novel equilibrium and non-equilibrium tools often treat electromagnetic radiation and matter at 8 [email protected]; (driven) topological phases and light-induced super­ different levels of approximation, thereby limiting their use [email protected] conductivity9. Over the years, the development and pro- to only particular cases. Usually, photons are considered doi:10.1038/s41570-018-0118 gress of specific experimental spectroscopic techniques only as an external perturbation (often treated classically) Published online 7 Mar 2018 have resulted in remarkable spatial and time resolutions3,10, that probes matter (treated quantum mechanically),

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whereas the self-consistent back-reaction of matter on equation with the Maxwell equations. The spectrum is photons is neglected. Standard theoretical modelling interpreted as transitions between atomic states, each one can thus be insufficient when the degrees of freedom of interacting or dressed with the classical Maxwell field. photons and matter become equally important. Strong The use of the dressed-electron picture has been known light–matter interaction, as in the case of polariton conden- for a long time, and the coupled Schrödinger–Maxwell sates14, provides a direct example. However, the interaction or Dirac–Maxwell equations form the basic framework between photons and matter, beyond the semi-classical to describe molecular and solid-state spectros­copies. description, is also expected to play a role under other However, it is also clear that these equations are simplifi- conditions and could lead, for instance, to the emergence cations of an even more complete theory, which explicitly of collective responses in ensembles of molecules. takes into account the electromagnetic degrees of freedom In this Review, we take a step back from the ever- on a quantized level. Indeed, many experimental results increasing specialization of each of the current theoretical exist that require the simultaneous consideration of the approaches and scrutinize them as different low-energy quantum nature of electrons and the electro­magnetic approximations of the general framework of quantum degrees of freedom (see FIG. 1 for a schematic of the evo- electrodynamics (QED). In particular, we focus on lution of our understanding of combined light–matter non-relativistic QED, which is applicable within the typical systems). These include the modification of atomic energy and timescales of molecular and solid-state energy levels in high‑Q cavities23, the emergence of quasi‑ spectros­copies (from a few millielectronvolts to a few particles24 such as polaritons, and spectroscopies that thousand electronvolts and from attoseconds to milli­ make use of the quantum nature of light22. Traditionally, seconds). A great amount of theoretical and experimental many of these electromagnetic effects (for example, the techniques have been developed to study photon–matter Lamb shift or the finite lifetime of an excited molecular interactions in these energetic and temporal regimes, and state) have been investigated in atomic and small molec- it is beyond the scope of this Review to describe them all or ular systems23,25,26. Although usually small for isolated to provide a comprehensive list of examples; however, we atoms, these electromagnetic effects have an important refer to the relevant literature where possible. We consider role in molecular systems and even more in solid-state mainly optical and electronic spectroscopies, focusing on systems, leading, for example, to the following: the electron–photon phenomena in molecular and solid- screening, polarization and retardation effects observed state systems; we do not explicitly examine direct proton– in light–matter energy transfer induced by attosecond photon interactions but instead describe the nuclei laser pulses27; the emergence of quasi-particles that do (consisting of protons and neutrons) as effective particles. not have a classical counterpart, such as polaritons or Similar considerations can be extended straightforwardly axions28,29; and the formation of novel states of matter, to this additional case. We highlight the limitations of the such as hybrid photon–matter states30,31, exciton–polariton theoretical tools used to describe photon–matter inter- condensates14,32 or light-induced topological states8. Well- actions but also identify new phenomena and hidden known electromagnetic effects in quantum chemistry aspects of these interactions that can be revealed with and solid-state physics33 are long-range intermolecular the development of new tailored spectroscopic tools. interactions and Casimir–Polder forces34,35, Förster reso- 36,37 Polariton When matter and photons are strongly correlated, novel nance energy transfer or the hydrodynamical regime A polariton is a bosonic effects appear, such as changes in the chemical proper- of quantum light38. However, most common theoretical quasi-particle formed by an ties of molecules in optical cavities and the emergence of approaches do not treat these effects self-consistently, excitation, such as an exciton polaritonic chemistry15,16,17,18, strong photon coupling in light- meaning that they do not consider the back-reaction of or plasmon coupled (‘dressed’) 19 with photons. harvesting complexes or attraction between photons matter on the electromagnetic field and vice versa. due to quantum matter20. Such effects have the potential QED provides a general framework to treat electro- Polaritonic chemistry to directly challenge QED in the low-energy regime. We magnetic and matter degrees of freedom on equal quan- Molecular systems strongly finally envision that it will be possible to extract new rele- tized footing, as QED seamlessly combines quantum coupled to light show the (FIG. 2) emergence of polaritonic vant spectroscopic information for photon–matter systems mechanics and the Maxwell equations to describe states, which can change the by directly probing correlated photon–matter observables the full coupling of photons and matter. Indeed, in addi- chemical properties of through, for instance, entangled photon spectroscopy21,22. tion to describing the direct interaction between charged the molecules. Before going into detail, let us demonstrate why particles and light, QED also captures matter–matter there is a need to go beyond the conventional descrip- interactions induced by the electromagnetic field and Electromagnetic vacuum If all harmonic oscillators of the tion of light–matter interactions, which is based on the photon–photon interactions resulting from the presence quantized electromagnetic solution of the Schrödinger or Dirac equation (describ- of matter (FIG. 3). QED accounts for the absorption and field are in their ground state, ing matter) possibly coupled with the classical Maxwell emission of photons as well as effects beyond the ‘bare’ the number of photons is zero, equations (describing photons). It is well known that electronic quantum mechanics description of matter (for corresponding to the bare electromagnetic vacuum. the optical spectral lines of a dilute atomic gas are well example, the Lamb shift of energy levels due to quantum 25 However, when coupled to a represented by transitions between the eigenstates of the fluctuations of the electromagnetic vacuum ). matter system, the vacuum isolated atomic electronic Hamiltonian. However, as the fluctuations induce changes in gas density increases (for example, by applying pressure), A brief sketch of QED the matter system, which lead the observed atomic-gas spectrum deviates from simple Before discussing the most common theoretical spectros­ to the Lamb shift. This coupling forms the basis of atomic transitions, and its theoretical description requires copic tools, we give a brief overview of QED and its vacuum-mediated (‘dark’) the inclusion of the missing polarization, retardation low-energy limit. Charge densities and currents are polaritonic chemistry. and other effects by coupling the Schrödinger or Dirac the sources of electromagnetic fields. Classically, this is

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Quantum theory of molecules: Quantum theory of condensed matter: Spectroscopic measurements like the On the basis of the concept of ones of Raman (1930) provided the input quasi-particles (Landau, 1962), many Quantum theory of light: for the basic theoretical developments by effects in condensed-matter In order to explain the Debye (1936), Pauling (1954) and physics could be explained, e.g. photoelectric effect Mulliken (1966). More refined spectro- superconductivity (Bardeen, Cooper and (Milliken, 1923), Einstein scopic results, e.g. by Schawlow, Schrieffer, 1972) or Bose–Einstein proposed that light is Bloemberg and Siegbahn (1981), condensates (Cornell, Weiman and quantized (1921). This was provided the basis of, e.g. computational Ketterle, 2001). This has even led to the supported experimentally quantum chemistry (Kohn and Pople, discovery of new phases of matter by Compton (1927). 1998) and femtochemistry (Zewail, 1999). (Thouless, Haldane and Kosterlitz, 2016).

1902 1921 1927 1930 1933 1955 1962 1999 2012 2016

Quantum theory of atoms: Quantum theory of light and matter: On the basis of the spectra Based on spectroscopic measurements of the measured by Lorentz and hyperfine structure (Lamb and Kusch, 1955), a Zeeman (1902) and the complete theory of quantized light and matter was quantum hypothesis (Planck, developed (Feynman, Tomonaga and Schwinger, 1918), Bohr (1922) linked the 1965). With the development of the laser (Basov, spectral lines to the quantized Prokhorov and Townes, 1964), the spectroscopic nature of the hydrogen atom. methods became more advanced, which allowed This was later confirmed (Hertz control of the interaction between light and matter and Franck, 1925) and put into (Chu, Cohen-Tanoudji and Phillips, 1997). Further a rigorous form by Heisenberg developments in quantum optics (Glauber, Hall and (1932) and Schrödinger and Hänsch, 2005), allowed the control of individual Dirac (1933). quantum systems (Haroche and Wineland, 2012).

