Quantum electrodynamical time-dependent density functional theory on a lattice
QQuantum electrodynamical time-dependent density functional theory on a lattice
M. Farzanehpour ∗ and I. V. Tokatly
1, 2, † Nano-Bio Spectroscopy group and ETSF Scientific Development Centre,Departamento de F´ısica de Materiales, Universidad del Pa´ıs Vasco UPV/EHU, E-20018 San Sebasti´an, Spain IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain (Dated: November 27, 2018)We present a rigorous formulation of the time-dependent density functional theory for interactinglattice electrons strongly coupled to cavity photons. We start with an example of one particle ona Hubbard dimer coupled to a single photonic mode, which is equivalent to the single mode spin-boson model or the quantum Rabi model. For this system we prove that the electron-photon wavefunction is a unique functional of the electronic density and the expectation value of the photoniccoordinate, provided the initial state and the density satisfy a set of well defined conditions. Thenwe generalize the formalism to many interacting electrons on a lattice coupled to multiple photonicmodes and prove the general mapping theorem. We also show that for a system evolving from theground state of a lattice Hamiltonian any density with a continuous second time derivative is locally v -representable. PACS numbers: 31.15.ee, 71.10.Fd
I. INTRODUCTION
Time-dependent density functional theory (TDDFT)is a formulation of quantum many-body problem basedon the one-to-one map from the time-dependent densityto the driving external potential . The main practi-cal outcome of this map, which underlies the popularityof TDDFT, is a possibility to reproduce the dynamicsof the exact density in the interacting many-body sys-tem by solving an auxiliary problem for non-interactingKohn-Sham (KS) system driven by a properly adjustedself-consistent potential. This dramatically reduces thecomplexity of the problem and necessary computationalresource to study the dynamics of many-particle systems.The standard TDDFT assumes that the system isdriven by a classical time-dependent electromagneticfield . This approach is indeed sufficient for most typi-cal situations in quantum chemistry and condensed mat-ter physics. However, in the recent years, with the im-pressive progress in the fields of cavity and circuit quan-tum electrodynamics (QED) it has been made possible toexperimentally study systems interacting strongly withquantum light, like atoms in optical cavity , supercon-ducting qubits and quantum dots , trapped ions and molecules interacting with cavity photons .Recently a generalization of TDDFT for quantummany-electron systems coupled to cavity photons hasbeen proposed . This theory, which can be namedQED-TDDFT, relies on a generalized mapping theoremstating that there exists a unique map from the time-dependent electronic density and the expectation valueof the photonic filed to the total electron-photon wavefunction. In Ref. 15 the uniqueness of this generalizedmapping has been demonstrated using the Taylor expan-sion technique under the standard in TDDFT assump-tion of analyticity in time . The question of existenceof the density-to-potential map, which is known in DFTas the v -representability problem, is much more difficult. In fact, in the standard TDDFT for continuum systemsthis question is still not fully resolved, although a signif-icant progress has been made recently .Currently a complete and rigorous formulationof TDDFT, including the resolution of the v -representability problem, is available for lattice many-body systems . The aim of the present paper is toextend the uniqueness and existence theorems of the lat-tice TDDFT to systems strongly interacting with a quan-tized electromagnetic field.To make the idea of the proof more transparent westart with the simplest nontrivial system of one electronon a two-site lattice (a Hubbard dimer) coupled to a sin-gle photonic mode. It is worth noting that formulationof TDDFT for this system has it own value. Indeed,the dimer coupled to a quantum Bose field is (unitary)equivalent to such well known and popular models asthe quantum Rabi model and the spin-boson model which have a wide variety of applications ranging fromquantum optics and molecular physics to the mag-netic resonance in solid state physics . There is alsoa natural connection to the Dicke model . For thissystem we prove that, provided some well-defined con-ditions are fulfilled, there exists a unique mapping fromthe time-dependent on-site density and the expectationvalue of the bosonic coordinate to the wave function andthe external