From a quantum-electrodynamical light matter description to novel spectroscopies

Transcription

1 From a quantum-electrodynamical light matter descrition to novel sectroscoies Michael Ruggenthaler *, Nicolas Tancogne-Dejean *, Johannes Flick,3 *, Heiko Ael * and Angel Rubio,2 * Abstract Insights from sectroscoic exeriments led to the develoment of quantum mechanics as the common theoretical framework for describing the hysical and chemical roerties of atoms, molecules and materials. Later, a full quantum descrition of charged articles, electromagnetic radiation and secial relativity was develoed, leading to quantum electrodynamics (QED). This is, to our current understanding, the most comlete theory describing hoton matter interactions in correlated many-body systems. In the low-energy regime, simlified models of QED have been develoed to describe and analyse sectra over a wide satiotemoral range as well as hysical systems. In this Review, we highlight the interrelations and limitations of such theoretical models, thereby showing that they arise from low-energy simlifications of the full QED formalism, in which antiarticles and the internal structure of the nuclei are neglected. Taking molecular systems as an examle, we discuss how the breakdown of some simlifications of low-energy QED challenges our conventional understanding of light matter interactions. In addition to high-recision atomic measurements and simulations of article hysics roblems in solid-state systems, new theoretical features that account for collective QED effects in comlex interacting many-article systems could become a material-based route to further advance our current understanding of light matter interactions. Max Planck Institute for the Structure and Dynamics of Matter and Center for Free- Electron Laser Science, Hamburg, Germany. 2 Center for Comutational Quantum Physics (CCQ), The Flatiron Institute, New York, NY, USA. Present address: 3 John A. Paulson School of Engineering and Alied Sciences, Harvard University, Cambridge, MA, USA. * michael. ruggenthaler@msd.mg.de; nicolas.tancogne-dejean@ msd.mg.de; johannes.flick@msd.mg.de; heiko.ael@msd.mg.de; angel.rubio@msd.mg.de doi:0.038/s Published online 7 Mar 208 Sectroscoy (from the Latin seciō look at or view and Greek skoéō to see ) investigates the various roerties of hysical systems over different sace, time and energy scales by looking at their resonse to different erturbations. The different articles (electrons, rotons and neutrons) taking art in a hysico-chemical rocess, electro magnetic and thermal radiations or mechanical deformations can be investigated and simultaneously used as erturbations. In atomic, molecular and solidstate hysics, sectroscoic methods are secifically used to robe and control various asects of couled hoton matter systems, such as the quantum nature of light, matter matter interactions 2, electronic 3 and nuclear degrees of freedom 4 and collective effects in molecular and solid-state systems 5. The study of these hoton matter interactions has informed us, for examle, on unknown states of matter 6,7, novel equilibrium and non-equilibrium (driven) toological hases 8 and light-induced suerconductivity 9. Over the years, the develoment and rogress of secific exerimental sectroscoic techniques have resulted in remarkable satial and time resolutions 3,0, now reaching sub-angstroms and atto seconds. The different static and time-resolved sectro scoic techniques encomass otical (vibrational, rotational and electronic), magnetic, magnetic-resonance (nuclear and electron sin-resonance), energy-loss, mechanical (atomic force), transort (electronic, sin or heat), and imaging (diffraction, scanning-tunnelling microscoy and holograhy) sectroscoies. Therefore, it is no longer ossible to refer to sectroscoy without secifying the technique or the satial and temoral regime. Diverse sectroscoic techniques are alied in different fields from atomic and molecular hysics to solidstate hysics 2 and biochemistry 3 and the analysis and interretation of the comlex exerimental data in each of these fields are routinely erformed by means of urose-built theoretical tools. These theoretical tools often treat electromagnetic radiation and matter at different levels of aroximation, thereby limiting their use to only articular cases. Usually, hotons are considered only as an external erturbation (often treated classically) that robes matter (treated quantum mechanically), NATURE REVIEWS CHEMISTRY VOLUME 2 ARTICLE NUMBER 08

2 Polariton A olariton is a bosonic quasi-article formed by an excitation, such as an exciton or lasmon couled ( dressed ) with hotons. Polaritonic chemistry Molecular systems strongly couled to light show the emergence of olaritonic states, which can change the chemical roerties of the molecules. Electromagnetic vacuum If all harmonic oscillators of the quantized electromagnetic field are in their ground state, the number of hotons is zero, corresonding to the bare electromagnetic vacuum. However, when couled to a matter system, the vacuum fluctuations induce changes in the matter system, which lead to the Lamb shift. This couling forms the basis of vacuum-mediated ( dark ) olaritonic chemistry. whereas the self-consistent back-reaction of matter on hotons is neglected. Standard theoretical modelling can thus be insufficient when the degrees of freedom of hotons and matter become equally imortant. Strong light matter interaction, as in the case of olariton condensates 4, rovides a direct examle. However, the interaction between hotons and matter, beyond the semi-classical descrition, is also exected to lay a role under other conditions and could lead, for instance, to the emergence of collective resonses in ensembles of molecules. In this Review, we take a ste back from the everincreasing secialization of each of the current theoretical aroaches and scrutinize them as different low-energy aroximations of the general framework of quantum electrodynamics (QED). In articular, we focus on non-relativistic QED, which is alicable within the tyical energy and timescales of molecular and solid-state sectros coies (from a few millielectronvolts to a few thousand electronvolts and from attoseconds to milliseconds). A great amount of theoretical and exerimental techniques have been develoed to study hoton matter interactions in these energetic and temoral regimes, and it is