Figure 1 | Schematic evolution of our understanding of quantized coupled light–matter systems.Nature Reviews | Chemistry Representative Nobel prizes in physics and chemistry are highlighted (years in parenthesis).

formalized by the Maxwell equations39, and Dirac, Pauli Feynman, for which they were awarded the Nobel prize and Heisenberg were the first to provide a formulation for in physics in 1965 (REF. 44). the quantization of the electromagnetic field, thus giving Within the non-relativistic limit for the matter sub- birth to QED40. Electromagnetic fields induce dynamics system, the full QED Hamiltonian can be simplified to in a system of charged particles. Therefore, simultane- the Pauli Hamiltonian (for the matter subsystem) describ- ously treating photons and charged particles requires tak- ing the evolution of charged particles in spinor representa- ing into account the influence that charged matter and tion, which are coupled through the charge-density electromagnetic radiation have on each other. In QED, and charge-current operators to the quantized photon 41,42 34,42 this is done via the minimal coupling prescription , field . For a system of Ne electrons and Nn nuclei, the such that the charge current in the Dirac equation corresponding Hamiltonian, known as the Pauli–Fierz becomes the source of the quantized electromagnetic Hamiltonian43,45, is field, which at the same time modifies the momentum Ne |e| 2 of the Dirac fields. The quantized electromagnetic field Ĥ (t) = 1 σ · –iħ + Âtot(r , t) + PF Σ 2m [ l ( rl c l )] is described by individual quantum harmonic oscillators l = 1 ⊥ N n (n /2) 2 for each allowed mode and polarization, which gives rise 1 l Zl|e| tot Sl · –iħ R – Â (Rl , t) + Σ 2M [ ( l c )] Pauli Hamiltonian to the electromagnetic vacuum. Unfortunately, without l = 1 l ⊥ The Pauli Hamiltonian further modifications, the straightforward QED for- Ne Nn 1 1 comprises the standard mulation in terms of local photon–matter interactions w rl –rm + ZlZmw Rl –Rm + 2 Σ (| |) 2 Σ (| |) Schrödinger Hamiltonian and (that is, particles interact by sending photons back and l ≠ m l ≠ m the Pauli (Stern–Gerlach) term 41 Ne Nn forth) leads to unphysical results , because even a purely † σ ·B(r), which describes the Z w r –R + ħω a a – Σ Σ m (| l m|) Σ k k, λ k, λ coupling between the electron perturbative treatment of the coupled system diverges l = 1 m = 1 k, λ 43 spin (characterized by a vector beyond the first order. To solve this issue, Bethe intro- of the usual Pauli matrices σ) duced the concept of energy cut-offs and counter terms In this case, the internal structure of the nuclei is and the magnetic field B(r). — for instance, a diverging ‘electromagnetic’ mass result- neglected (as the individual protons and neutrons are

Spinor representation ing from the interaction with the bare electromagnetic not resolved), and the Coulomb gauge (photons are To represent the spin of a vacuum — that absorb the infinities when the cut-offs allowed to have only transversal polarization, λ = 1, 2) has quantum particle a vector of are removed. This renormalization procedure has been been chosen so that the longitudinal part of the photon wavefunctions can be used, in investigated in detail in high-energy physics via a per- field is given explicitly in terms of the charged particles which each entry corresponds turbative treatment of scattering events41, and its use has (electrons and effective nuclei) only41. The interaction to a specific spin state of the particle. A spin one-half led to results in excellent agreement with those of exper- between the charge-density operator and the longitudinal particle has two such entries, iments. The formal development of this renormalized part of the photon field then gives rise to the Coulomb that is, spin up and spin down. 2 QED theory was done by Tomonaga, Schwinger and interaction, w (| r − rʹ |) = e /4πε0| r − rʹ |, among particles

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3 (third, fourth and fifth terms in equation 1). Here, σl ∑l − |e|υext(rl, t) + ∑l Zl|e|υext(Rl, t)−∫d r (r)∙jext(r, t)/c is a vector of the usual Pauli matrices, reflecting the to the Hamiltonian in equation 1. Although,⊥ in this one-half value of the electron spin, m and |e| denote the case, only the electromagnetic radiation is described mass and the absolute value of the elementary charge, by a quantum field, the local coupling still gives rise to respectively, and the total transversal vector potential divergencies already at the level of perturbation theory. Âtot(r, t) =  (r) + Aext(r, t) contains both the quantized However, by introducing a physically reasonable energy internal⊥ transversal⊥ photon degrees of freedom repre- cut-off 43 for the photon modes, a formulation similar to sented by the transversal vector-potential operator  (r) the one of quantum mechanics46 based on self-adjoint (REF. 41) and any possible classical external vector poten⊥ - operators47 is possible. The emerging Hamiltonian tial Aext(r, t) (describing the interaction with external fulfils the variational principle, and the ground state of (nl/2) classical electro­magnetic fields). Further, Sl denotes the combined photon–matter system is mathematically 48–50 43 a vector of spin nl/2 matrices (where nl is even for well defined . As already pointed out by Bethe , in the

even-mass-number nuclides and nl is odd for odd-mass- non-relativistic regime, such a frequency cut-off (usually

number nuclides), reflecting the nl/2 value of the spin of taken at the rest-mass energy of the electron, which is the l‑th nucleus. For instance, to describe a nucleus with roughly 0.5 MeV) is physically reasonable because it an even mass number and consequently even effective ensures the underlying particle description in the Pauli– spin, for example, spin 1, a vector with three components Fierz Hamiltonian. In certain limits, the cut-off can even

of 3 × 3 spin-matrices is used. Furthermore, Ml denotes the be removed, and an exactly renormalized Hamiltonian 51 mass of the l‑th nucleus, and Zl denotes the corresponding can be defined . These are the main differences between effective positive charge. The last term in equation 1 gives non-relativistic and full (also charged particles treated the total energy of the quantized electro­magnetic field, fully relativistically and second quantized) QED, which † where ˆak,λ and ˆak,λ are the usual bosonic creation and anni- is usually formulated in terms of a perturbation theory hilation operators, respectively, for mode k and polar- for scattering events. ization λ. For notational simplicity, we do not include the possible coupling of the charged particles to a sca- Approximations to non-relativistic QED

lar external classical potential υext(r, t) or the coupling The relevant physical and chemical processes of interest of the photons to a classical external charge current in this Review correspond to an energy range well below

