Quantum Electrodynamical TDDFT: From basic theorems to approximate functionals

Transcription

1 Quantum Electrodynamical TDDFT: From basic theorems to approximate functionals Ilya Tokatly NanoBio Spectroscopy Group - UPV/EHU San Sebastiàn - Spain IKERBASQUE, Basque Foundation for Science - Bilbao - Spain European Theoretical Spectroscopy Facility (ETSF) TDDFT workshop Benasque, 2016

2 Outline 1 Motivation 2 Setting up the mathematical problem 3 Electron-photon TDDFT 4 Implications for the theory of open quantum systems 5 Summary

3 Outline 1 Motivation 2 Setting up the mathematical problem 3 Electron-photon TDDFT 4 Implications for the theory of open quantum systems 5 Summary

4 Cavity and circuit QED: Beginning of the story I. Cavity QED (1980s 1990s): Rydberg atoms in optical cavities II. Circuit QED (2004): SC qubits in transmission line resonators Cooper pair box in a microwave resonator Nature 431, 162 (2004)

5 Circuit QED: New generation of quantum circuits III. Quantum dot strictures integrated with resonators ( ) SWNTs-based QDs 2DEG-based DQDs InAs nanowire-based DQDs PRL 107, (2011) PRL 108, (2012) Nature 490, 380 (2012) Possibility to experimentally study many-body physics and transport in interacting systems strongly coupled to cavity photons

6 Cavity QED: Chemistry-in-cavity IV. Molecules in optical cavities (2012) Modifying Chemical Landscapes by Coupling to Vacuum Fields Angew. Chem. Int. Ed. 51, 1592 (2012); Nature Mat. 11, 272 (2012)

7 Cavity QED: Chemistry-in-cavity IV. Molecules in optical cavities (2012) Modifying Chemical Landscapes by Coupling to Vacuum Fields Angew. Chem. Int. Ed. 51, 1592 (2012); Nature Mat. 11, 272 (2012) QED-TDDFT: IVT, Phys. Rev. Lett. 110, (2013) Generalization of TDDFT for the first principle description of non-relativistic many-electron systems interacting with (or driven by) quantum electro-magnetic fields

8 Outline 1 Motivation 2 Setting up the mathematical problem 3 Electron-photon TDDFT 4 Implications for the theory of open quantum systems 5 Summary

9 Getting started: Classical dipole in classical cavity Relative coordinate: x = x + x Newton equation for dipole dynamics ẍ(t) = Ω 2 x(t) + e m E (r 0, t) Density of electric current: j(r, t) = eẋ(t)δ(r r 0 ) = t P(r, t) Maxwell equations for transverse electromagnetic field, E = 0 t B = c E t E = c B + 4π(j + j ext) Transverse part of the current j = t P couples to the cavity modes P (r, t) = ex(t)δ (r r 0 ) = e ( 4π x(t) ) r r 0

10 Classical cavity (II): the Hamiltonian structure Electric displacement field: D = E + 4πP Maxwell equations in the Hamiltonian form t B = c (D 4πP ) i[h e-m, B] t D = c B i[h e-m, D ] { H e-m = 1 8π dr E 2 + B 2} { = 1 8π dr (D 4πP ) 2 + B 2} [B i (r), D j (r )] = iε ijk k δ(r r )

11 Classical cavity (II): the Hamiltonian structure Electric displacement field: D = E + 4πP Maxwell equations in the Hamiltonian form t B = c (D 4πP ) i[h e-m, B] t D = c B i[h e-m, D ] { H e-m = 1 8π dr E 2 + B 2} { = 1 8π dr (D 4πP ) 2 + B 2} [B i (r), D j (r )] = iε ijk k δ(r r ) Projection on cavity modes E α (r): c 2 2 E α = ω 2 αe α, E α = 0 D = 4π q α ω α E α, B = 4π c p α E α [q α, p β ] = iδ αβ ω α α α H e-m = 1 [ 2 α ωα( 2 qα λα ω α x ) 2 + p 2 α ], λ α = e 4πE α (r 0 )