beyond the scoe of this Review to describe them all or to rovide a comrehensive list of examles; however, we refer to the relevant literature where ossible. We consider mainly otical and electronic sectroscoies, focusing on electron hoton henomena in molecular and solidstate systems; we do not exlicitly examine direct roton hoton interactions but instead describe the nuclei (consisting of rotons and neutrons) as effective articles. Similar considerations can be extended straightforwardly to this additional case. We highlight the limitations of the theoretical tools used to describe hoton matter interactions but also identify new henomena and hidden asects of these interactions that can be revealed with the develoment of new tailored sectroscoic tools. When matter and hotons are strongly correlated, novel effects aear, such as changes in the chemical roerties of molecules in otical cavities and the emergence of olaritonic chemistry 5,6,7,8, strong hoton couling in lightharvesting comlexes 9 or attraction between hotons due to quantum matter 20. Such effects have the otential to directly challenge QED in the low-energy regime. We finally envision that it will be ossible to extract new relevant sectroscoic information for hoton matter systems by directly robing correlated hoton matter observables through, for instance, entangled hoton sectroscoy 2,22. Before going into detail, let us demonstrate why there is a need to go beyond the conventional descrition of light matter interactions, which is based on the solution of the Schrödinger or Dirac equation (describing matter) ossibly couled with the classical Maxwell equations (describing hotons). It is well known that the otical sectral lines of a dilute atomic gas are well reresented by transitions between the eigenstates of the isolated atomic electronic Hamiltonian. However, as the gas density increases (for examle, by alying ressure), the observed atomic-gas sectrum deviates from simle atomic transitions, and its theoretical descrition requires the inclusion of the missing olarization, retardation and other effects by couling the Schrödinger or Dirac equation with the Maxwell equations. The sectrum is interreted as transitions between atomic states, each one interacting or dressed with the classical Maxwell field. The use of the dressed-electron icture has been known for a long time, and the couled Schrödinger Maxwell or Dirac Maxwell equations form the basic framework to describe molecular and solid-state sectros coies. However, it is also clear that these equations are simlifications of an even more comlete theory, which exlicitly takes into account the electromagnetic degrees of freedom on a quantized level. Indeed, many exerimental results exist that require the simultaneous consideration of the quantum nature of electrons and the electro magnetic degrees of freedom (see FIG. for a schematic of the evolution of our understanding of combined light matter systems). These include the modification of atomic energy levels in high Q cavities 23, the emergence of quasiarticles 24 such as olaritons, and sectroscoies that make use of the quantum nature of light 22. Traditionally, many of these electromagnetic effects (for examle, the Lamb shift or the finite lifetime of an excited molecular state) have been investigated in atomic and small molecular systems 23,25,26. Although usually small for isolated atoms, these electromagnetic effects have an imortant role in molecular systems and even more in solid-state systems, leading, for examle, to the following: the screening, olarization and retardation effects observed in light matter energy transfer induced by attosecond laser ulses 27 ; the emergence of quasi-articles that do not have a classical counterart, such as olaritons or axions 28,29 ; and the formation of novel states of matter, such as hybrid hoton matter states 30,3, exciton olariton condensates 4,32 or light-induced toological states 8. Wellknown electromagnetic effects in quantum chemistry and solid-state hysics 33 are long-range intermolecular interactions and Casimir Polder forces 34,35, Förster resonance energy transfer 36,37 or the hydrodynamical regime of quantum light 38. However, most common theoretical aroaches do not treat these effects self-consistently, meaning that they do not consider the back-reaction of matter on the electromagnetic field and vice versa. QED rovides a general framework to treat electromagnetic and matter degrees of freedom on equal quantized footing, as QED seamlessly combines quantum mechanics and the Maxwell equations (FIG. 2) to describe the full couling of hotons and matter. Indeed, in addition to describing the direct interaction between charged articles and light, QED also catures matter matter interactions induced by the electromagnetic field and hoton hoton interactions resulting from the resence of matter (FIG. 3). QED accounts for the absortion and emission of hotons as well as effects beyond the bare electronic quantum mechanics descrition of matter (for examle, the Lamb shift of energy levels due to quantum fluctuations of the electromagnetic vacuum 25 ). A brief sketch of QED Before discussing the most common theoretical sectroscoic tools, we give a brief overview of QED and its low-energy limit. Charge densities and currents are the sources of electromagnetic fields. Classically, this is 2 ARTICLE NUMBER 08 VOLUME 2

3 Quantum theory of light: In order to exlain the hotoelectric effect (Milliken, 923), Einstein roosed that light is quantized (92). This was suorted exerimentally by Comton (927). Quantum theory of molecules: Sectroscoic measurements like the ones of Raman (930) rovided the inut for the basic theoretical develoments by Debye (936), Pauling (954) and Mulliken (966). More refined sectroscoic results, e.g. by Schawlow, Bloemberg and Siegbahn (98), rovided the basis of, e.g. comutational quantum chemistry (Kohn and Pole, 998) and femtochemistry (Zewail, 999). Quantum theory of condensed matter: On the basis of the concet of quasi-articles (Landau, 962), many effects in condensed-matter hysics could be exlained, e.g. suerconductivity (Bardeen, Cooer and Schrieffer, 972) or Bose Einstein condensates (Cornell, Weiman and Ketterle, 200). This has even led to the discovery of new hases of matter (Thouless, Haldane and Kosterlitz, 206) Quantum theory of atoms: On the basis of the sectra measured by Lorentz and