jext(r, t). However, these can be simply included by adding 1 MeV. In this energetic regime, the creation of electron– positron pairs can be completely neglected, and a simpler first-quantized (particle-number conserving) descrip- tion of matter based on either the Pauli or higher-order approximations to the Dirac equation can be adopted. a Properties of light b Properties of matter The electromagnetic field is kept fully quantized, and, Theory: Theory: in contrast to charged particles, photons can be created classical electrodynamics quantum mechanics and destroyed. Thus, we focus the remaining part of the Dynamical variables: Dynamical variables: electromagnetic fields electrons and effective nuclei Review on the discussion of this non-relativistic QED formulation (see equation 1). c Quantum nature of light The low-energy QED formulation has three basic Theory: constituents — nuclei, electrons and photons — in which quantum optics Dynamical variables: the latter constitute the quantized electromagnetic field. photons and effective particles It is commonly believed that these three components are sufficient to explain all spectroscopic results within Properties of charged particles and photons the low-energy range. In this framework, the nuclei are Theory: non-relativistic QED Dynamical variables: electrons, effective nuclei and photons treated as positive point charges with a nuclear spin and mass instead of a combined system of protons and Properties of electron–positron fields and photons neutrons. This is justified by the fact that in the low- Theory: renormalized QED energy regime, only the nuclear motion — not the internal Dynamical variables: electrons, positrons and photons structure — is relevant. Properties of matter fields and their gauge bosons It is, of course, very appealing to directly challenge the Theory: standard model of particle physics non-relativistic QED theory with spectroscopic measure- Dynamical variables: quarks, leptons, gauge and scalar bosons ments of molecules, nanostructures and extended systems in order to test its accuracy, as done in high-energy physics Nature Reviews | Chemistry Figure 2 | Theoretical description of photon–matter interacting systems. Different studies. In the high-energy regime, fundamental aspects properties of photon–matter interacting systems are grouped and associated with the are investigated in detail, such as the instability of the dynamical variables and theoretical approaches that are currently used for their QED vacuum for extremely-high-intensity laser fields52 description. The properties of light and matter are treated within different frameworks (classical and quantum theories, part a, b and c respectively). Their combination results in that leads to a modification of the Maxwell equations 53 54 quantum electrodynamics (QED), which describes all fundamental quantum aspects of in vacuum , the electroweak interaction or QED effects in 55 electrons, positrons and photons. The currently most complete description of matter is plasmas . Similar effects are also investigated in materials the standard model of particle physics, which also includes a description of the nuclei, science studies, where a real material is used to simulate electroweak interactions, and others. In the low-energy regime, non-relativistic QED can the mixed axial–gravitational anomaly56, or Higgs57, be employed. Majorana58 or axion physics59. Other complications appear

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when one goes beyond scattering theory to determine, for constant. The coupling between the (transversal) charge example, the ground state and the spatially and temporally current and the electromagnetic field can be expressed resolved dynamics of electron–nucleus–photon systems. by using only the total dipole moment of the system In practice, approximations are introduced to treat the and the uniform electric field23,42. This approximation interacting particles as well as to model and interpret is commonly used in conjunction with the restriction of experimental results. Those approximations strongly the full-photon field to only a few contributing modes depend on the character of the light–matter system that that are in or near resonance with the selected energy is probed. The corresponding approximate equations levels of the isolated matter system that is not coupled might differ and provide access to different processes and to the light field. It is also possible to restrict the matter interactions. The different fundamental aspects of QED degrees of freedom to a few energy levels, leading to the that are usually probed can be grouped into three broad few-level and few-mode approximation23. The simplest categories (FIG. 2): the quantum nature of light investigated form, known as the Rabi model, comprises a two- in quantum optics23 or cavity QED60; the spatial and tem- level system coupled to one mode of the photon field, poral properties of the electromagnetic field investigated which is described by a harmonic oscillator. If it is fur- in photonics61 or plasmonics62; and the matter degrees of ther assumed that the absorption of a photon can only freedom investigated in solid-state physics or quantum excite the sample and emission can only de‑excite the chemistry63. The available (and new) spectroscopies can sample, the resulting simplification is called the Jaynes– then be categorized based on the property investigated. Cummings model, which effectively ignores quickly The first category corresponds to spectroscopies that oscillating terms (the rotating-wave approximation)23. directly probe the quantum nature of light. A typical This level of approximation to the Pauli–Fierz experimental setup includes microwave64 or optical65 Hamiltonian of non-relativistic QED is employed, for cavities and well-characterized matter systems, such instance, to describe single-atom lasers69,70, which were as Rydberg atoms or quantum dots. In such well- experimentally realized for the first time with a caesium controlled systems, the intricate behaviour of photons atom in a high‑Q optical cavity71. Often, these model and their interplay with matter can be directly observed. Hamiltonians are solved as open quantum systems72 Experiments performed in these conditions include to more accurately account for losses in the real situa- single-photon measurements in quantum optics66 or tion when photons can leave the cavity. Such few-level quantum information67, single-atom masers (microwave approximations are used not only in quantum optics amplification by stimulated emission of radiation) in but also in quantum chemistry and solid-state physics, cavity QED60 and electromagnetically induced trans­ for example, in the context of light-harvesting systems73 parency68. Most theoretical descriptions rely on the dipole or in nuclear magnetic resonance74. However, such a approximation, also known as the long-wavelength or reduced treatment can often be insufficient75,76, especially optical limit, which assumes that the relevant wavelength if one is interested in more than just the simple observ- of the electromagnetic field is much larger than the spatial able that the model was designed to describe, such extension of the matter subsystem. In this case, the as the dipole moment, and the physical implications spatial non-uniformity of the field at the relevant frequen- of few-level models are debated in the literature, for cies at any instant in time is neglected, meaning that the example, the superradiance phase transition due to the mode functions for the light field are approximated by a Dicke model77.

a Matter–matter b Photon–matter c Photon–photon

e– p p′ ωk ωk′ ωk p γ γ′ p (e–, p) + γ′ + p – – e e + – – e (e–, p) (e–, p) e γ – p p (e , p) γ – + + – – e p – – e e ′ + e e ′ γ γ′

Chemical bonding Photoionization γ γ′ Photon blockade

Figure 3 | Schematic description of the different components of the light–matter QED Hamiltonian.Nature Reviews | Chemistry The Hamiltonian describing the light–matter interaction in quantum electrodynamics (QED) accounts for matter–matter interactions (part a), photon–matter interactions (part b) and photon–photon interactions (part c). The straight lines in the diagrams of perturbation theory for each interaction represent charged particles (nuclei p and electrons e−), and the wiggly lines represent photons, γ. Part a highlights how matter–photon coupling induces effective matter–matter interactions, for example, the Coulomb interaction between electrons and nuclei, which is responsible for chemical bonding of a dimer. Part b shows a direct photon–matter interaction, for example, photoionization of a dimer. Part c highlights how matter–photon coupling can lead to effective photon–photon interactions responsible for, for

example, photon blockade in an optical cavity, where ωk is the cavity frequency. We note that despite the employed perturbative picture, QED naturally includes the self-consistent back-reaction of light on matter and vice versa, but this back-reaction is commonly neglected.

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In the second category, the quantum nature of light permittivity and permeability. For optical and electronic is considered not to be relevant because the number spectroscopies of molecules and solids, this decoupling of photons involved is usually large, for example, such of matter and light is usually employed. as in vibrational or ultra-fast laser spectroscopies78. In The third category usually provides the input to such cases, a semi-classical treatment of the electro- the above constitutive relations from a matter-only magnetic field of the combined light–matter system is perspective. The photon field is taken into account only by usually sufficient, which means that the non-relativistic the re­normalized masses (bare plus electro­magnetic46) QED description will include the classical Maxwell and the (longitudinal) Coulomb interaction among the field instead of the quantized photon field. This leads charged particles, as well as possible QED corrections. to a coupled Maxwell–Pauli equation (BOX 1), which is The matter-only description can be further simplified by employed for the characterization of ultra-fast electron decoupling the electronic and nuclear degrees of freedom dynamics in solids79,80, X‑ray single-molecule imag- and by means of conditional wavefunction expansions it ing81 or molecular nanopolaritonics82 or for the study is then possible to obtain Born–Oppenheimer surfaces of the spatial and temporal properties of the light field or quantized nuclear motion approximated by phonon (for example, in the context of fibre optics83, optical anten- modes. In its simplest form, the nuclear motion is approx- nas84, near-field spectroscopies85, optical tomography86, imated by semi-classical trajectories (Ehrenfest dynam- interferometry87 and holography88). In the latter case, it ics) coupled to the many-electron Schrödinger equation. is usually assumed that the light field leaves the matter This approach is indeed similar to the de­­coupling properties unchanged. This directly results in the macro­ scheme that leads to the Maxwell–Pauli equation, scopic Maxwell equations, where the matter degrees of where the classical equation for photons is solved in freedom (arising from the source term J (r, t) in BOX 1) conjunction with the many-body Schrödinger equation are used to define the electric displacement⊥ and magnet- (BOX 1). In particular, when only the electronic degrees izing fields. Constitutive relations, such as the depend- of freedom are considered, such as in the electron- ence of the polarization and magnetization on external spin-resonance or Ramsey technique90, the nuclei are fields, are then usually determined from a matter-only assumed to be clamped and usually treated as classi- theory (that is, the third category discussed below). This cal external Coulomb potentials. A good example of level of approximation to the Pauli–Fierz Hamiltonian such a simplified treatment of QED in the context of + of non-relativistic QED is employed, for instance, molecular systems is the Hamiltonian for H2 (or other in calculating the local fields of plasmonic structures89. In simple molecules26,91), which includes the Lamb shift, the limit of linear optics, the matter degrees of freedom the fine structure and the hyperfine structure (see are further simplified and reduced to an effective REF. 92 for higher-order contributions). Neglecting all