12 Quantum many-electron system in quantum cavity Many-body electron-photon wave function Ψ({x j }, {q α }, t) N electrons {x j } N j=1 and M photon modes {q α} M α=1 Hamiltonian in a multipolar (PWZ) gauge Ĥ = [ ] j 1 2m 2 j + V ext(x j, t) + i>j W x i x j + [ ) ] α q α α( ω2 qα λα ω α ˆX 1 ω α Pext α 2 ˆX = N j=1 x j is the c. m. coordinate of the electronic subsystem i t Ψ({x j }, {q α }, t) = ĤΨ({x j}, {q α }, t) Many-electron system harmonically coupled to a set of harmonic oscillators (cavity photons)

13 Photon induced electron-electron interaction 1 ( 2 ω2 α q α λ ) 2 α ˆX ω αˆq α λ αˆx j + 1 (λ αˆx i )(λ αˆx j ) ω α 2 j i,j }{{}}{{} (â +â)ψ ψ v(x,x )ˆn(x)ˆn(x ) Effective interaction Weff(x, α t; x, t ) = λ [ αˆx ωα ˆq 2 α (t)ˆq α (t ) + δ(t t ) ] λ αˆx }{{} D(t t )

14 Photon induced electron-electron interaction 1 ( 2 ω2 α q α λ ) 2 α ˆX ω αˆq α λ αˆx j + 1 (λ αˆx i )(λ αˆx j ) ω α 2 j i,j }{{}}{{} (â +â)ψ ψ v(x,x )ˆn(x)ˆn(x ) Effective interaction Weff(x, α t; x, t ) = λ [ αˆx ωα ˆq 2 α (t)ˆq α (t ) + δ(t t ) ] λ αˆx }{{} D(t t ) D(ω) = ω 2 α ω 2 ω 2 α }{{} D D ω +1 = ω 2 ω 2 ω 2 α }{{} E E ω ẍ Physical electron-electron interaction is mediated by the electric field. Only accelerated electron generates the transverse electric field felt by another electron!

15 Outline 1 Motivation 2 Setting up the mathematical problem 3 Electron-photon TDDFT 4 Implications for the theory of open quantum systems 5 Summary

16 Direct map: {Ψ 0, V ext, Pext} α {Ψ, n, Q α } i t Ψ({x j }, {q α }, t) = ĤΨ({x j}, {q α }, t); Ψ(t = 0) = Ψ 0 Ĥ = j + α [ 2 ] j 2m + V ext(x j, t) + W xi x j i>j [ q α }{{} B 2 ( q α λ α 1 ) 2 ] ˆX Pext α 2 ω α ω }{{ α } E 2 + ω2 α Basic variables: mean photon coodinates and the electronic density Q α (t) = Ψ(t) q α Ψ(t) electric displacement n(x, t) = Ψ(t) ˆn(x) Ψ(t) ˆX = xˆn(x)dx, R(t) = Ψ(t) ˆX Ψ(t) = xn(x, t)dx

17 Inverse map: {Ψ 0, n, Q α } {Ψ, V ext, Pext} α via NLSE Starting point: Equations of motion for basic variables Maxwell : Force balance : Qα + ω 2 αq α ω α λ α R = ω α P α ext, m n + F str + α f α = [ n( V ext + λ α P α ext) ], F str (x, t) = im Ψ [ ˆT + Ŵ, ĵ p ] Ψ = Π stress force f α (x, t) = λ α Ψ (ω α q α λ α ˆX)ˆn(x) Ψ force from α-mode

18 Inverse map: {Ψ 0, n, Q α } {Ψ, V ext, Pext} α via NLSE Starting point: Equations of motion for basic variables Maxwell : Force balance : Qα + ω 2 αq α ω α λ α R = ω α P α ext, m n + F str + α f α = [ n( V ext + λ α P α ext) ], F str (x, t) = im Ψ [ ˆT + Ŵ, ĵ p ] Ψ = Π stress force f α (x, t) = λ α Ψ (ω α q α λ α ˆX)ˆn(x) Ψ force from α-mode