Zeeman (902) and the quantum hyothesis (Planck, 98), Bohr (922) linked the sectral lines to the quantized nature of the hydrogen atom. This was later confirmed (Hertz and Franck, 925) and ut into a rigorous form by Heisenberg (932) and Schrödinger and Dirac (933). Quantum theory of light and matter: Based on sectroscoic measurements of the hyerfine structure (Lamb and Kusch, 955), a comlete theory of quantized light and matter was develoed (Feynman, Tomonaga and Schwinger, 965). With the develoment of the laser (Basov, Prokhorov and Townes, 964), the sectroscoic methods became more advanced, which allowed control of the interaction between light and matter (Chu, Cohen-Tanoudji and Phillis, 997). Further develoments in quantum otics (Glauber, Hall and Hänsch, 2005), allowed the control of individual quantum systems (Haroche and Wineland, 202). Figure Schematic evolution of our understanding of quantized couled light matter systems. Reresentative Nobel rizes in hysics and chemistry are highlighted (years in arenthesis). Pauli Hamiltonian The Pauli Hamiltonian comrises the standard Schrödinger Hamiltonian and the Pauli (Stern Gerlach) term σ B(r), which describes the couling between the electron sin (characterized by a vector of the usual Pauli matrices σ) and the magnetic field B(r). Sinor reresentation To reresent the sin of a quantum article a vector of wavefunctions can be used, in which each entry corresonds to a secific sin state of the article. A sin one-half article has two such entries, that is, sin u and sin down. formalized by the Maxwell equations 39, and Dirac, Pauli and Heisenberg were the first to rovide a formulation for the quantization of the electromagnetic field, thus giving birth to QED 40. Electromagnetic fields induce dynamics in a system of charged articles. Therefore, simultaneously treating hotons and charged articles requires taking into account the influence that charged matter and electromagnetic radiation have on each other. In QED, this is done via the minimal couling rescrition 4,42, such that the charge current in the Dirac equation becomes the source of the quantized electromagnetic field, which at the same time modifies the momentum of the Dirac fields. The quantized electromagnetic field is described by individual quantum harmonic oscillators for each allowed mode and olarization, which gives rise to the electromagnetic vacuum. Unfortunately, without further modifications, the straightforward QED formulation in terms of local hoton matter interactions (that is, articles interact by sending hotons back and forth) leads to unhysical results 4, because even a urely erturbative treatment of the couled system diverges beyond the first order. To solve this issue, Bethe 43 introduced the concet of energy cut-offs and counter terms for instance, a diverging electromagnetic mass resulting from the interaction with the bare electromagnetic vacuum that absorb the infinities when the cut-offs are removed. This renormalization rocedure has been investigated in detail in high-energy hysics via a erturbative treatment of scattering events 4, and its use has led to results in excellent agreement with those of exeriments. The formal develoment of this renormalized QED theory was done by Tomonaga, Schwinger and Feynman, for which they were awarded the Nobel rize in hysics in 965 (REF. 44). Within the non-relativistic limit for the matter subsystem, the full QED Hamiltonian can be simlified to the Pauli Hamiltonian (for the matter subsystem) describing the evolution of charged articles in sinor reresentation, which are couled through the charge-density and charge-current oerators to the quantized hoton field 34,42. For a system of N e electrons and N n nuclei, the corresonding Hamiltonian, known as the Pauli Fierz Hamiltonian 43,45, is N e 2m e c Ĥ PF (t) = Σ [σ l ( iħ rl  tot (r l, t))] 2 N n 2M l l = Z l e c (n Σ [S l ( iħ Rl  tot (R l, t))] 2 l /2) l = 2 N e w( r l r m ) Σ Z l Z m w( R l R m ) l m N e N n N n 2 Σ l m Σ Σ Z m w( r l R m ) Σ l = m = k, λ ħω k â k, λâ k, λ In this case, the internal structure of the nuclei is neglected (as the individual rotons and neutrons are not resolved), and the Coulomb gauge (hotons are allowed to have only transversal olarization, λ =, 2) has been chosen so that the longitudinal art of the hoton field is given exlicitly in terms of the charged articles (electrons and effective nuclei) only 4. The interaction between the charge-density oerator and the longitudinal art of the hoton field then gives rise to the Coulomb interaction, w ( r rʹ ) = e 2 /4πε 0 r rʹ, among articles NATURE REVIEWS CHEMISTRY VOLUME 2 ARTICLE NUMBER 08 3

4 a Proerties of light Theory: classical electrodynamics Dynamical variables: electromagnetic fields (third, fourth and fifth terms in equation ). Here, σ l is a vector of the usual Pauli matrices, reflecting the one-half value of the electron sin, m and e denote the mass and the absolute value of the elementary charge, resectively, and the total transversal vector otential tot  (r, t) =  (r) A ext (r, t) contains both the quantized internal transversal hoton degrees of freedom reresented by the transversal vector-otential oerator  (r) (REF. 4) and any ossible classical external vector otential A ext (r, t) (describing the interaction with external (n classical electro magnetic fields). Further, S l /2) l denotes a vector of sin n l /2 matrices (where n l is even for even-mass-number nuclides and n l is odd for odd-massnumber nuclides), reflecting the n l /2 value of the sin of the l th nucleus. For instance, to describe a nucleus with an even mass number and consequently even effective sin, for examle, sin, a vector with three comonents of 3 3 sin-matrices is used. Furthermore, M l denotes the mass of the l th nucleus, and Z l denotes the corresonding effective ositive charge. The last term in equation gives the total energy of the quantized electro magnetic field, where â k,λ and â k,λ are the usual bosonic creation and annihilation oerators, resectively, for mode k and olarization λ. For notational simlicity, we do not include the ossible couling of the charged articles to a scalar external classical otential υ ext (r, t) or the couling of the hotons to a classical external charge current j ext (r, t). However, these can be simly included by adding c Quantum nature of light Theory: quantum otics Dynamical variables: hotons and