Time-dependent density functional theory for many-electron systems interacting with cavity photons
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Time-dependent density functional theory for many-electron systems interacting withcavity photons
I. V. Tokatly ∗ 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 Sebast´ıan, Spain andIKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain
Time-dependent (current) density functional theory for many-electron systems strongly coupled toquantized electromagnetic modes of a microcavity is proposed. It is shown that the electron-photonwave function is a unique functional of the electronic (current) density and the expectation values ofphotonic coordinates. The Kohn-Sham system is constructed, which allows to calculate the abovebasic variables by solving selfconsistent equations for noninteracting particles. We suggest possibleapproximations for the exchange-correlation potentials and discuss implications of this approach forthe theory of open quantum systems.
PACS numbers: 31.15.ee, 42.50.Pq, 03.65.Yz, 71.15.Mb
Time-dependent density functional theory (TDDFT)is a theoretical framework which, similarly to the groundstate DFT [1], relies on the one-to-one mapping of thedensity of particles to the external potential [2]. Theunique density-potential correspondence implies a possi-bility to calculate the exact time-dependent density bysolving Hartree-like equations for fictitious noninteract-ing Kohn-Sham (KS) particles. This tremendous simpli-fication of the problem makes TDDFT one of the mostpopular ab initio approaches for describing quantum dy-namics of realistic many-body systems [3, 4].Standard TDDFT is formulated for systems of quan-tum particles driven by classical electromagnetic fields[2–4], which covers most traditional problems in physicsand chemistry. However, nowadays the experimental sit-uation is rapidly changing. Progress in the fields of cav-ity and circuit quantum electrodynamics (QED) opensa possibility to study many-electron systems stronglyinteracting with quantum light. Notable examples areatoms in optical cavities in cavity-QED [5–7], or meso-scopic systems such as superconducting qubits [8–10] andquantum dots [8–10] in circuit-QED. Recently a strongcoupling of molecular states to microcavity photons andthe modification of chemical landscapes by cavity vac-uum fields have been reported [11–13]. Obviously, theclassical treatment of external fields prevents applicationof TDDFT to this new interesting class of problems.This paper presents TDDFT for systems of electronsstrongly coupled to (or driven by) a quantized electro-magnetic field [14]. We prove the generalized mappingtheorems for TDDFT and for time-dependent currentdensity functional theory (TDCDFT). In both cases weanalyze the structure and properties of the exchange cor-relation (xc) potentials and discuss possible approxima-tion strategies. Finally we make a connection of thepresent theory to TDDFT for open quantum systems.Consider a system of N electrons, e. g., an atom or amolecule, placed inside a cavity hosting M photon modes.In the Schr¨odinger picture the configuration of the system is specified by the positions { x j } Nj =1 of the electrons andthe set { q α } Mα =1 of photonic coordinates. The full sys-tem is described by the wave function Ψ( { x j } , { q α } , t ).Assuming as usual [15] that the wavelength of relevantphoton modes is much larger than the size of the elec-tronic system we adopt the dipole approximation for theelectron-photon coupling. The Hamiltonian of the sys-tem takes the following formˆ H = N X j =1 m h i ∇ j + A ext ( x j , t ) + X α λ α q α i + X i>j W x i − x j + M X α =1 (cid:20) − ∂ q α + 12 ω α q α − J α ext ( t ) q α (cid:21) (1)where W x i − x j is the electron-electron interaction, ω α arethe frequencies of the photon modes, and λ α describescoupling to the α -mode. The electron and photon sub-systems can be driven externally by the classical vectorpotential A ext ( x , t ) and the external “currents“ J α ext ( t ).The vector potential describes (in a temporal gauge)forces from the ions in atoms and molecules and all otherpossible classical fields. The currents J α ext ( t ) allow for anexternal excitation of the cavity modes. The time evolu-tion from a given initial state Ψ ( { x j } , { q α } ) is governedby the Schr¨odinger equation i∂ t Ψ( { x j } , { q α } , t ) = ˆ H Ψ( { x j } , { q α } , t ) . (2)Solution of this equation gives a complete description ofthe system for a fixed configuration of the external fields.All DFT-like approaches assume that the state of thesystem is also uniquely determined by a small set of ba-sic observables, such the density in TDDFT, the cur-rent in TDCDFT, and possibly something else in othergeneralizations of the theory. Below we construct twogeneralizations of TDDFT for electron-photon systemsdescribed by the Hamiltonian of Eq. (1).Let us start with a technically simpler current-basedtheory. Consider the electronic current j ( x , t ) and expec-tation values Q α ( t ) of photon coordinates as the basicvariables. These variables are defined as follows Q α = h Ψ | q α | Ψ i , (3) j = h Ψ | ˆ j p ( x ) | Ψ i − nm A ext − X α λ α m h Ψ | q α ˆ n ( x ) | Ψ i , (4)where n ( x , t ) = h Ψ | ˆ n ( x ) | Ψ i is the electronic density, andˆ n ( x ) = P j δ ( x − x j ) and ˆ j p ( x ) = − i m P j {∇ j , δ ( x − x j ) } are the density and paramagnetic current operators.Equations of motion for Q α follow Eqs. (3) and (2):¨ Q α + ω α Q α = λ α J ( t ) + J α ext ( t ) (5)where J ( t ) = R j ( x , t ) d x is the space-averaged electroniccurrent. Equation (5) is simply the Maxwell equation forthe cavity vector potential projected on the α -mode.Equations (1)–(4) determine the wave function Ψ( t ),and the basic variables j and Q α as functionals of the ini-tial state Ψ , and the external fields A ext and J α ext . Thisdefines a unique map { Ψ , A ext , J α ext } 7→ { Ψ , j , Q α } . TD-CDFT assumes the existence of a unique ”inverse map” { Ψ , j , Q α } 7→ { Ψ , A ext , J α ext } . That is, given the initialstate and the basic observables one can uniquely recoverthe full wave function and the external fields that gener-ate the prescribed dynamics of the basic variables.To prove the uniqueness of the inverse map we followthe nonlinear Schr¨odinger equation (NLSE) approach [16,17]. Assume that j ( x , t ) and Q α ( t ) are given, and expressthe external fields from Eqs. (4) and (5) as follows, A ext = mn h Ψ | ˆ j p ( x ) − j | Ψ i − X α λ α n h Ψ | q α ˆ n ( x ) | Ψ i , (6) J α ext = ¨ Q α + ω α Q α − λ α J . (7)This defines the external fields as explicit functionals ofthe observables j ( x , t ) and Q α ( t ), and the instantaneousstate Ψ( t ). Substitution of Eqs. (6) and (7) into theHamiltonian (1) turns Eq. (2) into the many-body NLSE i∂ t Ψ( t ) = ˆ H [ j , Q α , Ψ]Ψ( t ) , (8)where ˆ H [ j , Q α , Ψ] is an instantaneous functional of Ψ( t ),which depends parametrically on j ( x , t ) and Q α ( t ). Theuniqueness of a solution to Eq. (8) can be proven eas-ily under the usual in TD(C)DFT assumption of t -analyticity [2, 18, 19]. Assuming that j ( x , t ) and Q α ( t )are analytic functions in time, we represent them andunknown Ψ( t ), by the Taylor series j ( t ) = ∞ X k =0 j ( k ) t k , Q α ( t ) = ∞ X k =0 Q ( k ) α t k , Ψ( t ) = ∞ X k =0 Ψ ( k ) t k . After inserting these series into Eq. (8) one observes thatall coefficients Ψ ( k ) with k > j ( k ) , Q ( k ) α , and Ψ (0) ≡ Ψ . The simplereason for this is that the right hand side in Eq. (8) is an instantaneous functional of Ψ( t ), while the left handside ∼ ∂ t Ψ( t ). As the recursion