Box 1 | Maxwell–Pauli equation If we make a mean-field ansatz for the matter–photon coupling in non-relativistic QED (see equation 1), we can approximate the correlated matter–photon wavefunction Ψ as the product of the matter wavefunction ψ and photon wavefunction ϕ that is, Ψ ≈ ψ ϕ. This ansatz, which is similar to the Born–Oppenheimer ansatz in electron–nuclear dynamics, enables us to rewrite⊗ the correlated problem as two coupled equations181

∂ iħ ψ(t)=Ĥ (t)ψ(t) ∂t P and 1 2 ∂ – ∇2 A (r, t) = μ cJ (r, t) ( c2 t2 ) 0 ∂ ⊥ ⊥ where the Pauli Hamiltonian for Ne electrons and Np nuclei is given by

Ne 2 Nn 2 1 |e| tot 1 Zl|e| tot Ĥ (t)= σ · –iħ + A (r , t) + S (nl /2) · –iħ – A (R , t) P Σ [ l ( rl l )] Σ [ l ( Rl l )] l = 1 2m c ⊥ l = 1 2Ml c ⊥ Ne Nn Ne Nn 1 1 + w(|rl –rm|) + ZlZmw(|Rl –Rm|) – Zmw(|rl –Rm|) 2 lΣ≠ m 2 lΣ≠ m lΣ= 1 mΣ= 1

Here, σl is a vector of Pauli matrices, m and |e| denote the mass and the absolute value of the elementary charge, respectively, and the total transversal vector potential Atot (r, t) = A (r, t) + Aext(r, t) + Aext(r, t) contains both the transversal External fields Maxwell field A (r, t) coming from the coupled Maxwell equation⊥ and⊥ any classical external vector potential Aext(r, t). This Fields that are externally is the mean-field⊥ approximation to the matter–photon coupling in the Pauli–Fierz Hamiltonian of equation 1. Further, controlled, fixed perturbations, (nl/2) such as pump and probe Sl denotes a vector of spin nl/2 matrices, Ml denotes the mass of the l‑th nucleus, and Zl denotes the corresponding 2 pulses or in the clamped nuclei effective positive charge. The interaction terms are given by the Coulomb interaction w(| r − rʹ |) = e /4πε0| r − rʹ |. Further, 41 96 approximation the nuclear J (r, t) is the induced transversal charge current of the matter system , . If there is no induced transversal charge current, attractive potential, whose the⊥ two equations decouple, and we are left with the usual many-body Pauli equation in the Coulomb approximation. sources are not included in Note, however, that in general time-dependent problems, the Coulomb approximation does not account for relativistic theoretical description. These causality. Local changes are felt instantaneously everywhere. Causality of the field is restored by including the neglected fields are usually classical but transversal part of the current density (that is, retardation effects or memory effects). can also be of quantum nature.

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the higher-order contri­butions from QED leads to the are also applicable when photon and matter degrees of usual electronic Schrödinger approximation, which for freedom are equally important96,99. The price to pay is instance, well describes resonance energy transfer and that approximations are needed for both the unknown the 1/R6 law, which was experimentally verified for the exchange–correlation functionals in density functional first time in tryptophyl peptides93. theory and the self-energies in Green’s function methods. It is possible, however, to approximate non-relativistic The recently developed approximations100 in the context QED in a different way. Instead of simplifying the of quantum-electrodynamical density functional theory Hamiltonian, one can reformulate the full problem in (QEDFT; BOX 2) can accurately treat explicitly coupled terms of reduced quantities that avoid unaffordable matter–photon situations99,17. These approximations also explicit calculations of the wavefunction. Here, we can provide a promising scheme to study realistic complex follow well-known strategies routinely employed in molecular systems and solids in quantum cavities (FIG. 4) quantum chemistry and solid-state physics, in which and address the appearance of novel states and phases the ground-state and time-dependent many-body that would have been inaccessible otherwise (including Schrödinger problem is reformulated in terms of density control of chemical reactions, energy transfer, and so on). functional theory94 or Green’s function theory95 to make The above approximations of non-relativistic QED this problem affordable for numerical computations have proved to be exceptionally successful. Predicted and simulations. Indeed, the common matter-only transition frequencies for small atomic or molecular density functional theory and Green’s function theory systems have been found to be in good agreement with approaches (applicable only to the third category above) high-precision spectroscopic measurements and are are approximations to density functional and Green’s used as benchmarks for fundamental constants and function reformulations of non-relativistic QED96–98 (in the accuracy of QED in the low-energy regime101–104. terms of densities and currents or equations of motion for Although the accuracy105 of these theoretical predic- the Green functions and self-energies). These formally tions decreases for more complex systems, such as bio­ exact reformulations of the Pauli–Fierz Hamiltonian molecules or solids, it is still possible to qualitatively treat light and matter as equally quantized and hence capture most of the relevant physico-chemical processes.

Box 2 | Quantum-electrodynamical density functional theory (QEDFT) Density functional theories94 are exact reformulations of the many-body problem in terms of an exact quantum-fluid description, in which only the total densities of the system appear. Instead of modelling the momentum stress and interaction stress tensors explicitly, one usually employs the Kohn–Sham construction, which uses the corresponding expressions of a non-interacting reference system and approximates the differences between the interacting and non-interacting quantum fluids with exchange–correlation fields. The original formulation of a purely electronic Hamiltonian with scalar external classical fields has been extended to very different physical situations, including superconductivity182 and general external classical electromagnetic fields183,184. In the latter case, instead of an exchange– correlation potential, an exchange–correlation vector-potential is needed to model the missing forces resulting from the Coulomb interaction. In the case of a coupled matter–photon system described by equation 1, the exact density functional reformulation96,185–188 includes photons in addition to the charged particles (electrons and nuclei), and thus, the particle subsystem can enact forces on the photon subsystem and vice versa. The corresponding Kohn–Sham scheme employs the expressions of a non-interacting multi-particle system and an uncoupled photon field, with exchange– correlation contributions describing the interaction due to the longitudinal photons and a new exchange–correlation contribution resulting from the interaction mediated by the transversal photons99. Without loss of generality, the original coupled fermion–boson problem (equation 1) can be exactly rewritten in terms of self-consistent coupled Maxwell– Kohn–Sham–Pauli equations188

2 ∂ 1 iħ ψ(t)=Ĥ (t)ψ(t) and ∂ –∇2 A (r,t)=μ cJ (r,t) ∂t MKS ( c2 t2 ) 0 ∂ ⊥ ⊥ where N N e |e| 2 n 2 1 tot xc 1 (n /2) Zl |e| tot xc Ĥ (t)= · –iħ + (A (r ,t)+A (r ,t)) + S l · –iħ – (A (R , t) + A (R , t)) MKS [σl ( rl l l )] [ l ( R l l )] Σl=1 2m c Σl=1 2Ml l c ⊥ ⊥ Here, Atot (r, t) = A (r, t) + Aext(r, t) and Axc[J, A ] are the exchange–correlation vector potentials that capture the missing internal forces⊥ from⊥ the photon–matter coupling⊥ and depend implicitly on J calculated from ψ(t) (REFS 41,96),

on A , and on the initial many-body and Kohn–Sham states. Further, σl is a vector of Pauli matrices, m and |e| denote the ⊥ (nl/2) mass and the absolute value of the elementary charge, respectively, Sl denotes a vector of spin nl/2 matrices, Ml

denotes the mass of the l‑th nucleus, and Zl denotes the corresponding effective positive charge. Here, J (r, t) is the transversal part of the induced charge current of the matter system. The Maxwell–Kohn–Sham–Pauli equations⊥ can be further decoupled into a set of self-consistent single-particle equations for the electrons and nuclei96. We note that by making approximations to quantum-electrodynamical density functional theory (QEDFT), we recover the exact reformulations of simplified models of non-relativistic QED. For example, by assuming the dipole approximation, we reduce to an exact reformulation of non-relativistic dipole QED96. Neglecting the A (r, t) and the corresponding exchange–correlation terms decouples the electromagnetic and matter sectors and⊥ leads to the common matter-only density functional theory96,187.