effective articles Proerties of charged articles and hotons Theory: non-relativistic QED Dynamical variables: electrons, effective nuclei and hotons Proerties of electron ositron fields and hotons Theory: renormalized QED Dynamical variables: electrons, ositrons and hotons Proerties of matter fields and their gauge bosons Theory: standard model of article hysics Dynamical variables: quarks, letons, gauge and scalar bosons b Proerties of matter Theory: quantum mechanics Dynamical variables: electrons and effective nuclei Figure 2 Theoretical descrition of hoton matter interacting systems. Different roerties of hoton matter interacting systems are groued and associated with the dynamical variables and theoretical aroaches that are currently used for their descrition. The roerties of light and matter are treated within different frameworks (classical and quantum theories, art a, b and c resectively). Their combination results in quantum electrodynamics (QED), which describes all fundamental quantum asects of electrons, ositrons and hotons. The currently most comlete descrition of matter is the standard model of article hysics, which also includes a descrition of the nuclei, electroweak interactions, and others. In the low-energy regime, non-relativistic QED can be emloyed. l e υ ext (r l, t) l Z l e υ ext (R l, t) d3 râ (r) j ext (r, t)/c to the Hamiltonian in equation. Although, in this case, only the electromagnetic radiation is described by a quantum field, the local couling still gives rise to divergencies already at the level of erturbation theory. However, by introducing a hysically reasonable energy cut-off 43 for the hoton modes, a formulation similar to the one of quantum mechanics 46 based on self-adjoint oerators 47 is ossible. The emerging Hamiltonian fulfils the variational rincile, and the ground state of the combined hoton matter system is mathematically well defined As already ointed out by Bethe 43, in the non-relativistic regime, such a frequency cut-off (usually taken at the rest-mass energy of the electron, which is roughly 0.5 MeV) is hysically reasonable because it ensures the underlying article descrition in the Pauli Fierz Hamiltonian. In certain limits, the cut-off can even be removed, and an exactly renormalized Hamiltonian can be defined 5. These are the main differences between non-relativistic and full (also charged articles treated fully relativistically and second quantized) QED, which is usually formulated in terms of a erturbation theory for scattering events. Aroximations to non-relativistic QED The relevant hysical and chemical rocesses of interest in this Review corresond to an energy range well below MeV. In this energetic regime, the creation of electron ositron airs can be comletely neglected, and a simler first-quantized (article-number conserving) descrition of matter based on either the Pauli or higher-order aroximations to the Dirac equation can be adoted. The electromagnetic field is ket fully quantized, and, in contrast to charged articles, hotons can be created and destroyed. Thus, we focus the remaining art of the Review on the discussion of this non-relativistic QED formulation (see equation ). The low-energy QED formulation has three basic constituents nuclei, electrons and hotons in which the latter constitute the quantized electromagnetic field. It is commonly believed that these three comonents are sufficient to exlain all sectroscoic results within the low-energy range. In this framework, the nuclei are treated as ositive oint charges with a nuclear sin and mass instead of a combined system of rotons and neutrons. This is justified by the fact that in the lowenergy regime, only the nuclear motion not the internal structure is relevant. It is, of course, very aealing to directly challenge the non-relativistic QED theory with sectroscoic measurements of molecules, nanostructures and extended systems in order to test its accuracy, as done in high-energy hysics studies. In the high-energy regime, fundamental asects are investigated in detail, such as the instability of the QED vacuum for extremely-high-intensity laser fields 52 that leads to a modification of the Maxwell equations in vacuum 53, the electroweak interaction 54 or QED effects in lasmas 55. Similar effects are also investigated in materials science studies, where a real material is used to simulate the mixed axial gravitational anomaly 56, or Higgs 57, Majorana 58 or axion hysics 59. Other comlications aear 4 ARTICLE NUMBER 08 VOLUME 2

5 when one goes beyond scattering theory to determine, for examle, the ground state and the satially and temorally resolved dynamics of electron nucleus hoton systems. In ractice, aroximations are introduced to treat the interacting articles as well as to model and interret exerimental results. Those aroximations strongly deend on the character of the light matter system that is robed. The corresonding aroximate equations might differ and rovide access to different rocesses and interactions. The different fundamental asects of QED that are usually robed can be groued into three broad categories (FIG. 2): the quantum nature of light investigated in quantum otics 23 or cavity QED 60 ; the satial and temoral roerties of the electromagnetic field investigated in hotonics 6 or lasmonics 62 ; and the matter degrees of freedom investigated in solid-state hysics or quantum chemistry 63. The available (and new) sectroscoies can then be categorized based on the roerty investigated. The first category corresonds to sectroscoies that directly robe the quantum nature of light. A tyical exerimental setu includes microwave 64 or otical 65 cavities and well-characterized matter systems, such as Rydberg atoms or quantum dots. In such wellcontrolled systems, the intricate behaviour of hotons and their interlay with matter can be directly observed. Exeriments erformed in these conditions include single-hoton measurements in quantum otics 66 or quantum information 67, single-atom masers (microwave amlification by stimulated emission of radiation) in cavity QED 60 and electromagnetically induced transarency 68. Most theoretical descritions rely on the diole aroximation, also known as the long-wavelength or otical limit, which assumes that the relevant wavelength of the electromagnetic field is much larger than the satial extension of the matter subsystem. In this case, the satial non-uniformity