Correlated Light-Matter Interactions in Cavity Quantum Electrodynamics

Transcription

1 Correlated in Cavity Quantum Electrodynamics [1] Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin [2] NanoBio Spectroscopy group and ETSF, Universidad del País Vasco, San Sebastián, Spain DPG Frühjahrstagung Dresden, 2 nd April 2014

2 Introduction In condensed matter physics and quantum chemistry Quantum mechanics for electrons - classical electromagnetic fields Ab-initio methods: (Time-dependent) density functional theory (TDDFT), Hartree-Fock, Coupled-cluster theory, Green s function theory... In quantum optics Simplified model for matter (Dicke model), quantum light (exactly solvable) model systems, density matrix formulation In this work: Generalization of TDDFT: Quantum Electrodynamical Density Functional Theory (QEDFT): Interacting electrons coupled with photon modes in cavity. Example: Jaynes-Cummings-Hubbard model system driven by external scalar and vector potentials.

3 Experiments Matter-light interactions in the single atom and single photon limit. The Nobel Prize in Physics 2012 (Image source: Manipulating individual quantum systems - Nature 492, 55 (2012)).

4 QEDFT: Quantum electrodynamical density functional theory Electron-photon many-body Hamiltonian: Ĥ = j + α [ ] 2 j 2m + Vext(x j, t) + W xi x j i>j [ p α + ω2 α 2 ( ) 2 p α λα ˆX + J α ] ext(t) p α, ω α ω α with the dipole polarization operator ˆX = N x j, j ω α the frequency of the photon mode α, and λ α is the coupling constant related to the α-mode. In QEDFT, the basic variables are: electron density: n(x, t) = Ψ n(ˆx) Ψ photon momenta: P α(t) = Ψ p α Ψ I. V. Tokatly, Phys. Rev. Lett., 110, (2013). M. Ruggenthaler, F. Mackenroth, and D. Bauer, Phys. Rev. A 84, (2011). M. Ruggenthaler, J.Flick, et. al, arxiv: to be submitted before DPG.

5 Jaynes-Cummings-Hubbard model Hamiltonian ω t x Two-level system Single-field limit Cavity, dipole coupling Many-body Hamiltonian: 1> 2> ( Ĥ(t) = t x ˆσ x +ωâ â + λˆσ z â + â ) ( +j ext(t) â + â ) + v ext(t)ˆσ z

6 Jaynes-Cummings-Hubbard model Hamiltonian Ĥ(t) = t x ˆσ x + ωâ â + λˆσ z (â + â ) ( + j ext(t) â + â ) + v ext(t)ˆσ z t x : electron hopping constant, ω: mode frequency, λ: coupling strength Electronic operators (Pauli-matrices): {σ a, σ b } = 2δ ab I Photonic operators: [a i, a j ] = δ i,j, [a i, a j ] = [a i, a j ] = 0 Pair of conjugated variables: electron density σ z (t) v ext(t) external (classical) potential photon density A(t) = â + â (t) j ext(t) external (classical) potential Equations of motion (using Heisenberg s EOM): 2 2 σz (t) = 4tx {tx σz (t) + vext(t)σx ([σz (t), A(t)]; t) + λ Aσx ([σz (t), A(t)]; t)} t 2 2 t A(t) = ω2 A(t) 2ω (λσ z (t) + j ext(t))

7 Kohn-Sham approach to the model Hamiltonian To solve the presented coupled equations in practice, one needs approximations for the terms, which contain the densities implicitly. Kohn-Sham construction: One possible way: Coupled quantum problem replaced by uncoupled quantum problem (matter and photon field decouples). Uncoupled quantum system chosen to exactly reproduce physical densities σ z (t) and A(t). Shared features of real and auxiliary quantum system lead to reliable approximations. Kohn-Sham dynamics using single-particle wavefunctions: Atom: i t φ el (t) = [ t x ˆσ x + v KS (t)ˆσ z ] φ el (t) v KS (t) = v ext(t) + v MF (t) + v xc(t). Field: i ( [ωâ t φpt(t) = â + j KS (t) â + â )] φ pt(t)

8 Mean-field approximation vs. exact potential Mean-field approximation is identical to Schrödinger-Maxwell propagation σ x ([Ψ 0, σ z, A]; t) σ x ([Φ 0, σ z ]; t) ˆσx ([Ψ 0, σ z, A]; t) A(t) σ x ([Φ 0, σ z ]; t) = A(t)σ x ([Φ 0, σ z ]; t) Adiabatic approximation (no memory) and mean-field approximation. v MF KS ([A, vext] ; t) = vext(t) + λa(t) j KS (t) = λσ z (t) + j ext(t) Compare to exact potential, obtained by Fixed-point iteration: M. Ruggenthaler, et. al, Phys. Rev. A 85, (2012) Iterative equation (self-consistency): v k+1 KS (t) = σz (t) + 4t2 x σ z (t) 8t x vks k ( (t) 4t x σx ([vks k ]; t) + 2)

9 Mean-field - weak-coupling limit - Rabi oscillation t x =, ω = 1, v ext(t) = j ext(t) = 0. Initial state: Ψ 0 = Φ 0 = 1 0 σ x (t) [a.u.] σ z (t) [a.u.] v KS (t) [a.u.] (a) (b) (c) λ = t [a.u.]

10 Mean-field - weak-coupling limit - Rabi oscillation t x =, ω = 1, v ext(t) = j ext(t) = 0. Initial state: Ψ 0 = Φ 0 = 1 0 v KS (t) [a.u.] σ z (t) [a.u.] λ = (a) (b) exact mean-field t [a.u.] j KS (t) [a.u.] A(t) [a.u.] (c) (d) t [a.u.]

11 Mean-field - weak-coupling limit - coherent states t x =, ω = 1, v ext(t) = j ext(t) = 0. Initial state: Ψ 0 = Φ 0 = e α, α = f n(α) n, with f n(α) = αn exp ( 1 n! 2 α 2) n=0 σ x (t) [a.u.] σ z (t) [a.u.] v KS (t) [a.u.] (a) 0.0 λ = 0.01 <n > = 4 (b) (c) t [a.u.]

12 Mean-field - weak-coupling limit - coherent states t x =, ω = 1, v ext(t) = j ext(t) = 0. Initial state: Ψ 0 = Φ 0 = e α, α = f n(α) n, with f n(α) = αn exp ( 1 n! 2 α 2) n=0 v KS (t) [a.u.] σ z (t) [a.u.] λ = 0.01 <n > = (a) (b) 0.0 exact mean-field t [a.u.] j KS (t) [a.u.] A(t) [a.u.] (c) (d) t [a.u.] Better functionals, beyond mean-field approximation, are needed!

13 Summary and Outlook Outlook: QEDFT is capable of reducing computational costs in correlated photon-matter problems. Already in the weak coupling limit, the mean-field approximation shows clear deviations from the exact results. Improved approximation available with optimized effective potential (OEP) scheme: See following talk by Camilla Pellegrini (O 47.3). Scale approach to larger system size. Fixed-point iteration for larger/two-dimensional system size.

14 Acknowledgements Camilla Pellegrini (San Sebastia n) Michael Ruggenthaler (Innsbruck) Rene Jesta dt (FHI) Ilya Tokatly (San Sebastia n) Angel Rubio (San Sebastia n + FHI) Heiko Appel (FHI) Johannes Flick1, Heiko Appel1, and Angel Rubio1,2

15 Summary and Outlook Outlook: QEDFT is capable of reducing computational costs in correlated photon-matter problems. Already in the weak coupling limit, the mean-field approximation shows clear deviations from the exact results. Improved approximation available with optimized effective potential (OEP) scheme: See following talk by Camilla Pellegrini (O 47.3). Scale approach to larger system size. Fixed-point iteration for larger/two-dimensional system size. Thank you for your attention!