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a b c –5 Photon OEP n(r) –3 (Photon OEP–OEP) Δn(r) –6 ×10 n(r) and Δn(r) ×10 ×10 5 15 8.0 15 2.0

) n(r)/2000

-3 4 10 10 Δn(r) 6.0 3 5 5 1.0 2 -3 -3

0 ( Å ) 0 Å 4.0 Å x ( Å ) x 1 –5 –5 0 2.0 0.0 –10 –10

density ( Å Electron –1 –15 0.0 –15 –2 –15 –10 –5 0 5 10 15 –15 –10 –5 0 5 10 15 –15 –10 –5 0 5 10 15 y (Å) y (Å) y (Å)

Figure 4 | Numerical example for a QEDFT calculation: study of a 3D sodium dimer in an opticalNature cavity. Reviews | Chemistry The electronic ground-state density n(r) in the xy plane (z = 0) of a sodium dimer strongly coupled to the mode of an optical high‑Q cavity calculated with quantum-electrodynamical density functional theory (QEDFT) in dipole approximation using the optimized effective potential (OEP) approach193 (part a), the difference between the electron density in the xy plane of the coupled and the ‘bare’ sodium dimer Δn(r) (part b), and the difference between coupled and bare density in blue against the coupled density in grey in the xy plane and summed along the x axis (part c). Note that the latter has been reduced by a factor of 1/2,000. We note that changes in the ground-state density are small, that is, they are the result of a cavity-induced Lamb shift. For this setup, we find 7.86 × 10−4 photons in the cavity. In other observables and in the excited state (see also FIG. 5), the changes can become very large (see REF. 193 for further details on the accuracy of the photon OEP approach). These graphs were obtained using the parameters for a sodium dimer given in REF. 194. The energy of the

3s–3p transition was chosen in resonance to the optical cavity frequency, that is, ħωk= 2.19 eV. Further, the photon field is 1/2 −1 polarized along the y direction with a strength of λk = 2.95 eV nm ey. The real-space grid is sampled as 31.75 × 31.75 × 31.75 Å3 with a grid spacing 0.265 Å.

Electronic and optical properties phonon–phonon scattering processes). Moreover, the We have introduced the intrinsic approximations com- Maxwell and time-dependent Schrödinger equations can monly used to describe and understand the different be decoupled to determine how the molecule reacts to any spectroscopies. However, accounting for correlated external perturbation or the absorption and/or emission matter requires further approximations. Such approx- of light. Despite the dipole approximation for the external imations include, for instance, the truncation of the field, the field induced by the fluctuations of the charge Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy106, and current densities (source terms in BOX 1) needs to be renormalization-group techniques107–109, stochastic equa- determined. This field induced by the polarized electronic tions such as master and Fokker–Planck equations110 cloud of the molecule tends to compensate the external and many others23,111–118. An in‑depth discussion of each perturbation and can give rise to the emission of photons, of these is beyond the scope of this Review. However, in possibly at higher frequencies (this corresponds to the order to analyse the limitations of the chosen intrinsic case of nonlinear optics as high-harmonic generation). To approximations, we need to consider specific systems. avoid full self-consistency, the calculated electronic charge Here, we choose to consider the electronic and optical and current densities are used as fixed inputs for the properties of molecular systems. In this way, we can iden- Maxwell equations79,119. Still, these calculations are cum- tify physical situations that highlight QED effects beyond bersome and can be further simplified by assuming that the applied intrinsic approximation in molecular sys- the molecules radiate like dipoles. This finally enables one tems, which we can use to directly challenge the QED to completely neglect the Maxwell equations and consider framework. In the molecular case, focus is placed on the only the time-dependent Schrödinger equation to compute matter degrees of freedom (FIG. 2b). The following dis- the reaction to spatially and temporally arbitrary external cussion will further highlight that there are several addi- perturbations. tional approximations involved along with those leading In many instances, instead of solving the full time- to the approximation to the Pauli–Fierz Hamiltonian. dependent Schrödinger equation, one can evaluate Isolated gas-phase molecules provide an ideal frame- specific linear and higher-order response functions (BOX 3) work to test the limits of common approximations, even using perturbation theory. The idea of response functions if experiments are far from being trivial. In most exper- used in time-dependent perturbation theory closely iments, many photons are involved, and the mean-field resembles that of a spectroscopic experiment where approximation for the photon field is well justified. This an external probe — radiation or particles — perturbs means that the non-relativistic QED description is replaced the system and a signal — radiation fields (electric and by a description based on the Maxwell–Pauli equation magnetic), particles or both (coincidence experiments) (BOX 1). Assuming clamped nuclei, the many-body system — is detected. As an example, the absorption spectrum is simplified to a many-electron system, which can be (dipole response of the system due to an external electric modelled by the standard many-electron Schrödinger dipole field) for an isolated gas-phase model dimer is equation. Usually, the dipole approximation is also shown in FIG. 5a in red. However, in order to describe the applied to external fields. Furthermore, electron–phonon nonlinear dynamics of driven systems, one has to solve coupling is often treated perturbatively (including the time-dependent equations described above. Similar

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a Absorption spectra b Light–matter spectra c 2D absorption spectra 16 ħωk ħωk Electronic 14 4 excitations 12 Vibronic ħΩR sidebands 0 10

8 –4

6 log [I (a.u.)] –8 log (Intensity) (a. u.) log (Intensity) (a. u.) 4 Absorption energy (eV) Absorption energy

2 –12 Rabi splitting ħΩ ħΩ R R 0 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16 0 2 4 6 8 10 12 14 16

Absorption energy (eV) Energy (eV) Cavity energy ħωk (eV)

Figure 5 | Calculated spectra for a 1D model dimer. a | From top to bottom: electronic excitationsNature (red) Reviews with clamped | Chemistry

nuclei, electronic excitations dressed by vibronic sidebands (green), and Rabi splitting ΩR of electronic excitations (blue) for

the dimer in an optical cavity with frequency ωk with increasing effective coupling strength (see BOX 4 for more detail). b | Light–matter spectra, as defined in BOX 5. The negative and positive amplitudes are plotted separately. c | 2D absorption

spectra for the dimer in a cavity, where we scan through different cavity frequencies ωk. Results obtained using the clamped-nuclei approximation are shown in blue, whereas the results obtained considering the quantized electron–

nucleus–photon system are reported in red. For the spectra in parts a and b, we chose a cavity frequency ħωk of 11.02 eV and

electron–photon coupling strengths gk /ħωk of (0.425, 0.85, 1.70)/nm; the same parameter for the spectra reported in part c

was chosen to be gk/ħωk = ħωk ×0.3843/nm eV. The spectra in parts a and c were calculated using equation S21 from 2 REF. 17, and for those reported in part b, we replaced in the same equation | Ψ0|R|Ψk | → Ψ0|R|Ψk Ψk|q |Ψ0 , where Ψ0 represents the ground state of the coupled matter–photon system and Ψk represents⟨ ⟩ the excited⟨ states.⟩⟨ The system⟩ is ˆ ˆ ˆ identical to the first example in REF. 17 and can be described by the Hamiltonian Ĥ=Te +TN +Ŵee +ŴNN +ŴeN +ĤP , where Te ˆ ˆ ˆ ˆ and TN represent the standard electronic and nuclear kinetic energy operators, respectively; Wee , WNN and WeN correspond to the electron–electron, nuclear–nuclear and nuclear–electron interactions, respectively, with the soft-Coulomb interaction (|r – r |) = e2/4π (r – r )2 + 1 2 2 2 · 2 w ε0 (where r and rʹ are the positions); and ĤP =[–∂ /∂q +ωk(q+λ eR/ωk) ]/2, where ( †) ˆq= ħ/2ʹ ωk aˆ+aˆ is the photonicʹ displacement coordinate, λ is the transversal polarization vector times the dipole- approximation coupling strength (| | = g 2/ħ ), and R = Z R – r is the total dipole operator and Z denotes the λ ωk Σl l l Σl l l nuclear charge.