deriving potentials. Afterwards we extendthe QED-TDDFT mapping theorem to the general caseof N interacting electrons on an M -site lattice coupled tomultiple photonic modes. We also prove that, similarlyto the standard lattice TDDFT , the local existence/ v -representablity is guaranteed if the dynamics starts fromthe ground state of a lattice Hamiltonian.The structure of the paper is the following. In Sec. IIwe present a complete formulation of QED-TDDFT forthe Hubbard dimer coupled to a single photonic mode.We derive the equation of motion for the expectationvalue of the field and the force balance equation and con- a r X i v : . [ c ond - m a t . s t r- e l ] S e p struct the corresponding universal nonlinear Schr¨odingerequation (NLSE). Then we prove the maping theoremof QED-TDDFT for this model by applying the knownresults from the theory of semilinear partial differentialequations (PDE) . In Sec. III we generalize the for-malism to the system of many particles on a many-sitelattice which is coupled to multiple photonic modes. Wederive the corresponding NLSE the many-body systemand then formulate and prove the general existence anduniqueness theorem for the lattice QED-TDDFT. Sec-tion IV presents a practically important case of a systemevolving from its ground state. The main outcome ofthis section is the theorem of a local v -representabilityfor the initial ground state. In Conclusion we summarizeour results. II. QED-TDDFT FOR A HUBBARD DIMERCOUPLED TO A SINGLE PHOTONIC MODE
To make our approach more transparent and clearwe consider first a simple system one quantum parti-cle on a two-site lattice, which is coupled to a single-mode photonic field. The state of the system at time t is characterized by the electron-photon wave function ψ i ( p ; t ) = (cid:104) i, p | Ψ( t ) (cid:105) , where the index i = { , } corre-sponds to the particle “coordinate” and takes values onthe lattice sites, and the real continuum variable p de-scribes the photonic degree of freedom.In this model the electronic density is coupled to theexternal on-site potential v i ( t ) which acts on the individ-ual sites, while the photonic subsystem can be driven(exited) independently by an external time-dependent“current” J ex ( t ). Assuming that the wavelength of thephoton field is much larger than the size of the system,we adopt the dipole approximation. Figure 1 shows aschematic view of a two-site lattice in a quantum cavity.The following time-dependent Schr¨odinger equa-tion governs the time evolution of the electron-photonwave function ψ i ( p ; t ) from a given initial state ψ i ( p, t )i ∂ t ψ ( p ; t ) = − T ψ ( p ; t ) (1a)+ (cid:32) − ∂ p ω p J ex ( t ) p + λp + v ( t ) (cid:33) ψ ( p ; t ) , i ∂ t ψ ( p ; t ) = − T ψ ( p ; t ) (1b)+ (cid:32) − ∂ p ω p J ex ( t ) p − λp + v ( t ) (cid:33) ψ ( p ; t ) , where the real coefficient T corresponds to the rate ofhopping from one site to the other, ω is the frequency ofthe photon mode and λ is the electron-photon couplingconstant (see figure 1).Formally Eqs. (1) describes a driven two-level systemcoupled to a quantum harmonic oscillator. Because of thegauge invariance the physics is not changed if we add tothe potential a global time-dependent constant. There-fore without loss of generality we can adopt the gauge FIG. 1. A schematic view of a two-site lattice in a cavity.The electron, which can tunnel from one site to the otherwith the hopping rate T , experiences the on-site potential v i ( t ) specific to that site. The photonic field in the cavityis driven by a time-dependent external current J ex ( t ). Thewave length of the electromagnetic field 2 π/ω is proportionalto the cavity size and assumed to be much larger than thelattice size, so that we can adopt the dipole approximationfor the electron-photon interaction. condition v + v = 0 and define the on-site potential asfollows v ( t ) = v ( t ) = − v ( t ). With this definition theHamiltonian in Eq. (1) takes to formˆ H ( t ) = − T ˆ σ x + v ( t )ˆ σ z + λp ˆ σ z − ∂ p ω p J ex ( t ) p. (2)ˆ σ x , and ˆ σ z are the Pauli matrices, and a 2 × . Therefore the subsequent discus-sion and all results of this section are directly applicableto these models. We also note that a detailed derivationof Hamiltonian Eq. (2) for a nonrelativistic system in acavity can be found in Ref. 16 (see Appendix E).Let us now turn to the formulation of QED-TDDFT.In general all DFT-like approaches assume that the stateof the system is uniquely determined by small set of basicobservables, such as the density in TDDFT, the currentin TDCDFT. Below we prove a theorem which general-izes the lattice-TDDFT of Ref. 20 to the system coupledto a quantum oscillatoric degree of freedom as definedin Eq. (1). Namely, we will prove that, provided somewell defined conditions are fulfilled, the electron-photonwave function Ψ( t ) is uniquely determined by the on-sitedensity n i and the expectation value of the photonic co-ordinate P = (cid:104) p (cid:105) .In our formulation we follow the NLSE approach toTDDFT and adopt the same general logic as inRef. 