19 Inverse map: {Ψ 0, n, Q α } {Ψ, V ext, Pext} α via NLSE Starting point: Equations of motion for basic variables Maxwell : Force balance : Qα + ω 2 αq α ω α λ α R = ω α P α ext, m n + F str + α f α = [ n( V ext + λ α P α ext) ], F str (x, t) = im Ψ [ ˆT + Ŵ, ĵ p ] Ψ = Π stress force f α (x, t) = λ α Ψ (ω α q α λ α ˆX)ˆn(x) Ψ force from α-mode V ext [n, Ψ], P α ext[q α, n] : Ĥ[V ext, P α ext] Ĥ[n, Q α, Ψ] = NLSE This NLSE defines the TDDFT map: {Ψ 0, n, Q α } {Ψ, V ext, P α ext} QED-TDDFT mapping theorem Ψ(t), V ext (x, t), and P α ext(t) are unique functionals of the initial state Ψ 0 and the basic observables n(x, t), and Q α (t).

20 Comments on mathematical issues Nonlinear many-body problem for QED-TDDFT i t Ψ = Ĥ[V ext, Pext] Ψ, α Ψ(t = 0) = Ψ 0 Q α + ωαq 2 α ω α λ α R = ω α Pext α m n + F str [Ψ] + f α [Ψ] = [ n( V ext + λ α Pext) α ] α Solve for Ψ(t), V ext (x, t), and P α ext(t), given Ψ 0, n(x, t), and Q α (t) Existence of a unique solution: QED-TDDFT mapping theorem Ψ(t), V ext (x, t), and P α ext(t) are unique functionals of the initial state Ψ 0 and the basic observables n(x, t), and Q α (t).

21 Comments on mathematical issues Nonlinear many-body problem for QED-TDDFT i t Ψ = Ĥ[V ext, Pext] Ψ, α Ψ(t = 0) = Ψ 0 Q α + ωαq 2 α ω α λ α R = ω α Pext α m n + F str [Ψ] + f α [Ψ] = [ n( V ext + λ α Pext) α ] α Solve for Ψ(t), V ext (x, t), and P α ext(t), given Ψ 0, n(x, t), and Q α (t) Existence of a unique solution: QED-TDDFT mapping theorem Ψ(t), V ext (x, t), and P α ext(t) are unique functionals of the initial state Ψ 0 and the basic observables n(x, t), and Q α (t). Currently we have (assuming a finite number of photon modes): Standard proof of uniqueness under the assumption of t-analyticity [IVT, PRL 110, (2013)] Rigorous proof of uniqueness and existence for lattice electrons [M. Farzanepour and IVT, PRB 90, (2014)]

22 Kohn-Sham construction for the QED-TDDFT Maxwell-Schrödinger dynamics for N noninteracting KS particles [ i t φ j = 2 2m φ j + V S + ] α (ω αq α λ α R Pext)λ α α x φ j, Q α + ω 2 αq α ω α (λ α R + P α ext) = 0 The KS density reproduces the physical density, n S (x, t) = n(x, t), if V S = V ext + V el Hxc [n, Q α] + α V α xc[n, Q α ] The electronic V el Hxc [n, Q α] and photonic V α xc[n, Q α ] xc potentials (n V el Hxc) = (F S str F str ) = ( ΠHxc), (n V α xc) = λ α Ψ (λ α ˆX ω α q α ) ˆn Ψ