produces the uniqueTaylor series for Ψ( t ), the many-body wave function is aunique functional of the initial state and the basic vari-ables, Ψ[Ψ , j , Q α ]. By substituting this wave functioninto Eq. (6) we find the functional A ext [Ψ , j , Q α ], whichcompletes the proof of the TDCDFT mapping theorem.The KS system for this theory is constructed as follows.Consider a system of N noninteracting particles coupledto the photon modes at the mean field level. This systemis described by a set of N one-particle KS orbitals ϕ j ( x , t )which satisfy the following equations i∂ t ϕ j = 12 m (cid:16) i ∇ + A S + X α λ α Q α (cid:17) ϕ j , (9)where Q α ( t ) is the solution to Eq. (5) with J ( t ) beingreplaced by the space-average of the KS current density j S = 1 m X j Im( ϕ ∗ j ∇ ϕ j ) − nm (cid:16) A S + X α λ α Q α (cid:17) . (10)Using the above NLSE argumentation (or the standardTDCDFT mapping [19]) we find that ϕ j ( x , t ) are uniquefunctionals of the KS current j S ( x , t ) and the KS initialstate Ψ S . A comparison of Eqs. (4) and (10) shows thatthe KS current reproduces the physical current j S = j if A S in Eq. (9) is defined as A S = A ext + A Hxc , where A Hxc = 1 n h X j Im( ϕ ∗ j ∇ ϕ j ) − m h Ψ | ˆ j p ( x ) | Ψ i + X α λ α h Ψ | ∆ q α ∆ˆ n ( x ) | Ψ i i (11)Here ∆ q α = q α − Q α ( t ), and ∆ˆ n ( x ) = ˆ n ( x ) − n ( x , t )are the fluctuation operators for the photonic coordinatesand the electronic density, respectively. By construction,for given initial states Ψ and Ψ S , the potential A Hxc isa gauge invariant universal functional of j and Q α .Therefore the current and the photonic coordinates canbe calculated from a system of Eqs. (9), (5) describ-ing noninteracting fermions driven by a selfconsistentfield and coupled to a set of classical harmonic oscilla-tors. There is a deep reason for explicitly separating the”mean-field” part A mf = P α λ α Q α of the selfconsistentpotential in Eq. (9). This part accounts for the net forceexerted on electrons from the photons. The remainingHxc-part A Hxc does not produce a global force,
Z (cid:2) j × ( ∇ × A Hxc ) − n∂ t A Hxc (cid:3) d x = 0 , (12)which can be checked directly using Eqs. (11), (2), and(9). Apparently both Hartree and xc contributions to A Hxc = A H + A xc satisfy the identity of Eq. (12) indepen-dently. Equation (12) is a generalization of the zero-forcetheorem [20] for the considered electron-photon system.In this regard A xc is similar to the xc potential in theusual TDCDFT for closed purely electronic systems.If the electrons are driven by a scalar external poten-tial V ext ( x , t ), one can choose the density n ( x , t ) as thebasic variable for electronic degrees of freedom. Let usidentify the photonic basic observables for the electron-photon TDDFT. First, we transform of the photon fieldin Eq. (1) from the velocity to the length gauge [15].Then we perform the canonical transformation of pho-ton variables, i∂ q α ω α p α , q α
7→ − iω − α ∂ p α , which“exchanges” the photon coordinates and momenta whilepreserving their commutation relations. The system isnow described by the wave function Φ( { x j } , { p α } , t ) thatis the Fourier transform of Ψ( { x j } , { q α } , t ). The finaltransformed Hamiltonian takes the formˆ H = X j " − ∇ j m + V ext ( x j , t ) + X i>j W x i − x j + X α h − ∂ p α + ω α (cid:16) p α − λ α ω α ˆ X (cid:17) + ˙ J α ext ( t ) ω α p α i , (13)where ˆ X = P Nj =1 x j . The structure of Eq. (13) suggeststhat the proper basic variables are the density n ( x , t ) andthe expectation values P α ( t ) of the photon momenta n ( x , t ) = h Φ | ˆ n ( x ) | Φ i , P α ( t ) = h Φ | p α | Φ i . (14)Equations of motion for the basic variables read¨ P α + ω α P α − ω α λ α R = − ˙ J α ext /ω α , (15) m ¨ n + ∇ F str + X α ∇ f α = ∇ ( n ∇ V ext ) , (16)where R ( t ) = h Φ | ˆ X | Φ i = R x n ( x , t ) d x is the expecta-tion value of the center of mass coordinate, and F str = im h Φ | [ ˆ T + ˆ W , ˆ j p ] | Φ i = −∇ Π ↔ is the usual electronic stressforce which is equal to the divergence of the electronicstress tensor. The force f α ( x , t ) exerted from the α -photon mode on electrons is given by the expression f α ( x , t ) = λ α h Φ | ( ω α p α − λ α ˆ X )ˆ n ( x ) | Φ i . (17)Now we are ready to prove the uniqueness of the map { Φ , n, P α } 7→ { Φ , V ext , J α ext } from the initial state andthe observables to the time-dependent wave function andthe external fields. The corresponding many-body NLSEis constructed using Eqs. (15) and (16).By solving Eqs. (15) and (16) for ˙ J α ext and V ext we getthe external fields as functionals of the basic variablesand the instantaneous wave function Φ( t ), V ext [Φ , n ] and˙ J α ext [ n, P α ]. Inserting these functionals into Eqs. (13)we obtain a Φ-dependent Hamiltonian ˆ H [ n, P α , Φ] of themany-body NLSE that determines the wave function fora given initial state Φ , the density n ( t ) and photon mo-menta P α ( t ). The uniqueness of a solution to this NLSEis demonstrated in exactly the same way as for the above TDCDFT case, provided the standard t -analyticity con-ditions are fulfilled. This proves the generalized TDDFTmapping theorem: the many-body wave function and theexternal fields are the unique functionals of the basic vari-ables, n and P α , and the initial state [21].The KS system can be again constructed explicitly.Consider a system of noninteracting particles describedby N KS orbitals which satisfy the equations i∂ t φ j = − ∇ m φ j + h V S + X α ( ω α P α − λ α R ) λ α x i φ j , (18)where the second term in the square brackets is the mean-field analog of the electron-photon interaction term inEq. (13). The force balance equation for this systemtakes the form m ¨ n + ∇ F S str + ∇ X α λ α ( ω α P α − λ α R ) n = ∇ ( n ∇ V S ) , (19)where F S str = im h Φ S | [ ˆ T , ˆ j p ] | Φ S i = −∇ Π ↔ S is the kineticstress force of noninteracting fermions [Φ S ( t ) is the KSSlater determinant]. By applying the NLSE argumentsto Eqs. (18)-(19) we conclude that ϕ j and V S are uniquefunctionals of n ( x , t ), P α ( t ), and the KS initial state Φ S .Then, from Eqs. (16) and (19) one finds that the KSdensity reproduces the exact density if V S is of the form V S = V ext + V elHxc + X α V α xc , (20)where the universal functionals V elHxc [ n, P ] and V α xc [ n, P ]are defined via the following Sturm-Liouville problems ∇ ( n ∇ V elHxc ) = ∇ ( F S str − F str ) = ∇ ( ∇ Π ↔ Hxc ) , (21) ∇ ( n ∇ V α xc ) = ∇ λ α h Φ | ( λ α ∆ ˆ X − ω α ∆ p α )∆ˆ n | Φ i , (22)with ∆ p α = p α − P α ( t ) and ∆ ˆ X = ˆ X − R ( t ) being thefluctuation operators for the photon momenta and thecenter of mass coordinate of the electrons.Interestingly, in TDDFT the total xc potential is nat-urally separated into the usual electronic stress contribu-tion and the contributions assigned to each photon mode.It is obvious from Eqs. (21) and (22) that each contribu-tion to the total xc potential satisfies the zero-force the-orem, R n ∇ V elHxc d x = R n ∇ V α xc d x = 0. The net photonforce exerted on electrons is fully captured by the meanfield electron-photon potential in Eq. (18). The zero forcetheorem is a consequence of the harmonic potential theo-rem (HPT) [20, 22], which also holds true here as photonsform a set of harmonic oscillators coupled bilinearly tothe electronic center of mass.In practice any DFT-type approach requires approxi-mations for xc potentials. In the present generalizationof the theory we succeeded to define the xc potential insuch a way that it has the same general properties, andobeys the same set of constraints as the xc potential inthe usual purely electronic TDDFT. This suggests natu-ral strategies for constructing approximations.