20. We start with defining the basic observables forour two-site model. The first basic variable, is the on-sitedensity n i – the number of particles on the site in i ( t ) = (cid:90) p. | ψ i ( p ; t ) | . (3)For the electron-photon system the second, photon-related variable is required. The most natural choice is the expectation value P of the photonic coordinate pP = (cid:90) p. ( | ψ ( p ; t ) | + | ψ ( p ; t ) | ) p. (4)The next step is to find the equations which relate thebasic observables, the density n i and the field average P , to the “external potentials”, the on-site potential v i and the external current J ex . Therefore we proceed toderiving the equations of motion for the two fundamentalvariables.In order to derive the relevant equation of motionfor n i we calculate the time-derivative of (3) and thensubstitute the derivatives of the wave function from theSchr¨odinger equation (1). The result takes the followingform ˙ n ( t ) = − T ρ ( t )] (5)where ˙ n = ∂ t n , and ρ ( t ) is the density matrix , ρ = (cid:90) ρ ( p ; t )p. = (cid:90) ψ ∗ ( p ; t ) ψ ( p ; t )p. . (6)The conservation of the particles dictates that the changein the density in one site is equal to minus the change inthe other site ˙ n = 2Im T ρ . Obviously Eq. (5) is a lat-tice version of the continuity equation for site 1. Since inthe left hand side of Eq. (5) we have the time derivativeof the on-site density, the right hand side should be iden-tified with a current flowing along the link connecting thetwo sites J ( t ) = 2Im[ T ρ ( t )] . (7)Differentiating the continuity equation (5) with respectto time and replacing the derivative of the wave functionfrom the Schr¨odinger equation we get an equation whichconnects the on-site density n i to the on-site potential v i ¨ n ( t ) = − T (cid:16) Re[ ρ ]( v ( t ) − v ( t ))+ T ( n − n ) + 2 λ (cid:90) Re[ ρ ( p )] p p. (cid:17) . (8)Physically this equation can be interpreted as the (dis-crete) divergence of the force balance equation for thetwo-site model .A special role of Eq. (8) for TDDFT follows fromthe fact that it explicitly relates the potential disbal-ance v ( t ) − v ( t ) to the density n i ( t ) and its deriva-tives. Like before, the conservation of the particle im-poses ¨ n = − ¨ n . Hence the force balance equation for n ( t ) is obtained from Eq. (8) by changing the sign inthe right hand side. It is worth noting that the coefficient of the potentialdisbalance, v ( t ) − v ( t ), in the force balance equation(8) is the kinetic energy k = 2 T Re[ ρ ] therefore forthe current J and kinetic k we have: K + i J = 2 T ρ (9)Importantly, Eq. (8) contains only the potential disbal-ance v − v which reflects the well known gauge redun-dancy of TDDFT. For a given density the force balanceequation fixes the on-site potential up to a constant. Inorder to resolve this issue we fix the gauge by consideringon-site potentials which sum up to zero v = − v = v (10)This can be interpreted as a switching from the wholetwo dimensional space of all allowed potentials to the onedimensional space of equivalence classes for physicallydistinct potentials.Next, we need to derive a similar equation for P . Sowe differentiate (4) with respect to t and simplify theright hand side using the Schr¨odinger equation (1) andthe result is as follows:˙ P = Im (cid:2) (cid:90) (cid:16) ψ ∗ ∂ p ψ + ψ ∗ ∂ p ψ (cid:17) p. (cid:3) (11)For brevity we suppressed the explicit p - and t -dependence of the wave function.By differentiating Eq. (11) with respect to timeand again substituting the time derivatives from theSchr¨odinger equation (1) we get an equation which re-lates J ex to P , ¨ P and n i ¨ P = − ω P − λ ( n − n ) − J ex ( t ) . (12)This equation is, in fact, the inhomogeneous Maxwellequation projected on the single photon mode .In the next subsection we will use the force balanceequation (8) and equation of motion (12) to analyze theexistence of a TDDFT-like theory for a two-site latticecoupled to a photonic field. Statement of the mathematical problem and thebasic existence theorem