approximations are also found in photoemission spec­ microscopic response of the electronic system is used troscopy120,121, which is usually described by either a one- as an input into the macroscopic Maxwell equations to step or a three-step model. The latter model is based on obtain the macroscopic response of the system, which cor- the difference between the intrinsic effects, which are responds to the experimentally observed radiation. This described by the purely electronic Hamiltonian and the approach is valid within the limits of weak perturbation corresponding spectral function, and the extrinsic effects. and when the back-reaction of light on matter is neglected The extrinsic effects account for polarization effects, (that is, decoupled Maxwell and Schrödinger equa- effects arising from electron propagation in the matter, tions). In this way, certain deficiencies of the intrinsic interactions between electrons and the created holes Schrödinger equation can be overcome by including and their propagation through the surface. The one-step some of the extrinsic effects. For instance, modifica- model is simpler and relies on the sudden approximation tion of the optical properties of a gas of molecules upon for which the electron is assumed to reach the detec- increasing its density is well explained and captured by tor instantaneously, without any interaction with other this microscopic–macroscopic connection, which incor- (quasi-)particles nor light nor the surface of the material, porates the polarization effects absent in basic quantum thus neglecting all the aforementioned extrinsic effects. mechanics. Similar extrinsic effects, due to coupling with When the electrons are not treated as independent the microscopic fluctuations of the induced light field particles, their interaction leads to complex fluctuations (local-field effects), can be included in the modelling of in the electronic charge and current densities. The cor- photoemission experiments120,121. responding induced fields should be computed from the The study of real molecular, nanostructured and Maxwell equations and added to the external field. This extended systems requires practical and accurate approx- can be incorporated in the formalism of perturbation imations to the intrinsic electronic equation. This is a theory by means of a correction term, leading to what central topic of modern quantum physics and chemistry is called the microscopic–macroscopic connection, and has led to the development of several theo­retical which provides a practical framework to incorporate the methods, including quantum chemistry methods with effects of the induced current and density fluctuations configuration interaction63,126, coupled-cluster127,128, back into the Maxwell equations122–125. In this case, the quantum Monte Carlo129, tensor network approaches130,

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Box 3 | Linear and higher-order response functions Non-relativistic QED can help to overcome this defi- ciency. For instance, when increasing the density of For a weak external perturbing field F, the change in an observable O can be expressed atomic potassium vapour while measuring the optical as a sum of different responses, coming from different orders in the perturbation spectra, polarization and local-field effects become (1) (2) (3) Oind = χ F + χ FF + χ FFF +…. important. From non-relativistic QED, we know that When the perturbation is small enough such that |χ(1) F| »|χ(2) FF| »…, this series is in such cases, we need to include at least the back- convergent. This is the perturbative regime, in which many experiments are conducted. reaction from the classical electromagnetic field124. If the In such an expansion, χ(n) are called n‑th order response functions. The response same potassium vapour is inside a high‑Q optical cavity, 189 functions can be computed recursively, using the so‑called ‘2n + 1’ theorem . These a coupled Maxwell–Pauli equation might no longer pro- response functions are extremely useful, as they determine the response of the system vide an accurate model, and we have to include some to any weak perturbation. To make this more explicit, we consider the first-order- of the discarded quantized photon modes explicitly142. induced density n(1) of a system that is perturbed by an external classical vector ind In this way, we are able to directly test this higher-lying potential Aext(r, t) starting at time t = 0. The linear or first-order response and the perturbation are related by theory. Beyond these three obvious reasons lies the pos- t sibility that we probed something that is not contained in (1) 3 (1) ext nind(r, t)= dt d r χ n, J(r,t; r ,t )·A (r ,t ) ∫0 ∫ our current formulation of non-relativistic QED, that is, ʹ ʹ ʹ ʹ ʹ ʹ In this expression, we consider the so‑called density–current response function122, something in addition to photons, non-relativistic electrons but similar expressions can be obtained for any combinations of perturbations and and nuclei. Spectroscopic experiments and theory can responses, for example, density–density or magnetization–density responses. In a work hand‑in‑hand to test quantum field theory in the general case in which we have two observables, such as the charge n(r) and current low-energy regime and provide an alternative route to J (r, t), the density–current response function reads gain new insight into the fundamentals of light–matter

(1) ˆ interactions. χ n, J(r,t; r ,t )=–iθ(t–t ) [ˆnI(r, t), JI(r , t )]

ʹ ʹ ʹ ⟨ ʹ ʹ ⟩ where we use the notations, Ô ψ0|Ô|ψ0 , where ψ0 is the initial state Towards novel spectroscopies many-electron wavefunction, Ô⟨ I denotes⟩=⟨ the ⟩interaction picture, and the Heaviside The approximations to low-energy QED often fail to model 95 function θ(t) guarantees the correct time ordering, that is, causality . experiments that can directly test specific QED effects. If the probed system is stationary in time, for example, in the ground state, then the In these experiments, the fully coupled light–matter response depends on the time difference between the moment of the perturbation and character of QED becomes apparent and has a direct the moment when the system is probed (χ(1) (r, t; rʹ, tʹ) = χ(1) (r, rʹ; t − tʹ)). This implies that the first-order response functions depend only on a single frequency. However, for an influence on physical observables and physico-chemical initial state that is not an eigenstate of the electronic system or time-dependent phenomena. Effects due to the strong light–matter Hamiltonians, the response functions are non-equilibrium response functions, coupling can be observed in different scenarios: for depending on both time coordinates t and tʹ independently. These are response instance, when the coupling between atoms and photons functions relevant, for instance, for describing pump–probe spectroscopies. is mediated by nanoscopic dielectric devices143; in cyanine dyes, where it is possible to reach non-radiative energy transfer well beyond the Förster limit between a spatially dynamical mean-field theory131, hierarchical equations of separated donor and acceptor144; in living green sulfur motion approaches132–134, electronic structure theory with bacteria, in which the level structure can be modified by density functional theory94, many-body perturbation strongly coupling to a cavity mode, allowing for novel theory and others135,136. Many of these approaches have insight into photosynthesis145; or in state-selective chem- also been extended to a relativistic two-component or istry at room temperature achieved by strong vacuum– four-component treatment137–139. All these methods matter coupling18. Novel spectroscopic tools that make have certain strengths and weaknesses and are often explicit use of the entanglement of electromagnetic radi- used to complement and test each other. However, not ation and matter can be proposed to tackle those effects, all the above methods have an extension to dynamical but they pose limits to the current theoretical approaches. properties owing to the inherent complexity of the time- The proper treatment of electronic correlations is dependent many-body dynamical problem95,109,140,141. In one of the limitations of common theoretical modelling this respect, time-dependent density-functional-based based on the intrinsic many-electron Schrödinger equa- methods (and QEDFT, described in BOX 2) offer an tion. Strongly correlated or weakly (dispersion-like) inter­ alternative exact framework to determine the quantum acting charged particles are difficult to describe accurately. non-equilibrium dynamics and response functions of Common approximation strategies often fail to correctly interacting many-body systems. capture charge-transfer, double and multiple excitations Reasons for disagreement between theoretical models or autoionizing resonances. Although certain quantum and experiments can arise from the approximate solution approaches, for example, Monte Carlo, dynamical mean- of the intrinsic equation, the modelling of the spectro- field and embedding theories or coupled-cluster methods, scopic experiment and/or that the approximate intrinsic can provide better results, they are often not easily extend- equation itself is not accurate enough. The most inter- able to describe response functions or non-equilibrium esting case is observed when the intrinsic equation systems. The correlation problem becomes even more itself (here, the many-electron Schrödinger equation) severe if we also have to account for the quantum nature does not provide a sufficient description of the system. of the nuclei. In this case, the full molecular Hamiltonian In this case, it is the decoupling of light and matter that that couples electrons and nuclei needs to be solved. A leads to disagreement between theory and experiment, typical spectrum for a dimer model that results from this not a poor description of the electron–electron correlation. approach is given in FIG. 5a in green, in which the nuclei