of the field at the relevant frequencies at any instant in time is neglected, meaning that the mode functions for the light field are aroximated by a constant. The couling between the (transversal) charge current and the electromagnetic field can be exressed by using only the total diole moment of the system and the uniform electric field 23,42. This aroximation is commonly used in conjunction with the restriction of the full-hoton field to only a few contributing modes that are in or near resonance with the selected energy levels of the isolated matter system that is not couled to the light field. It is also ossible to restrict the matter degrees of freedom to a few energy levels, leading to the few-level and few-mode aroximation 23. The simlest form, known as the Rabi model, comrises a twolevel system couled to one mode of the hoton field, which is described by a harmonic oscillator. If it is further assumed that the absortion of a hoton can only excite the samle and emission can only de excite the samle, the resulting simlification is called the Jaynes Cummings model, which effectively ignores quickly oscillating terms (the rotating-wave aroximation) 23. This level of aroximation to the Pauli Fierz Hamiltonian of non-relativistic QED is emloyed, for instance, to describe single-atom lasers 69,70, which were exerimentally realized for the first time with a caesium atom in a high Q otical cavity 7. Often, these model Hamiltonians are solved as oen quantum systems 72 to more accurately account for losses in the real situation when hotons can leave the cavity. Such few-level aroximations are used not only in quantum otics but also in quantum chemistry and solid-state hysics, for examle, in the context of light-harvesting systems 73 or in nuclear magnetic resonance 74. However, such a reduced treatment can often be insufficient 75,76, esecially if one is interested in more than just the simle observable that the model was designed to describe, such as the diole moment, and the hysical imlications of few-level models are debated in the literature, for examle, the suerradiance hase transition due to the Dicke model 77. a Matter matter b Photon matter c Photon hoton e e γ e e γ γ e e γ γ e e ω k e e ω k ω k (e, ) (e, ) (e, ) (e, ) γ γ Chemical bonding Photoionization γ γ Photon blockade Figure 3 Schematic descrition of the different comonents of the light matter QED Hamiltonian. The Hamiltonian describing the light matter interaction in quantum electrodynamics (QED) accounts for matter matter interactions (art a), hoton matter interactions (art b) and hoton hoton interactions (art c). The straight lines in the diagrams of erturbation theory for each interaction reresent charged articles (nuclei and electrons e ), and the wiggly lines reresent hotons, γ. Part a highlights how matter hoton couling induces effective matter matter interactions, for examle, the Coulomb interaction between electrons and nuclei, which is resonsible for chemical bonding of a dimer. Part b shows a direct hoton matter interaction, for examle, hotoionization of a dimer. Part c highlights how matter hoton couling can lead to effective hoton hoton interactions resonsible for, for examle, hoton blockade in an otical cavity, where ω k is the cavity frequency. We note that desite the emloyed erturbative icture, QED naturally includes the self-consistent back-reaction of light on matter and vice versa, but this back-reaction is commonly neglected. NATURE REVIEWS CHEMISTRY VOLUME 2 ARTICLE NUMBER 08 5

6 In the second category, the quantum nature of light is considered not to be relevant because the number of hotons involved is usually large, for examle, such as in vibrational or ultra-fast laser sectroscoies 78. In such cases, a semi-classical treatment of the electromagnetic field of the combined light matter system is usually sufficient, which means that the non-relativistic QED descrition will include the classical Maxwell field instead of the quantized hoton field. This leads to a couled Maxwell Pauli equation (BOX ), which is emloyed for the characterization of ultra-fast electron dynamics in solids 79,80, X ray single-molecule imaging 8 or molecular nanoolaritonics 82 or for the study of the satial and temoral roerties of the light field (for examle, in the context of fibre otics 83, otical antennas 84, near-field sectroscoies 85, otical tomograhy 86, interferometry 87 and holograhy 88 ). In the latter case, it is usually assumed that the light field leaves the matter roerties unchanged. This directly results in the macroscoic Maxwell equations, where the matter degrees of freedom (arising from the source term J (r, t) in BOX ) are used to define the electric dislacement and magnetizing fields. Constitutive relations, such as the deendence of the olarization and magnetization on external fields, are then usually determined from a matter-only theory (that is, the third category discussed below). This level of aroximation to the Pauli Fierz Hamiltonian of non-relativistic QED is emloyed, for instance, in calculating the local fields of lasmonic structures 89. In the limit of linear otics, the matter degrees of freedom are further simlified and reduced to an effective ermittivity and ermeability. For otical and electronic sectroscoies of molecules and solids, this decouling of matter and light is usually emloyed. The third category usually rovides the inut to the above constitutive relations from a matter-only ersective. The hoton field is taken into account only by the re normalized masses (bare lus electro magnetic 46 ) and the (longitudinal) Coulomb interaction among the charged articles, as well as ossible QED corrections. The matter-only descrition can be further simlified by decouling the electronic and nuclear degrees of freedom and by means of conditional wavefunction exansions it is then ossible to obtain Born Oenheimer surfaces or quantized nuclear motion aroximated by honon modes. In its simlest form, the nuclear motion is aroximated by semi-classical trajectories (Ehrenfest dynamics) couled to the many-electron Schrödinger equation. This aroach is indeed similar to the de couling scheme that leads to the Maxwell Pauli equation, where the classical equation for hotons is solved in conjunction with the many-body Schrödinger equation (BOX ). In articular, when only the electronic degrees of freedom are considered, such as in the