23 Kohn-Sham construction for the QED-TDDFT Maxwell-Schrödinger dynamics for N noninteracting KS particles [ i t φ j = 2 2m φ j + V S + ] α (ω αq α λ α R Pext)λ α α x φ j, Q α + ω 2 αq α ω α (λ α R + P α ext) = 0 The KS density reproduces the physical density, n S (x, t) = n(x, t), if V S = V ext + V el Hxc [n, Q α] + α V α xc[n, Q α ] The electronic V el Hxc [n, Q α] and photonic V α xc[n, Q α ] xc potentials (n V el Hxc) = (F S str F str ) = ( ΠHxc), (n V α xc) = λ α Ψ (λ α ˆX ω α q α ) ˆn Ψ Zero force theorem holds true for both xc potentials VHxc el and V xc α n V el Hxc dx = n Vxcdx α = 0

24 The problem of approximations We succeeded to define xc potentials in the electron-photon TDDFT in such a way, that they have similar general properties and satisfy similar constraints as V xc in the usual purely electronic TDDFT. I. Velocity gradient expansion : Natural approximation strategies At zero level we set V α xc = 0 and take V el xc = V ALDA xc. This 0-approximation correctly reproduces a HPT-type rigid motion with a uniform velocity analog of ALDA Quantum/nonadiabatic corrections v(x, t) similar to Vignale-Kohn construction in the electronic TDCDFT

25 The problem of approximations We succeeded to define xc potentials in the electron-photon TDDFT in such a way, that they have similar general properties and satisfy similar constraints as V xc in the usual purely electronic TDDFT. I. Velocity gradient expansion : Natural approximation strategies At zero level we set V α xc = 0 and take V el xc = V ALDA xc. This 0-approximation correctly reproduces a HPT-type rigid motion with a uniform velocity analog of ALDA Quantum/nonadiabatic corrections v(x, t) similar to Vignale-Kohn construction in the electronic TDCDFT II. OEP-strategy : Making a connection to the many-body theory via electron-photon generalization of the Sham-Schlütter equation.

26 Electron-photon Optimized Effective Potential [1] Formulation in terms of orbital xc functionals: V xc = δe xc /δn Ground state: Interpret Lamb shift as xc orbital functional E xc = 1 ɛ j ɛ k f k (1 f j ) φ j λ α x φ k φ k λ α x φ j 2 ɛ j ɛ k + ω α α,k,j Dynamics: Schwinger-Keldysh xc action functional A xc = α,k,j C dτ 1 dτ 2 f k (1 f j )W > (τ 1, τ 2 ) φ j λ α x φ k τ1 φ k λ α x φ j τ2 [1] C. Pellegrini, et. al., PRL 115, (2015)

27 Test case: 2-site molecule coupled to a singe mode Ĥ = T σ x + V ext (t)σ z 1 ( 2 2 q + ω2 q λ ) 2 2 ω σ z

28 Outline 1 Motivation 2 Setting up the mathematical problem 3 Electron-photon TDDFT 4 Implications for the theory of open quantum systems 5 Summary

29 Connection to the theory of open quantum systems Electron-photon dynamics without a photon driving field, P α ext(t) = 0 i t Ψ({x j }, {q α }, t) = ĤΨ({x j}, {q α }, t); Ψ(t = 0) = Ψ 0 Ĥ = j + [ α [ 2 ] j 2m + V ext(x j, t) + W xi x j i>j 1 ( 2 2 q α + ω2 α q α λ ) 2 ] α ˆX 2 ω α Many-electron system harmonically coupled to a set of harmonic oscillators (cavity photons) Caldeira-Leggett model of dissipative quantum systems [Ann. Phys. 149, 374 (1983)]

30 TDDFT for open quantum systems KS equations in the cavity TDDFT with P α ext(t) = 0 i t φ j = 2 2m φ j + [ V ext + V Hxc [n, Q α ] + α (ω αq α λ α R)λ α x ] φ j, Q α + ω 2 αq α = ω α λ α R(t), R(t) = xn(x, t)dx Tracing out photons (bath): Q α [n](t) = t 0 sin[ω α(t t )]λ α R(t )dt Closed KS equations for an open quantum system i t φ j = 2 2m φ j + ( V ext + V eff [n] ) φ j, V eff [n] = V Hxc [n, Q α [n]] + α (ω αq α [n] λ α R)λ α x