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Box 4 | Strong matter–photon coupling and hybrid light–matter states experimentally, for instance, for molecules inside a cavity that reach what is known as the strong matter–photon Single photons interact weakly with charged particles in free space, and the main coupling regime18 (BOX 4). At the interface between cavity contribution to the photon field in this case is captured by the longitudinal Coulomb QED and quantum chemistry, we can no longer neglect interaction. This is the usual case of an isolated molecule (dimer in part a of the figure). the quantum nature of radiation. The states of the com- This dimer has a specific electronic excitation frequency ω between its ground state |g bined light–matter system — hybrid light–matter states and its first excited state |e that can be probed spectroscopically (by a field γ). ⟩ However, if we enclose photons inside a cavity, they can interact many times, and the — exhibit a combined electronic, photonic and nuclear ⟩ 17 effective coupling strength increases. This effective coupling strength is inversely character . This ‘dressing’ of the bare quantities by extra proportional to the volume V of the cavity and proportional to the quality factor of the bosonic degrees of freedom can have a strong influence cavity, that is, how often the photons are reflected at the mirrors, as well as the number on the chemical properties of the system under investiga- of charged particles N inside the cavity and also depends on the spatial shape of the tion and the macroscopic response of molecular assem- modes23. In the simple Jaynes–Cummings or Dicke picture, the coupling strength is blies18. For instance, the vibrational frequencies of the proportional to the Rabi frequency free-space molecules can be shifted148, and these hybrid 149 2Nωk states can exhibit enhanced Raman scattering . These ΩR | g|d|e | × nph +1 √ ε0ħV √ are examples of polaritonic chemistry effects, which can ∝ ⟨ ⟩ where we assume that the photon mode is in resonance with the transition frequency even arise in the case of a dark cavity; that is, the molecules ω of the isolated molecule. Further, dˆ is the dipole operator for the respective mode, couple to only the electromagnetic vacuum of the optical cavity, and no real photons are present. Indeed, it has and nph describes the number of involved photons. In this case, the excited electronic state |e splits owing to the coupling to the photon field into two hybrid states, the been shown that strong vacuum-coupling can change upper |up⟩ and the lower |lp polariton, as depicted schematically in part b of the figure, chemical reactions, such as photoisomerization or a 23 15,150 which have⟩ matter–photon ⟩character . The splitting is proportional to ΩR, and in recent prototypical deprotection reaction of alkynylsilane . 76,148 molecular experiments , a splitting as large as ΩR/ωkk ≈ 0.25 was observed. This also The traditional distinction between electromagnetic changes the characteristic excitation frequency ωʹ between the ground state and the radiation and matter becomes inadequate in these first excited state. A different way to achieve strong matter–photon coupling is, for cases because the fact that matter–matter interactions example, by field enhancement in plasmonic nanostructures, which gives rise to are mediated by photons becomes apparent. A descrip- nanoplasmonic cavities156,164,190. tion based on the many-body Schrödinger equation, in a Isolated dimer b Dimer in cavity which the explicit photon degrees of freedom have been discarded, is therefore not adequate. Even a description (BOX 1) γ based on the coupled Maxwell–Pauli equation might not be able to reproduce the observed effects, and p p an explicit treatment of the joint electronic and photonic γ + + quantum degrees of freedom becomes necessary. In this e– e– case, fundamental physical processes, such as the mass- e– e– renormalization of charged particles due to the photon p p field, become accessible in the low-energy regime46,151. In + + addition, in quantum optics, there are many interesting quantum phenomena76 that cannot be described by the ω Jaynes–Cummings or Dicke models152,153. The develop- ment of methods that treat the coupled matter–photon E ω′ problem on equal footing in order to explore, understand ⎮up〉 and control these complex hybrid interacting systems ⎮e〉 ħΩR ⎮lp〉 represents an active field of research between quantum ħω ħω′ optics and materials science20,154. Our recently developed (BOX 2) ⎮g〉 ⎮g〉 theory of QEDFT provides a computationally accurate and tractable framework to deal with such mixed Nature Reviews | Chemistry fermion–boson systems. It is exactly when light and matter are strongly cor- are treated fully quantum mechanically and vibronic related that low-energy QED should provide adequate sidebands appear. Time-resolved spectroscopies for approximations that we can test. A reasonable approx- non-adiabatic molecular dynamics, for example, require imation of the non-relativistic QED Hamiltonian a treatment based on the full molecular electronic and in this case consists of using the dipole approximation ionic Hamiltonian146,147. In this case, however, the prob- for the matter–photon coupling and only a few modes for lem does not arise from an insufficient description of the description of the photon field76. This results in the light–matter coupling of QED but rather from the fact that standard molecular Hamiltonian being coupled to a few the intrinsic equation (here, the electronic Schrödinger harmonic oscillators17,155. Typical spectra showing such a equation, with ions treated as classical particles follow- coupled electron–nucleus–photon problem are reported ing Newton dynamics) cannot capture the quantum in FIG. 5a, where the blue spectra were obtained by tuning correlations between the electronic and nuclear degrees the effective coupling strengths in a dimer model. The of freedom. Therefore, we need to consider situations strong matter–photon coupling leads to Rabi splitting between category b and c in FIG. 2, where the quantum (BOX 4) of the electronic excitation, and new quasi-particle nature of light cannot be neglected. Such a situation arises excitations emerge as the light–matter coupling increases.

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The red spectrum shows the cavity system in the clamped- field, relies on the solution of the many-electron nuclei approximation, whereas the full dispersion rela- Schrödinger equation. In the linear-response regime, this tion of the coupled light–matter system becomes visible is described by a matter–matter response function (BOX 3). in the two-dimensional absorption spectrum shown in The resulting induced charge fluctuations are then used FIG. 5c. The red–yellow spectrum shows the rich structure as a source term for the Maxwell equations to determine of the electron–nucleus–photon system, in which multi- the induced classical radiation. However, more than just ple Rabi splittings for different cavity frequencies become matter–matter response functions should be considered visible. They strongly alter the spectrum and demonstrate to predict the response of the combined light–matter the polaritonic nature of the underlying states. Simple system to an external perturbation, especially in the case few-level approximations pose limitations for captur- of strong coupling or hybridized light–matter states. ing these effects in the full energy range, because such Indeed, for coupled light–matter systems, general photon– effects require a description in full real space. Although matter response functions (BOX 5) should be considered. semi-classical theories can describe hybridizations This allows one to directly determine the response of the and energy splittings to a first approximation in the quantized photon field or cross-correlation between the weak-coupling limit, an accurate treatment requires full matter and photon subsystems, for example, in FIG. 5b quantum theory. It should also be noted that control of where the matter-displacement response is shown. The matter–photon coupling by means of plasmonic-field use of photons and their quantum nature to probe the enhancement156 or collective excitations18 is still an open matter system to determine many-electron correlations22 experimental issue. or to trigger specific chemical reactions seems to be an The inclusion of electron–photon dressing induces interesting alternative to existing common techniques that some changes in the intrinsic equation such that the rely on classical fields and matter–matter response func- standard approximations employed in quantum chem- tions. The use of entangled photon pairs in spec­troscopy istry also need to be modified96,155,16. For instance, the enables one to go beyond the classical Fourier limit22; usual Born–Oppenheimer surfaces that give an indica- alternatively, photons can be used to imprint correlation tion of chemical reaction pathways can be modified155,157, onto matter17,158,159, or their radiation-pressure can be used and conical intersections can be influenced (displaced and to manipulate the sample, as done in optomechanics160,161. modified)17. More interestingly, when photon and matter In addition to the case of molecules in cavities, there degrees of freedom must be considered simultaneously, are other situations in which strong coupling to the alternative spectroscopic measurements can be pro- light field can occur162. This happens in nanoplasmonic posed to unravel cross-correlation response functions. devices or nanostructures, where strong exciton–plasmon The standard approach for the description of the optical coupling can be achieved156,163. The collective motion of the properties of molecular systems in the weak-coupling electrons that form plasmons in metallic nano­structures, condition, for example, the classical radiation emitted by for example, induces very large local fields that can then a quantum system owing to perturbation by a classical couple to molecules or molecular assemblies. In this case,