The standard TDDFT is based on the existence a one-to-one map between the time-dependent density and theexternal potential. In the case of the electron-photonsystem the map is slightly different. Here the two basicobservables, the on-site density n i and the expectationvalue of the field P , are mapped to the two external fields,the on-site potential v ( t ) and the external current J ex ( t ).Equations (1), (3) and (4) uniquely determine the in-stantaneous wave function Ψ( t ), the on-site density n i and the field average P as functionals of the initial stateΨ , the on-site potential v ( t ) and the external current J ex ( t ). This defines a unique direct map { Ψ , v, J ext } →{ Ψ , n, P } .The TDDFT formalism relies on the existence of aunique inverse map: { Ψ , n, P } → { Ψ , v, J ext } . Toprove the inverse map we follow the NLSE approach toTDDFT-type theories .Assuming n i ( t ) and P ( t ) are given functions of time,we express v and J ex from the equations (8) and (12) asfollows v = − ¨ n + 2 T ( n − n ) + 4 T λ (cid:82)
Re[ ρ ( p )] p p.4 T Re[ ρ ] (13) J ex = − ¨ P − ω P − λ ( n − n ) (14)where we assumed that Re[ ρ ] (cid:54) = 0.We note that at any time, including the initial time t = t , the given density has to be consistent with the wavefunction. At t = t this means that the right hand sides ofEq. (3) evaluated at the initial wave function Ψ has to beequal to n i ( t ) in the left hand side. The same follows forthe first derivative of the density ˙ n i and the field average P and its first derivative ˙ P . All of them should to beconsistent with the initial state Ψ trough Eqs. (5), (4)and (11) respectively. The consistency conditions whichshould be fulfilled are the following n i ( t ) = (cid:90) p. | ψ i ( p ; t ) | , (15a)˙ n ( t ) = − T ρ ( t )] (15b) P ( t ) = (cid:90) p. p (cid:0) | ψ ( p ; t ) | + | ψ ( p ; t ) | (cid:1) (15c)˙ P ( t ) = Im (cid:104) (cid:90) p. (cid:16) ψ ∗ ( p ; t ) ∂ p ψ ( p ; t )+ ψ ∗ ( p ; t ) ∂ p ψ ( p ; t ) (cid:17)(cid:105) (15d)The on-site potential v of Eq. (13) and the externalcurrent J ex of Eq. (14) can be substituted as a functionalsof n , P and Ψ in to the the Schr¨odinger equation (1). Theresult is a universal NLSE in which the Hamiltonian isa function of the instantaneous wave function and the(given) basic variablesi ∂ t Ψ( t ) = H [ n, P, Ψ]Ψ( t ) . (16)Now the question of existence of a unique QED-TDDFTmap { Ψ , n i , P } → { Ψ , v, J ext } can be formulated math-ematically as the problem of existence of a unique solu-tion to NLSE (16) with given n i ( t ), P ( t ) and Ψ . Theorem 1. (existence of QED-TDDFT for a Hub-bard dimer coupled to a photonic mode)
Assume thatthe on-site density n i ( t ) is a positive, continuous func-tion of time, which has a continuous second derivativeand add up to unity, n ( t ) + n ( t ) = 1. Consider P ( t )which is a continuous function of time with a continuoussecond derivative. Let Ω be a subset of the Hilbert spacewhere Re[ ρ ] (cid:54) = 0. If the initial state Ψ ∈ Ω , and theconsistency conditions of Eqs. (15) hold true, then:( i ) there is an interval around t in which NLSE (16)has a unique solution and, therefore, there exists a uniquemap { Ψ , n i , P } → { Ψ , v, J ext } . ( ii ) The solutions (i. e. the QED-TDDFT map) is notglobal in time if at some t ∗ > t the boundary of Ω isreached. Proof : By the condition of the theorem Ψ ∈ Ω whereRe[ ρ ] (cid:54) = 0. Therefore the on-site potential v can be ex-pressed in terms of the density and the wave functionas given by (13), and the Hamiltonian ˆ H [ n, P, Ψ] in theuniversal NLSE is well defined.Let us rewrite NLSE (16) in the following formi ∂ t Ψ = (cid:16) ˆ H + ˆ H [ n, P, Ψ] (cid:17) Ψ (17)where ˆ H is the time-independent (linear) part of theHamiltonian,ˆ H = − ∂ p + 12 ω p + λp ˆ σ z − T ˆ σ x , (18)and ˆ H contains all time-dependent, in particular non-linear, terms,ˆ H [ n, P, Ψ] = J ex [ n, P ] p + v [ n, P, Ψ]ˆ σ z . (19)Here J ex [ n, P ] and v [ n, P, Ψ] are defined by Eqs. (14) and(13), respectively.Since ˆ H is the Hamiltonian of the static shifted har-monic oscillator it defines a continuous propagator in theHilbert space of square integrable functions. ThereforeEq. (17) can be transformed to the following integralequation,Ψ( t ) = e − i ˆ H ( t − t ) Ψ (20) − i (cid:90) tt e − i ˆ H ( t − s ) ˆ H [ n ( s ) , P ( s ) , Ψ( s )]Ψ( s )s. . To prove the existence of solutions to this equation we canuse well established theorems from the theory of quasilin-ear PDE . In particular, we apply the the followingresult. Consider an integral equation of the form, u ( t ) = W ( t, t ) u + (cid:90) tt W ( t, s ) K s ( u ( s ))s. , (21)where W ( t, s ) is a continuous linear propagator on T =[ t , ∞ ) and the kernel K t ( u ) is continuous function oftime, which is