(i) The first possibility is a velocity gradient expan-sion. At zero level we set V α xc = 0 and take V elxc = V ALDAxc ,the xc potential in the standard adiabatic local densityapproximation (ALDA). This seemingly naive approx-imation exactly reproduces the correct HPT-type dy-namics as for the rigid motion with a uniform velocity v = j /n the effect of photons on the density dynamicsis exhausted by the mean-field contribution. The zerolevel approximation can be viewed as a generalizationof ALDA. The dynamical corrections should be propor-tional to velocity gradients. It should be possible to de-rive them perturbatively in the TDCDFT scheme alongthe lines of the Vignale-Kohn approximation [23, 24].(ii) Probably a more promising strategy is to make aconnection of the effective KS potential to the many-bodytheory [25]. Beyond the mean field level the electron-photon coupling generates a retarded photon-mediatedinteraction between the electrons. The correspondingphoton propagators will enter the diagrams for the elec-tronic self energy as additional interaction lines. Thenew contribution to the self energy can then be con-nected to the xc potential via Sham-Schl¨utter equation[26, 27]. In principle the corresponding xc potential canbe constructed perturbatively to any desired order in thecoupling constant [28]. However, already the simplestapproximation, generated by the exchange-like diagram,is expected to capture the important physics. Formallythis approximation for V α xc is an analog of the x-only opti-mized effective potential [27, 29]. Physically it should beresponsible for the Lamb shift effects and for the sponta-neous photon emission in nonequilibrium situations.Exploring practical performance of these approxima-tions is an interesting direction for the future research.If the functionals V Hxc [ n, P α ] and V α xc [ n, P α ] are known,the basic variables, n ( x , t ) and P α ( t ), can be calculatedby solving Eqs. (18), (15). In general the KS Eq. (18)should be solved numerically, while Eq. (15) always ad-mits an analytic solution. For example, for the equilib-rium initial state and J α ext = 0 this solution reads P α ( t ) = Z t sin[ ω α ( t − t ′ )] λ α R ( t ′ ) dt ′ . (23)By substituting Eq. (23) into Eq. (18) we eliminate thephoton variables and get the KS equation involving onlythe electronic density. Thus we obtain TDDFT for anopen quantum system – now the KS equation describes ina closed form only the electronic part of the full electron-photon system.Formally Eq. (13) is a version of the Caldeira-Leggett(CL) model [30, 31]. Therefore, as a byproduct we ob-tained TDDFT for open systems coupled to the CL bathof harmonic oscillators. Let us assume Ohmic spec-tral density of the bath, π P α λ µα λ να δ ( ω − ω α ) = 2 ηδ µν , where η is the friction constant. In this case the self-consistent potential in Eq. (18) reduces to the form V eff = V Hxc + ηN ˙ Rx . The last, mean field term is exactlythe potential in the phenomenological dissipative NLSEproposed by Albrecht [32, 33]. Hence already at zero levelwe recover one of the heuristic theories of quantum dis-sipation. Deficiencies of the Albrecht equation should becorrected by going beyond the zero level approximation.Currently there are several formulations of TDDFTfor open systems, based on the master equation for thedensity matrix [34–36], or on the stochastic Schr¨odingerequation [37, 38]. At the level of the final KS equationsour theory is similar to the formulation of Refs. [35, 36]which also allows for the unitary propagation of KS or-bitals. The conceptual difference is that in the presentcase both the TDDFT mapping and the approximationstrategies are universally valid for the cavity situationwith a few discrete photon modes and for the bath witha continuous