Box 5 | General photon–matter response functions Any coupled photon–matter system has, to first order, a general response function to a classical external perturbation that can be decomposed in terms of photon and matter contributions χ χ χ ≡ mm mp [ χpm χpp ]

where χmm corresponds to the response of the matter subsystem due to a perturbing field acting on the matter,

χmp corresponds to the response of the matter from the perturbation of the photonic subsystem and vice versa for χpm,

and χpp is the response of the photons due to a perturbation of the photonic subsystem. The latter would correspond to the Maxwell equations if there is no coupling to matter, but owing to the coupling to matter, the Maxwell equations are

modified. Especially interesting are the correlated matter–photon responses χmp. For instance, the induced density response

resulting from perturbing the photon field of the combined matter–photon wavefunction Ψ0 by an external current is

χn, A(r,t; r ,t ) –iθ(t–t ) [ˆnI(r,t); ÂI(r ,t )] ʹ ʹ = ʹ ⟨ ʹ ʹ ⟩ where we use the same notation as in BOX 3. Using the expression for the vector-potential operator Â(r) (REF. 41), employing the dipole approximation for the mode functions of the photon field and performing the length-gauge transformation, the above response function in a mode-resolved form leads to

χn, q(r,t; α, t ) –iθ(t–t ) [ˆnI(r,t); qˆαI(t )] ʹ = ʹ ⟨ ʹ ⟩ where qˆ = ħ/2 (aˆ + aˆ† ) is the photon displacement operator for mode (k, ) (REFS 23,187), and the α √ ωk α α α λ expectation value is taken with respect to the fully coupled matter–photon wavefunction.≡ In FIG. 5b, the absorption spectrum of the response function that quantifies matter–photon correlation is depicted for the case of a model dimer in a single-mode cavity. The perturbation can also be of quantum nature22,191,192. To model this, we can either include the perturbing photons or particles in the initial state, that is, we consider a combined evolution of the system plus quantum perturbation, or we generate the quantum perturbation by a source term41,42. Therefore, because a classical external charge current generates

photons, χmp can be interpreted as the response of the matter–photon system to a quantum perturbation by photons.

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the response of the combined systems can show a strong systems57, the envisioned novel spectroscopic situations hybrid character, and the usual approximations that com- provide an alternative way to challenge our understand- bine the individual responses become problematic164. ing of light–matter interactions and to test the predictions A similar effect is observed in surface-enhanced Raman of QED in the low-energy regime. scattering, where the atomic molecular dynamics is altered by strong coupling with plasmons and the usual Conclusion and outlook selection rules break down165,166. In this Review, we have considered the idea that using In the cases presented above, the discarded trans­ experimental spectroscopies to explore conditions or sys- versal photon degrees of freedom become important tems that are at the frontiers of current theoretical mod- owing to boundary effects, arising from the mirrors of a elling, where standard approximations to treat coupled cavity or plasmons on a surface167. However, similar sit- light–matter systems become inadequate, offers the uations can also occur in free space when many emitters opportunity to identify interesting physics. The cases are coherently coupled, for instance, when the density of illustrated here are clearly not the only ones possible, but molecules inside the gas chamber is increased or in solid- they are especially interesting because they make appar- state systems. Beyond the linear-response regime, the ent the approximate nature of the Coulomb interaction intrinsic equation should account also for the Maxwell and the standard strategies to model complex many-body field, for example, in the case of attosecond polarization systems in quantum chemistry. Though ignoring the spectroscopy, which measures the response of a solid transversal electromagnetic degrees of freedom is well jus- to strong few-cycle laser pulses27. Such situations are at tified in most cases, their inclusion can be advantageous the boundary between the matter-only and light-only to gain further insight into complex quantum systems categories (FIG. 2a,b). In small systems, inclusion of the by novel spectroscopic techniques employed in cavity or discarded degrees of freedom becomes necessary when circuit QED. We focused on low-energy QED but did not the interaction induced by transversal charge currents discuss statistical aspects such as temperature or indefi- or fields is no longer negligible, thus leading to sizeable nite number of particles. The simple reason for our choice retardation effects or induced magnetic fields. For is that we have infinitely many degrees of freedom instance, the dipole approximation can become inade- owing to the photon field, and thus, the usual descrip- quate in strong-field physics168, to capture the coupling to tion based on the Gibbs state becomes mathematically unusual light modes such as superchiral light169, ill defined174. Furthermore, even without coupling to the to describe the appearance of novel selection rules170 or to photon field, atomic, molecular and solid-state systems describe the emitted radiation171. The same is true for allow for the usual Gibbs states only if enclosed in a highly energetic photons in the case of X‑ray spec­ finite box175,176. troscopy, where multi-polar transitions and radiation are It is not only the advances in theoretical and compu- known to be important172. Here, the local field couples tational techniques that make exploring situations at the with the full transversal charge current of the system and interface between the different research fields of modern changes the dynamics81. quantum chemistry and physics interesting and timely By testing the limit of particle-only descriptions of but also the advances in experimental techniques and complex systems by a change in the environment or in novel light sources such as the Advanced Light Source non-equilibrium situations, it becomes evident that all in Berkeley, the X‑Ray Free-Electron Laser in Hamburg, degrees of freedom of non-relativistic QED can become the National Synchrotron Radiation Research Center in important and interact strongly. The complexity of the Hsinchu, the European Synchrotron Radiation Facility systems, the large amount of charges and their sizes in Grenoble, the SPRing‑8 Compact Free-Electron can enhance the interplay between the basic degrees of Laser in Japan and the pan-European Extreme Light freedom, that is, electrons, effective nuclei and photons. Infrastructure (ELI), to name but a few177. These facilities Consequently, in these cases, comprehensive intrinsic provide unprecedented possibilities to probe dynamical equations deduced from QED in the low-energy regime processes in chemistry, materials sciences and biology need to be used, and the resulting theoretical models need via ultra-fast X‑ray spectroscopies178–180. Together with to be compared with spectroscopic measurements. Along specialized light sources for ultra-intense laser fields with high-precision spectroscopy for simple atomic and that can probe the QED vacuum, such as ELI in Prague, molecular systems101–103, measurements of Casimir– these facilities will allow one to probe the most funda- Polder and related forces at the macroscopic scale173 mental aspects of light–matter systems from very low to and simulations of high-energy physics in solid-state ultra-high energies.

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Acknowledgements The authors acknowledge financial support from the European Research Council (ERC- 2015‑AdG‑694097). The authors thank Arunangshu Debnath, Klaas Giesbertz, Christian Schäfer and Martin Ruggenthaler for fruitful discussions.

Author contributions All authors contributed equally to the preparation of this manuscript.

Competing interests The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

How to cite this article Ruggenthaler, M., Tancogne-Dejean, N., Flick, J., Appel, H., and Rubio, A. From a quantum-electrodynamical light– matter description to novel spectroscopies. Nat. Rev. Chem. 2, 7 Mar (2018).

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