electronsin-resonance or Ramsey technique 90, the nuclei are assumed to be clamed and usually treated as classical external Coulomb otentials. A good examle of such a simlified treatment of QED in the context of molecular systems is the Hamiltonian for H 2 (or other simle molecules 26,9 ), which includes the Lamb shift, the fine structure and the hyerfine structure (see REF. 92 for higher-order contributions). Neglecting all Box Maxwell Pauli equation If we make a mean-field ansatz for the matter hoton couling in non-relativistic QED (see equation ), we can aroximate the correlated matter hoton wavefunction Ψ as the roduct of the matter wavefunction ψ and hoton wavefunction ϕ that is, Ψ ψ ϕ. This ansatz, which is similar to the Born Oenheimer ansatz in electron nuclear dynamics, enables us to rewrite the correlated roblem as two couled equations 8 External fields Fields that are externally controlled, fixed erturbations, such as um and robe ulses or in the clamed nuclei aroximation the nuclear attractive otential, whose sources are not included in theoretical descrition. These fields are usually classical but can also be of quantum nature. iħ ψ(t) = Ĥ P (t)ψ(t) t and ( 2 t 2 )A 2 (r, t) = μ 0 cj (r, t) c 2 where the Pauli Hamiltonian for N e electrons and N nuclei is given by N e N Ĥ P (t) = e n Σ [σ l ( iħ rl Z [S (n l l /2) ( iħ Rl l e l = 2m c l = 2M l c N e N n Σ l = m = w( r l r m ) Σ Σ Z l Z m w( R l R m ) Σ Z m w( r l R m ) 2 N e l m 2 N n l m A tot (r l, t))] 2 Σ A tot (R l, t))] 2 Here, σ l is a vector of Pauli matrices, m and e denote the mass and the absolute value of the elementary charge, resectively, and the total transversal vector otential A tot (r, t) = A (r, t) A ext (r, t) A ext (r, t) contains both the transversal Maxwell field A (r, t) coming from the couled Maxwell equation and any classical external vector otential A ext (r, t). This is the mean-field aroximation to the matter hoton couling in the Pauli Fierz Hamiltonian of equation. Further, S l (n l /2) denotes a vector of sin n l /2 matrices, M l denotes the mass of the l th nucleus, and Z l denotes the corresonding effective ositive charge. The interaction terms are given by the Coulomb interaction w( r rʹ ) = e 2 /4πε 0 r rʹ. Further, J (r, t) is the induced transversal charge current of the matter system 4, 96. If there is no induced transversal charge current, the two equations decoule, and we are left with the usual many-body Pauli equation in the Coulomb aroximation. Note, however, that in general time-deendent roblems, the Coulomb aroximation does not account for relativistic causality. Local changes are felt instantaneously everywhere. Causality of the field is restored by including the neglected transversal art of the current density (that is, retardation effects or memory effects). 6 ARTICLE NUMBER 08 VOLUME 2

7 the higher-order contri butions from QED leads to the usual electronic Schrödinger aroximation, which for instance, well describes resonance energy transfer and the /R 6 law, which was exerimentally verified for the first time in trytohyl etides 93. It is ossible, however, to aroximate non-relativistic QED in a different way. Instead of simlifying the Hamiltonian, one can reformulate the full roblem in terms of reduced quantities that avoid unaffordable exlicit calculations of the wavefunction. Here, we can follow well-known strategies routinely emloyed in quantum chemistry and solid-state hysics, in which the ground-state and time-deendent many-body Schrödinger roblem is reformulated in terms of density functional theory 94 or Green s function theory 95 to make this roblem affordable for numerical comutations and simulations. Indeed, the common matter-only density functional theory and Green s function theory aroaches (alicable only to the third category above) are aroximations to density functional and Green s function reformulations of non-relativistic QED (in terms of densities and currents or equations of motion for the Green functions and self-energies). These formally exact reformulations of the Pauli Fierz Hamiltonian treat light and matter as equally quantized and hence are also alicable when hoton and matter degrees of freedom are equally imortant 96,99. The rice to ay is that aroximations are needed for both the unknown exchange correlation functionals in density functional theory and the self-energies in Green s function methods. The recently develoed aroximations 00 in the context of quantum-electrodynamical density functional theory (QEDFT; BOX 2) can accurately treat exlicitly couled matter hoton situations 99,7. These aroximations also rovide a romising scheme to study realistic comlex molecular systems and solids in quantum cavities (FIG. 4) and address the aearance of novel states and hases that would have been inaccessible otherwise (including control of chemical reactions, energy transfer, and so on). The above aroximations of non-relativistic QED have roved to be excetionally successful. Predicted transition frequencies for small atomic or molecular systems have been found to be in good agreement with high-recision sectroscoic measurements and are used as benchmarks for fundamental constants and the accuracy of QED in the low-energy regime Although the accuracy 05 of these theoretical redictions decreases for more comlex systems, such as biomolecules or solids, it is still ossible to qualitatively cature most of the relevant hysico-chemical rocesses. Box 2 Quantum-electrodynamical density functional theory (QEDFT) Density functional theories 94 are exact reformulations of the many-body roblem in terms of an exact quantum-fluid descrition, in which only the total densities of the system aear. Instead of modelling the momentum stress and interaction stress tensors exlicitly, one usually emloys the Kohn Sham construction, which uses the corresonding exressions of a non-interacting reference system and aroximates the differences between the interacting and non-interacting quantum fluids with exchange correlation fields. The original formulation of a urely electronic Hamiltonian with scalar external classical fields has been extended to very different hysical situations, including suerconductivity 82 and general external classical electromagnetic fields 83,84. In the latter case, instead of an exchange correlation otential, an exchange correlation vector-otential is needed to model the missing forces resulting