locally Lipschitz in a Banach space B .Then there is an interval [ t , t ∗ ) where Eq. (21) has aunique continuous solution.In our case we consider L as the proper Banach space B . The kernel K t (Ψ) = ˆ H [ n ( t ) , P ( t ) , Ψ]Ψ in Eq. (20)is continuous and Lipschitz in L if n ( t ), ¨ n ( t ), P ( t ) and¨ P ( t ) are continuous functions of time, Ψ ∈ Ω, and theconsistency conditions Eqs. (15) are fulfilled. Hence ifall conditions of the theorem are satisfied Eq. (20) hasa unique solution. Moreover since in this case Ψ is inthe domain of H , Ψ ∈ D ( H ), there exists a uniquedifferentiable (strong) solution of Eq. (20) which provesthe statement ( i ) of the theorem.The extension theorems for quasilinear PDE implythat the local solution can not be extended beyond somemaximal existence time t ∗ > t only in two cases: first,at t → t ∗ the solution becomes unbounded or, second, at t → t ∗ it reaches the boundary of Ω. In our case the so-lution is guaranteed to be normalized and thus bounded.Therefore we are left only with the second possibility,which proves the statement ( ii ) and completes the proofof the theorem.The above theorem generalizes the results of Ref. 16where the uniqueness (but not the existence) of the map { Ψ , n i , P } → { Ψ , v, J ext } has been proven for analyticin time potentials using the standard Taylor expansiontechnique.The Theorem 1 can be straightforwardly generalized tothe case of multiple photon modes. The only differenceis that ˆ H in Eq. (18) becomes the Hamiltonian of amultidimensional shifted harmonic oscillator. The rest ofthe proof remains unchanged. This proves the existenceof QED-TDDFT for the spin-boson model in its standardform . A less obvious generalization for the system ofmany interacting electrons on a many-site lattice in ispresented in the next section. III. QED-TDDFT FOR MANY ELECTRONS ONMANY-SITE LATTICES INTERACTING WITHCAVITY PHOTONS
In the previous section we proved the QED-TDDFTexistence theorem for a system of one electron on a two-site lattice coupled to a photonic mode. Below we gen-eralize our results to the case of N interacting electronson a M -site lattice coupled to L photonic modes. Thestate of the system is described by an electron-photonwave function ψ ( r , · · · , r N ; { p } ) where coordinates r i ofthe particles ( i = 1 , , · · · , N ) take values on the lat-tice sites and { p } is the set of continuous coordinatesdescribing the photonic (oscillatoric) degreed of freedom { p } = { p , p , ..., p L } . Again we assume that the elec-tronic subsystem is driven by classical on-site potentials v ( r ; t ) and each photonic mode is coupled to correspond-ing external current J αex ( t ). As usual, assuming that thesize of the lattice is much smaller than the wave lengthof the photon field, we describe the electron-photon cou-pling at the level of the dipole approximation with λ α being the coupling constant to the α -photon.The following time-dependent Schr¨odinger equa-tion describes the time evolution of the system from the initial state ψ ( r · · · r N ; { p } )i ∂ t ψ ( r , ..., r N ; { p } ) = − N (cid:88) i =1 (cid:88) x i T r i , x i ψ ( ..., x i , ... ; { p } )+ N (cid:88) i =1 v ( r j ; t ) ψ ( r , ..., r N ; { p } ) + (cid:88) j>i w r i , r j ψ ( r , ..., r N ; { p } )+ K (cid:88) α =1 (cid:104) − ∂ p + 12 ω α p α + J αex ( t ) p α (cid:105) ψ ( r , ..., r N ; { p } )+ N (cid:88) i =1 K (cid:88) α =1 λ α · r i p α ψ ( r , ..., r N ; { p } ) (22)where the real coefficients T r , r (cid:48) = T r (cid:48) , r correspond to therate of hopping from site r to site r (cid:48) (for definiteness weset T r , r = 0), and w r , r (cid:48) is the potential of a pairwiseelectron-electron interaction.Following the logic of Sec. II we define the on-site den-sity n ( r ) and the expectation value of the field P α fora mode α , which are the basic variables for the QED-TDDFT n ( r ) = N (cid:88) r ,..., r N (cid:90) | ψ ( r , r , ..., r N ; { p } ) | p . , (23) P α = (cid:88) r ,..., r N (cid:90) p α | ψ ( r , ..., r N ; { p } ) | p . (24)where p . = p. · · · p. L .Similarly to the two-site case we derive the force bal-ance equation by calculating the second derivative of thedensity (23) and using the Schr¨odinger equation (22) tosimplify the terms with the time derivative of the wavefunction¨ n ( r ) = 2Re (cid:88) r (cid:48) T r , r (cid:48) ρ ( r , r (cid:48) )( v ( r (cid:48) ; t ) − v ( r ; t ))+ q ( r ; t )+ f ( r ; t )(25)where q ( r ; t ) is the lattice divergence of the internal forces q ( r ; t ) = − (cid:88) r (cid:48) , r (cid:48)(cid:48) T r , r (cid:48) (cid:104) T r (cid:48) , r (cid:48)(cid:48) ρ ( r , r (cid:48)(cid:48) ) − T r , r (cid:48)(cid:48) ρ ( r (cid:48) , r (cid:48)(cid:48) )+ ρ ( r , r (cid:48)(cid:48) , r (cid:48) )( w r , r (cid:48)(cid:48) − w r (cid:48) , r (cid:48)(cid:48) ) (cid:105) , (26)and f ( r ; t ) is the force exerted on electrons from the pho-tonic subsystem f ( r ; t ) = 2Re (cid:88) α (cid:88) r (cid:48) T r , r (cid:48) λ α · ( r (cid:48) − r ) (cid:90) p α ρ ( r , r (cid:48) ; p α )p. α . (27)Here ρ ( r , r (cid:48) ) is the one-particle density matrix ρ ( r , r (cid:48) ) = (cid:90) ρ ( r , r (cid:48) ; p α )p. α (28)= N (cid:88) r ,..., r N (cid:90) ψ ∗ ( r , r ..., r N ; { p } ) ψ ( r (cid:48) , r , ..., r N ; { p } ) p . . and ρ ( r , r (cid:48)(cid:48) , r (cid:48) ) is the two-particle density matrix ρ ( r , r (cid:48)(cid:48) , r (cid:48) ) = N ( N − (cid:88) r ,..., r N (cid:90) (cid:2) ψ ∗ ( r , r (cid:48)(cid:48) ..., r N ; { p } ) × ψ ( r (cid:48) , r (cid:48)(cid:48) , ..., r N ; { p } ) (cid:3) p . (29)The equation of motion for the field average P α (24)is derived in the same manner as in Sec. II by calcu-lating the second time derivative of P α and using theSchr¨odinger equation (22),¨ P α = − ω α P α − J αex − λ α · d (30)where d is the total dipole moment of the N -electronsystem d = (cid:88) r r n ( r ) . (31)The existence of QED-TDDFT is equivalent to the ex-istence of the inverse map { Ψ , n ( r ) , P α } → { Ψ , v, J αext } .To study this map we compile the universal NLSE by ex-pressing the on-site potential v ( r ) and the external cur-rent J αex in terms of the fundamental observables n ( r )and P α , and the wave function Ψ.To find the current J αext as a functional of the field av-erage P α and the density n ( r ) we only need to rearrangeEq. (30) J αex = − (cid:16) ¨ P α + ω α P α + λ α · d (cid:17) . (32)The problem of finding the potential v ( r ) as a functionalof n ( r ) and Ψ is more involved as we need to solve thesystem of M linear equations, Eq. (25), for v ( r ). Let usfirst rewrite (25) in a matrix form as followsˆ K [Ψ] V = S [¨ n, Ψ] , (33)where ˆ K is a real symmetric M × M matrix with elements k r , r (cid:48) [Ψ] = 2Re (cid:34) T r , r (cid:48) ρ ( r , r (cid:48) ) − δ r , r (cid:48) (cid:88) r (cid:48)(cid:48) T r , r (cid:48)(cid:48) ρ ( r , r (cid:48)(cid:48) ) (cid:35) . (34)and V and S are M -dimensional vectors with components v r = v ( r ) , (35a) s r [¨ n, Ψ] = − ¨ n ( r ) − q [Ψ]( r ) − f [Ψ]( r ) . (35b)The problem of inverting (solving for v ( r )) the forcebalance equation (33) for a general lattice has been an-alyzed in Ref. [20] in the context of the standard elec-tronic TDDFT. The same argumentation regarding theproperties of the matrix ˆ K is applicable in the presentcase. Solving Eq. (25) for the on-site potential v ( r ) isequivalent to multiplying both sides of Eq. (33) by in-verse of the ˆ K -matrix. Therefore the matrix ˆ K mustbe non-degenerate. At this point it is worth noting thatbecause of the gauge invariance ˆ K matrix (34) always has at least one zero eigenvalue that corresponds to aspace-constant eigenvector. Therefore if V is the M -dimensional space of lattice potentials v ( r ), then the in-vertibility/nondegeneracy of ˆ K should always refer to theinvertibility in an M − V , whichis orthogonal to a constant vector v C ( r ) = C . In morephysical terms this means that the force balance equation(34) determines the self-consistent potential v [ n, Ψ]( r )only up to an arbitrary constant, like Eq. (8) in sectionII. Therefore in the orthogonal subspace subspace of V we have V = ˆ K − S (36)where by ˆ K − we mean inversion in the subspace of V .Equation (36) the required functional v [ n, Ψ]( r ) whichcan be used to construct the universal NLSE.Finally, like in the two-site case (see Sec. II) the initialsate Ψ , the density n ( r ) and the field average P α shouldsatisfy the consistency conditions at t = t n ( r ; t ) = N (cid:88) r ,..., r N (cid:90) | ψ ( r , ..., r N ; { p } ) | p . , (37a)˙ n ( r ; t ) = − (cid:88) r (cid:48) T r , r (cid:48) ρ ( r , r (cid:48) )] (37b) P α ( t ) = (cid:88) r ,..., r N (cid:90) p α | ψ ( r , ..., r N ; { p } ) | p . (37c)˙ P α ( t ) = Im (cid:88) r ,..., r N (cid:90) (cid:2) ψ ∗ ( r , ..., r N ; { p } ) × ∂ p α ψ ( r , ..., r N ; { p } ) (cid:3) p . (37d)By substituting the on-site potential v ( r ) fromEq. (36), and the field average P α from Eq. (32) in tothe Schr¨odinger equation (22) we find the proper NLSEi ∂ t Ψ( t ) = H [ n, P, Ψ]Ψ( t ) (38)which we use to prove the existence of the unique inversemap { Ψ , n ( r ) , P α } → { Ψ , v ( r ) , J αext } and thus the exis-tence of the QED-TDDFT in a close analogy with