31 TDDFT for open quantum systems KS equations in the cavity TDDFT with P α ext(t) = 0 i t φ j = 2 2m φ j + [ V ext + V Hxc [n, Q α ] + α (ω αq α λ α R)λ α x ] φ j, Q α + ω 2 αq α = ω α λ α R(t), R(t) = xn(x, t)dx Tracing out photons (bath): Q α [n](t) = t 0 sin[ω α(t t )]λ α R(t )dt Closed KS equations for an open quantum system i t φ j = 2 2m φ j + ( V ext + V eff [n] ) φ j, V eff [n] = V Hxc [n, Q α [n]] + α (ω αq α [n] λ α R)λ α x Ohmic spectral density of the bath π α λµ αλ ν αδ(ω ω α ) = 2ηδ µν = V eff = V Hxc + ηnṙx In zero-level approximation we recover Albrecht s dissipative NLSE [Phys. Lett. B 56,127 (1975)]

32 Outline 1 Motivation 2 Setting up the mathematical problem 3 Electron-photon TDDFT 4 Implications for the theory of open quantum systems 5 Summary

33 Summary It is possible to formulate a rigorous TDDFT approach to address dynamics of many-electron systems strongly coupled to cavity/resonator photons General properties of corresponding xc potentials suggest several strategies for constructing approximations. We constructed and tested an electron-photon QED-OEP functional (work is still in progress) QED-TDDFT leads to a very natural formulation of TDDFT for open/dissipative quantum systems: (i) first set up TDDFT together with approximation, and (ii) then trace out the bath QED-TDDFT is naturally formulated in the dipole approximation. Beyond dipole approximation coupling to the photon s magnetic field has to be considered. Hence one should use the electron current j(x, t) as a basic variable QED-TDCDFT.

34 Summary It is possible to formulate a rigorous TDDFT approach to address dynamics of many-electron systems strongly coupled to cavity/resonator photons General properties of corresponding xc potentials suggest several strategies for constructing approximations. We constructed and tested an electron-photon QED-OEP functional (work is still in progress) QED-TDDFT leads to a very natural formulation of TDDFT for open/dissipative quantum systems: (i) first set up TDDFT together with approximation, and (ii) then trace out the bath QED-TDDFT is naturally formulated in the dipole approximation. Beyond dipole approximation coupling to the photon s magnetic field has to be considered. Hence one should use the electron current j(x, t) as a basic variable QED-TDCDFT.

35 Summary It is possible to formulate a rigorous TDDFT approach to address dynamics of many-electron systems strongly coupled to cavity/resonator photons General properties of corresponding xc potentials suggest several strategies for constructing approximations. We constructed and tested an electron-photon QED-OEP functional (work is still in progress) QED-TDDFT leads to a very natural formulation of TDDFT for open/dissipative quantum systems: (i) first set up TDDFT together with approximation, and (ii) then trace out the bath QED-TDDFT is naturally formulated in the dipole approximation. Beyond dipole approximation coupling to the photon s magnetic field has to be considered. Hence one should use the electron current j(x, t) as a basic variable QED-TDCDFT.

36 Summary It is possible to formulate a rigorous TDDFT approach to address dynamics of many-electron systems strongly coupled to cavity/resonator photons General properties of corresponding xc potentials suggest several strategies for constructing approximations. We constructed and tested an electron-photon QED-OEP functional (work is still in progress) QED-TDDFT leads to a very natural formulation of TDDFT for open/dissipative quantum systems: (i) first set up TDDFT together with approximation, and (ii) then trace out the bath QED-TDDFT is naturally formulated in the dipole approximation. Beyond dipole approximation coupling to the photon s magnetic field has to be considered. Hence one should use the electron current j(x, t) as a basic variable QED-TDCDFT.