from the Coulomb interaction. In the case of a couled matter hoton system described by equation, the exact density functional reformulation 96,85 88 includes hotons in addition to the charged articles (electrons and nuclei), and thus, the article subsystem can enact forces on the hoton subsystem and vice versa. The corresonding Kohn Sham scheme emloys the exressions of a non-interacting multi-article system and an uncouled hoton field, with exchange correlation contributions describing the interaction due to the longitudinal hotons and a new exchange correlation contribution resulting from the interaction mediated by the transversal hotons 99. Without loss of generality, the original couled fermion boson roblem (equation ) can be exactly rewritten in terms of self-consistent couled Maxwell Kohn Sham Pauli equations 88 iħ ψ(t) = Ĥ MKS (t)ψ(t) and t ( 2 c 2 t 2 )A 2 (r, t) = μ 0 cj (r, t) where Ne Ĥ MKS (t) = Σ l = e [σ l ( iħ rl (A tot (r l, t) A xc (r l, t)))] 2 Nn (n (A 2m tot (R c [S l, t) A xc (R l, t)))] 2 l /2) Z l ( iħ Rl l e Σl = 2M l c Here, A tot (r, t) = A (r, t) A ext (r, t) and A xc [J, A ] are the exchange correlation vector otentials that cature the missing internal forces from the hoton matter couling and deend imlicitly on J calculated from ψ(t) (REFS 4,96), on A, and on the initial many-body and Kohn Sham states. Further, σ l is a vector of Pauli matrices, m and e denote the mass and the absolute value of the elementary charge, resectively, S l (n l /2) denotes a vector of sin n l /2 matrices, M l denotes the mass of the l th nucleus, and Z l denotes the corresonding effective ositive charge. Here, J (r, t) is the transversal art of the induced charge current of the matter system. The Maxwell Kohn Sham Pauli equations can be further decouled into a set of self-consistent single-article equations for the electrons and nuclei 96. We note that by making aroximations to quantum-electrodynamical density functional theory (QEDFT), we recover the exact reformulations of simlified models of non-relativistic QED. For examle, by assuming the diole aroximation, we reduce to an exact reformulation of non-relativistic diole QED 96. Neglecting the A (r, t) and the corresonding exchange correlation terms decoules the electromagnetic and matter sectors and leads to the common matter-only density functional theory 96,87. NATURE REVIEWS CHEMISTRY VOLUME 2 ARTICLE NUMBER 08 7

8 a Photon OEP n(r) b 0 3 (Photon OEP OEP) Δn(r) c n(r) and Δn(r) n(r)/ Δn(r) y (Å) y (Å) y (Å) Figure 4 Numerical examle for a QEDFT calculation: study of a 3D sodium dimer in an otical cavity. The electronic ground-state density n(r) in the xy lane (z = 0) of a sodium dimer strongly couled to the mode of an otical high Q cavity calculated with quantum-electrodynamical density functional theory (QEDFT) in diole aroximation using the otimized effective otential (OEP) aroach 93 (art a), the difference between the electron density in the xy lane of the couled and the bare sodium dimer Δn(r) (art b), and the difference between couled and bare density in blue against the couled density in grey in the xy lane and summed along the x axis (art c). Note that the latter has been reduced by a factor of /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 setu, we find hotons in the cavity. In other observables and in the excited state (see also FIG. 5), the changes can become very large (see REF. 93 for further details on the accuracy of the hoton OEP aroach). These grahs were obtained using the arameters for a sodium dimer given in REF. 94. The energy of the 3s 3 transition was chosen in resonance to the otical cavity frequency, that is, ħω k = 2.9 ev. Further, the hoton field is olarized along the y direction with a strength of λ k = 2.95 ev /2 nm e y. The real-sace grid is samled as Å 3 with a grid sacing Å. x (Å) Å -3 x (Å) Å -3 Electron density (Å -3 ) Electronic and otical roerties We have introduced the intrinsic aroximations commonly used to describe and understand the different sectroscoies. However, accounting for correlated matter requires further aroximations. Such aroximations include, for instance, the truncation of the Bogoliubov Born Green Kirkwood Yvon hierarchy 06, renormalization-grou techniques 07 09, stochastic equations such as master and Fokker Planck equations 0 and many others 23, 8. An in deth discussion of each of these is beyond the scoe of this Review. However, in order to analyse the limitations of the chosen intrinsic aroximations, we need to consider secific systems. Here, we choose to consider the electronic and otical roerties of molecular systems. In this way, we can identify hysical situations that highlight QED effects beyond the alied intrinsic aroximation in molecular systems, which we can use to directly challenge the QED framework. In the molecular case, focus is laced on the matter degrees of freedom (FIG. 2b). The following discussion will further highlight that there are several additional aroximations involved along with those leading to the aroximation to the Pauli Fierz Hamiltonian. Isolated gas-hase molecules rovide an ideal framework to test the limits of common aroximations, even if exeriments are far from being trivial. In most exeriments, many hotons are involved, and the mean-field aroximation for the hoton field is well justified. This means that the non-relativistic QED descrition is relaced by a descrition based on the Maxwell Pauli equation (BOX ). Assuming clamed nuclei, the many-body system is simlified to a many-electron system, which can be modelled by the standard many-electron Schrödinger equation. Usually, the diole aroximation is also alied to external fields. Furthermore, electron honon couling is often treated erturbatively (including honon honon scattering rocesses). Moreover, the Maxwell and time-deendent Schrödinger equations can be decouled to determine how the molecule reacts to any external erturbation or the absortion and/or emission of light. Desite the diole aroximation for the external field, the field induced by the fluctuations of the charge and current densities (source terms in BOX ) needs to be determined. This field induced by the olarized electronic cloud of the molecule tends to comensate the external erturbation and can give rise to the emission of hotons, ossibly at higher frequencies (this corresonds to the case of nonlinear otics as high-harmonic generation). To