theTheorem 1. Theorem 2. (existence of the QED-TDDFT for lat-tice systems coupled to cavity photons) — Assume thata given time-dependent density n ( r ; t ) is nonnegative oneach lattice site, sums up to the number of particles N ,and has a continuous second time derivative ¨ n ( r ; t ). Alsoassume that P α ( t ) is a continuous function of t and hasa continuous second time derivatives ¨ P α ( t ). Let Ω be asubset of the N -particle Hilbert space H where the ma-trix ˆ K [Ψ] (34) has only one zero eigenvalue correspondingto the space-constant vector. If the initial state Ψ ∈ Ω,and at time t the consistency conditions of Eq. (37) arefulfilled, then:( i ) There is a time interval around t where the many-body NLSE (38) has a unique solution that defines thewave function Ψ( t ) and the external potentials, v ( t )and J αex ( t ), as unique functionals of the density n ( r ; t ), fieldaverage P α , and the initial state Ψ ;( ii ) The solution of item ( i ) is not global in time if andonly if at some maximal existence time t ∗ > t the bound-ary of Ω is reached.The proof of this theorem goes along the same lines asthe proof of Theorem 1 in Sec. II . We transform NLSE(38) to a multidimensional integral equation similar to(20) and then apply the general existence results forequations of the type of Eq. (21) to show that the state-ments ( i ) and ( ii ) are in fact true. We skip the detailsas the procedure is mostly a straightforward repetitionof the proof presented in Sec. II. IV. TIME-DEPENDENT v -REPRESENTABILITY FOR A SYSTEMEVOLVING FROM THE GROUND STATE In this section we will show that the ground stateof a quite general lattice Hamiltonian belongs to the v -representability subset Ω. This implies the map { Ψ , n, P α } → { Ψ , v, J αext } is guaranteed to exist if thedynamics starts from the ground state. The main the-orem of this section is a generalization of Theorem 2 inRef. 20.Consider the following the lattice Hamiltonian of manymutually interacting electrons coupled to photonic modesˆ H α,β,γ = ( ˆ T + ˆ V + α ˆ W e − e ) ⊗ ˆ I ph +ˆ I e ⊗ ˆ H ph + βH e − ph (39)where ˆ T is the usual lattice operator of the kinetic energy,ˆ V corresponds to the interaction with a local externalpotential, ˆ W describes the electron-electron interaction,ˆ H ph is the photonic Hamiltonian and H e − ph is the theHamiltonian for the interaction between electrons andthe photon modes, ˆ I ph and ˆ I e are, respectively, the unitmatrices in the photonic and the electronic sectors of theHilbert space, and α and β are real coefficients.Here we demonstrate that the ground state of theHamiltonian (39), for any α, β ∈ R and any on-site po-tential, belongs to the v -representability subset Ω if allterms in Eq. (39) except ˆ T commute with the densityoperator ˆ n r . The proof of this quite general statementclosely follows the proof of the Theorem 2 in Ref. [20].Therefore below we only briefly go through the main lineof arguments.Assume that Ψ k = | k (cid:105) form a complete set of eigen-states for the Hamiltonian (39) and let Ψ = | (cid:105) be theground state. We will show that the matrix ˆ K [Ψ ] eval-uated at the ground state is strictly negative definite inthe subspace of potentials that are orthogonal to a space-constant vector V C . That is, V T ˆ K [Ψ ] V ≡ (cid:88) r , r (cid:48) v ( r ) k r , r (cid:48) v ( r ) < , (40)for all M -dimensional vectors V = { v ( r ) } which are or- thogonal to the spatially constant potential V T V C = C (cid:88) r v ( r ) = 0 , (41)where V T stands for a transposed vector. Thereforeˆ K [Ψ ] is nondegenerate in the subspace orthogonal theto constant potentials.Using the f -sum rule and the spectral representationof the density-density response function (see, for exam-ple, Ref. 34) one can represent the elements of ˆ K -matrixEq. (34) as follows (see Ref.[20] for details) k r , r (cid:48) [Ψ ] = − (cid:88) k ω k (cid:104) | ˆ n r | k (cid:105)(cid:104) k | ˆ n r (cid:48) | (cid:105) . (42)where ω k = E k − E is excitation energy of the systemfrom the ground state to the state k .Substituting k r , r (cid:48) of Eq. (42) into the left hand side ofEq. (40) we find the following result V T ˆ K [Ψ ] V = − (cid:88) k ω k (cid:12)(cid:12)(cid:12) (cid:88) r v ( r ) (cid:104) | ˆ n r | k (cid:105) (cid:12)(cid:12)(cid:12) = − (cid:88) k ω k |(cid:104) | ˆ v | k (cid:105)| ≤ , (43)where ˆ v is an operator corresponding to the potential v ( r ), ˆ v = (cid:88) r v ( r )ˆ n r . (44)The equality in Eq. (43) holds only for a space-constantpotential v C ( r ) = C . Indeed, since each term in the sumin Eq. (43) is non-negative, the result of summation iszero if and