spectral density. The bath in traced out atthe very last step after setting up the TDDFT frameworktogether with approximations.In conclusion, TD(C)DFT for systems strongly cou-pled to the cavity photon fields is proposed. We provedthe corresponding generalizations of the mapping the-orems, established the existence of the KS system, andsuggested a few technically feasible approximation strate-gies. In the limit of dense spectrum of photon modes thisapproach naturally leads to TD(C)DFT for open quan-tum systems. This work is a step towards ab initio theoryof various cavity/circuit QED experiments, and practicalTDDFT for dissipative systems.This work was supported by the Spanish MEC(FIS2007-65702-C02-01). ∗ [email protected][1] P. Hohenberg and W. Kohn, Phys. Rev. , B864 (1964)[2] E. Runge and E. K. U. Gross, Phys. Rev. Lett. , 997(1984)[3] Fundamentals of Time-Dependent Density FunctionalTheory , edited by M. A. Marques, N. T. Maitra, F. M.Nogueira, E. Gross, and A. Rubio (Springer, Berlin,2012)[4] C. A. Ullrich,
Time-Dependent Density-Functional The-ory: Concepts and Applications (Oxford UniversityPress, New York, 2012)[5] H. Mabuchi and A. C. Doherty, Science , 1372 (2002)[6] J. M. Raimond, M. Brune, and S. Haroche,Rev. Mod. Phys. , 565 (2001)[7] H. Walther, B. T. Varcoe, B.-G. Englert, and T. Becker,Rep. Prog. Phys. , 1325 (2006)[8] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, Phys. Rev. A , 062320 (2004)[9] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S.Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J.Schoelkopf, Nature , 162 (2004)[10] J. Q. You and F. Nori, Nature , 589 (2011) [11] T. Schwartz, J. A. Hutchison, C. Genet, and T. W. Ebbe-sen, Phys. Rev. Lett. , 196405 (2011)[12] J. A. Hutchison, T. Schwartz, C. Genet, E. Devaux, andT. W. Ebbesen, Angew. Chem. Int. Ed. , 1592 (2012)[13] A. F. i Morral and F. Stellacci, Nat. Mat. , 272 (2012)[14] A recent fully relativistic QED-based formulation of TD-CDFT [39] hints at existence of such theories.[15] F. H. M. Faisal, Theory of Multiphoton Processes (Plenum Press, New York, 1987)[16] I. V. Tokatly, Chem. Phys. , 78 (2011)[17] N. T. Maitra, T. N. Todorov, C. Woodward, andK. Burke, Phys. Rev. A , 042525 (2010)[18] R. van Leeuwen, Phys. Rev. Lett. , 3863 (1999)[19] G. Vignale, Phys. Rev. A , 062511 (2008)[20] G. Vignale, Phys. Rev. Lett. , 3233 (1995)[21] The level of mathematical rigor and the conditions ofthe present mapping theorems are exactly the same asfor the standard purely electronic TD(C)DFT. Strictlyspeaking, no extra assumption is required if the numberof photon modes is arbitrary large, but finite.[22] J. F. Dobson, Phys. Rev. Lett. , 2244 (1994)[23] G. Vignale and W. Kohn, Phys. Rev. Lett. , 2037(1996)[24] G. Vignale, C. A. Ullrich, and S. Conti, Phys. Rev. Lett. , 4878 (1997) [25] M. Gatti, V. Olevano, L. Reining, and I. V. Tokatly,Phys. Rev. Lett. , 057401 (2007)[26] L. J. Sham and M. Schl¨uter, Phys. Rev. Lett. , 1888(1983)[27] R. van Leeuwen, Phys. Rev. Lett. , 3610 (1996)[28] I. V. Tokatly and O. Pankratov, Phys. Rev. Lett. ,2078 (2001)[29] C. A. Ullrich, U. J. Gossmann, and E. K. U. Gross, Phys.Rev. Lett. , 872 (1995)[30] A. Caldeira and A. Leggett, Physica A , 587 (1983)[31] A. Caldeira and A. Leggett, Ann. Phys. , 374 (1983)[32] K. Albrecht, Phys. Lett. B , 127 (1975)[33] R. W. Hasse, J. Math. Phys. , 2005 (1975)[34] K. Burke, R. Car, and R. Gebauer, Phys. Rev. Lett. ,146803 (2005)[35] J. Yuen-Zhou, C. Rodriguez-Rosario, and A. Aspuru-Guzik, Phys. Chem. Chem. Phys. , 4509 (2009)[36] J. Yuen-Zhou, D. G. Tempel, C. A. Rodr´ıguez-Rosario,and A. Aspuru-Guzik, Phys. Rev. Lett. , 043001(2010)[37] M. Di Ventra and R. D’Agosta, Phys. Rev. Lett. ,226403 (2007)[38] R. D’Agosta and M. Di Ventra, Phys. Rev. B , 165105(2008)[39] M. Ruggenthaler, F. Mackenroth, and D. Bauer,Phys. Rev. A84