avoid full self-consistency, the calculated electronic charge and current densities are used as fixed inuts for the Maxwell equations 79,9. Still, these calculations are cumbersome and can be further simlified by assuming that the molecules radiate like dioles. This finally enables one to comletely neglect the Maxwell equations and consider only the time-deendent Schrödinger equation to comute the reaction to satially and temorally arbitrary external erturbations. In many instances, instead of solving the full timedeendent Schrödinger equation, one can evaluate secific linear and higher-order resonse functions (BOX 3) using erturbation theory. The idea of resonse functions used in time-deendent erturbation theory closely resembles that of a sectroscoic exeriment where an external robe radiation or articles erturbs the system and a signal radiation fields (electric and magnetic), articles or both (coincidence exeriments) is detected. As an examle, the absortion sectrum (diole resonse of the system due to an external electric diole field) for an isolated gas-hase model dimer is shown in FIG. 5a in red. However, in order to describe the nonlinear dynamics of driven systems, one has to solve the time-deendent equations described above. Similar 8 ARTICLE NUMBER 08 VOLUME 2

9 a Absortion sectra ħω k b Light matter sectra ħω k c 2D absortion sectra 6 Electronic excitations 4 4 log (Intensity) (a. u.) Vibronic sidebands log (Intensity) (a. u.) Absortion energy (ev) ħω R log [I (a.u.)] Rabi slitting ħω R Absortion energy (ev) ħω R Energy (ev) Cavity energy ħω k (ev) Figure 5 Calculated sectra for a D model dimer. a From to to bottom: electronic excitations (red) with clamed nuclei, electronic excitations dressed by vibronic sidebands (green), and Rabi slitting Ω R of electronic excitations (blue) for the dimer in an otical cavity with frequency ω k with increasing effective couling strength (see BOX 4 for more detail). b Light matter sectra, as defined in BOX 5. The negative and ositive amlitudes are lotted searately. c 2D absortion sectra for the dimer in a cavity, where we scan through different cavity frequencies ω k. Results obtained using the clamed-nuclei aroximation are shown in blue, whereas the results obtained considering the quantized electron nucleus hoton system are reorted in red. For the sectra in arts a and b, we chose a cavity frequency ħω k of.02 ev and electron hoton couling strengths g k /ħω k of (0.425, 0.85,.70)/nm; the same arameter for the sectra reorted in art c was chosen to be g k /ħω k = ħω k /nm ev. The sectra in arts a and c were calculated using equation S2 from REF. 7, and for those reorted in art b, we relaced in the same equation Ψ 0 R Ψ k 2 Ψ 0 R Ψ k Ψ k q Ψ 0, where Ψ 0 reresents the ground state of the couled matter hoton system and Ψ k reresents the excited states. The system is identical to the first examle in REF. 7 and can be described by the Hamiltonian Ĥ = ˆ T e ˆT N Ŵ ee Ŵ NN Ŵ en Ĥ P, where Tˆe and TˆN reresent the standard electronic and nuclear kinetic energy oerators, resectively; Ŵ ee, Ŵ NN and Ŵ en corresond to the electron electron, nuclear nuclear and nuclear electron interactions, resectively, with the soft-coulomb interaction w ( r rʹ ) = e 2 /4πε 0 (r rʹ) 2 (where r and rʹ are the ositions); and Ĥ P = [ 2 / q 2 ω 2 k(q λ er/ω k ) 2 ]/2, where qˆ = ħ/2ω k (â â ) is the hotonic dislacement coordinate, λ is the transversal olarization vector times the diolearoximation couling strength ( λ = g 2/ħω k ), and R = ΣlZ l R l Σlr l is the total diole oerator and Z l denotes the nuclear charge. 2 aroximations are also found in hotoemission sectroscoy 20,2, which is usually described by either a oneste or a three-ste model. The latter model is based on the difference between the intrinsic effects, which are described by the urely electronic Hamiltonian and the corresonding sectral function, and the extrinsic effects. The extrinsic effects account for olarization effects, effects arising from electron roagation in the matter, interactions between electrons and the created holes and their roagation through the surface. The one-ste model is simler and relies on the sudden aroximation for which the electron is assumed to reach the detector instantaneously, without any interaction with other (quasi-)articles nor light nor the surface of the material, thus neglecting all the aforementioned extrinsic effects. When the electrons are not treated as indeendent articles, their interaction leads to comlex fluctuations in the electronic charge and current densities. The corresonding induced fields should be comuted from the Maxwell equations and added to the external field. This can be incororated in the formalism of erturbation theory by means of a correction term, leading to what is called the microscoic macroscoic connection, which rovides a ractical framework to incororate the effects of the induced current and density fluctuations back into the Maxwell equations In this case, the microscoic resonse of the electronic system is used as an inut into the macroscoic Maxwell equations to obtain the macroscoic resonse of the system, which corresonds to the exerimentally observed radiation. This aroach is valid within the limits of weak erturbation and when the back-reaction of light on matter is neglected (that is, decouled Maxwell and Schrödinger equations). In this way, certain deficiencies of the intrinsic Schrödinger equation can be overcome by including some of the extrinsic effects. For instance, modification of the otical roerties of a gas of molecules uon increasing its density is well exlained and catured by this microscoic macroscoic connection, which incororates the olarization effects absent in basic quantum mechanics. Similar extrinsic effects, due to couling with the microscoic fluctuations of the induced light field (local-field effects), can be included in the modelling of hotoemission exeriments 20,2. The study of real molecular, nanostructured and extended systems requires ractical and accurate aroximations to the intrinsic electronic equation. This is a central toic of modern quantum hysics and chemistry and has led to the develoment of several theo retical methods, including quantum chemistry methods with configuration interaction 63,26, couled-cluster 27,28, quantum Monte Carlo 29, tensor network aroaches 30, NATURE REVIEWS CHEMISTRY VOLUME 2 ARTICLE NUMBER 08 9