only if (cid:104) | ˆ v | k (cid:105) = 0 , for all k (cid:54) = 0 . (45)Assuming that Eq. (45) is fulfilled and expanding thevector ˆ v | (cid:105) in the complete set of states {| k (cid:105)} we getˆ v | (cid:105) = (cid:88) k | k (cid:105)(cid:104) k | ˆ v | (cid:105) = | (cid:105)(cid:104) | ˆ v | (cid:105) ≡ λ | (cid:105) . (46)Therefore the condition of Eq. (45) implies that theground state | (cid:105) is an eigenfunction of the operator ˆ v .Since ˆ v corresponds to a local multiplicative one-particlepotential this can happen only if the potential is spa-tially constant. Hence for all potentials which are or-thogonal to a constant in a sense of Eq. (41) the strictinequality in Eq. (43) takes place. This means that ma-trix ˆ K [Ψ ] is negative definite and thus invertible in the M − V orthogonal to a con-stant vector V C . In other words, the ground state of N -particle system on a connected lattice does belong tothe v -representability subset Ω. This result combinedwith the general existence theorem of Sec. III provesthe following particular version of the time-dependent v -representability theorem. Theorem 3. — Let the initial state Ψ for a time-dependent many-body problem on a connected latticecorresponds to a ground state of a Hamiltonian of theform (39). Consider continuous positive density n ( r ; t )and field average P ( t ) which satisfy the consistency con-ditions of Eqs. (37) and has a continuous second timederivative. Then there is a finite interval around t inwhich n ( r ; t ) and P ( t ) can be reproduced uniquely by atime evolution of Schr¨odinger equation (22) with sometime dependent on-site potential v i ( t ) and external cur-rent J ex ( t ).Note that Theorem 3 is valid for any Hamiltonian ofthe form of Eq. (39) as long as all the terms in the Hamil-tonian, except the kinetic part, commute with the den-sity operator ˆ n r , and, therefore, do not contribute to theˆ K -matrix. An important special case is when the ini-tial state is the interacting many-electron ground statewhich is decoupled from the photonic field, β = 0. Inthis case the ground state is a direct product of the elec-tronic ground state and photonic ground state. Anotherpractically relevant case of α = β = 0 corresponds to theinitial state in a form of the direct product of the non-interacting many-electron wave function (the Slater de-terminant) and the photonic vacuum. For all those casesthe local v -representability is guarantied by the aboveTheorem 3. V. CONCLUSION
In this work we extended the formalism of the latticeTDDFT and presented a rigorous proof of the mappingtheorem of QED-TDDFT for many-electron lattice sys-tems coupled to quantized photonic modes. First we con-sidered the simplest non-trivial model of a one-electronHubbard dimer coupled to a single photonic mode, whichis identical to the Rabi and spin-boson models. We iden-tified a pair of basic variables describing the electronicand photonic degrees of freedom, and showed that the existence of the QED-TDDFT map is equivalent to theexistence of a unique solution to a certain system of non-linear partial differential equations (the universal NLSE).We proved that the Cauchy problem for this NLSE in-deed has a unique solution and, therefore, the uniqueQED-TDDFT map exists, provided the basic variableshave a continuous second time derivative and the initialstate belongs to some well defined subset of the Hilbertspace (the v -representability subset). Further we general-ized the theory to many electrons and multiple photonicmodes. We proved that the same QED-TDDFT mappingcan be constructed for this generic interacting electron-photon system. Finally, we showed that the ground stateof a quite general electron-photon lattice Hamiltonian be-longs to the v -representability subset Ω of the Hilbertspace. Therefore if the system evolves from such groundstate the local v -representability is always guaranteed.The main difference and, in fact, the main mathe-matical difficulty of the present theory, as compared tothe purely electronic lattice TDDFT , is the exis-tence continuum variables describing photonic modes. Asa result the Hilbert space becomes an infinite dimen-sional functional space, and the universal NLSE turnsinto a system of PDE. In a certain sense the latticeQED-TDDFT is on half way between the electronic lat-tice TDDFT and the usual TDDFT in the continuumspace. Hopefully the present rigorous results will shednew light on remaining unresolved issues of TDDFT andthus deepen our understanding of this popular and prac-tically important theory. ACKNOWLEDGEMENTS
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