16 Experiments Wiring up quantum systems R. J. Schoelkopf and S. M. Girvin, Nature 451, 664 (2008). Cavity Optomechanics: Back-Action at the Mesoscale T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).

17 DFT General remarks on (normal) DFT for electrons Ĥ = [ ˆ 2 ] j 2m + Vext(x j, t) + W xi x j j i>j Ground-state DFT (Hohenberg-Kohn theorem): 1:1 v ext Ψ 0 1:1 n 0 or Ψ[v ext] 1:1 Ψ[n 0 ] or O[v ext] TDDFT: Bijective Mapping (one-to-one correspondence): Ψ([Ψ 0, v ext]; t) 1:1 Ψ([Ψ 0, n]; t) 1:1 O[n 0 ]

18 Generalization Ĥ = j + α [ ] 2 j 2m + Vext(x j, t) + W xi x j i>j [ p α + ω2 α 2 ( ) 2 p α λα ˆX + J α ] ext(t) p α, ω α ω α where ˆX = N x j and the basic variables are: j density : n(x, t) = Ψ n(ˆx) Ψ photon momenta : P α(t) = Ψ p α Ψ Equations of motion for photon momenta: 2 t 2 Pα + ω2 α Pα ωαλαr = J α ext /ωα Kohn-Sham dynamics for electrons: [ i tφ j = 2 j 2m φ j + V s + ] (ω αp α λ αr) λ αx φ j, α with V s = V ext + VHxc el + Vxc α (I. V. Tokatly, Phys. Rev. Lett., 110, (2013).) α

19 Electron-photon interactions To describe dynamics of particles coupled to photons, we solve an evolution equation of the form: where x 0 = ct and x = (ct, r) i c 0 Ψ(t) = Ĥ(t) Ψ(t) with Ψ(t 0 ) = Ψ 0, Ĥ(t) =ĤM + ĤEM + 1 d 3 r c Ĵµ(x)µ (x) + 1 ) d 3 r (Ĵµ(x)a µ ext c (x) +  µ (x)jµ ext (x) Ĵ µ(x) charge current  µ(x) Maxwell-field operator a µ ext (x) (classical) external vector potential j µ ext (x) (classical) external current

20 Wavefunction Typically, one chooses an initial state Ψ 0 and an external pair (v ext,j ext): 2 Ψ([Ψ 0, v ext, j ext]; t) = c xn(t) x n x=1 n=0 These initial settings determine all observables, especially the observables: σ z (t) = Ψ(t) ˆσ z Ψ(t) and A(t) = Ψ(t) ( â + â ) Ψ(t) Proof shows 1:1 correspondence between σ z (t) 1:1 v ext(t) A(t) 1:1 j ext(t)

21 Wavefunction Ψ([Ψ 0, v ext, j ext]; t) 1:1 Ψ([Ψ 0, σ z, p ]; t) Accordingly, every expectation value becomes a unique functional of the initial state Ψ 0 and the internal pair ( σ z (t), A(t) ) Thus, instead of trying to calculate the (numerically expensive) wave function, it is enough to determine the internal pair for a given initial state. For general observables: the explicit functional dependency on the densities might be unknown.

22 Direct connection between conjugated pairs For the present model system, it is rather straightforward to establish a direct connection between the conjugated pairs. Using Heisenberg s equation of motion, yields: 2 2 ˆσz = 4tx (tx ˆσz + vext(t)ˆσx + λˆpˆσx ) t 2 2 t ˆp = ω2ˆp 2ω (λˆσ z + j ext(t)) Now, we can write an equation for the expectation values: (Remember: all expectation values are by construction functionals of v ext and j ext for fixed initial state Ψ 0 ): 2 2 σz ([vext, jext]; t) = 4tx {tx σz ([vext, jext]; t) t +v ext(t)σ x ([v ext, j ext]; t) + λ pσ x ([v ext, j ext]; t)} 2 2 t p([vext, jext]; t) = ω2 p([v ext, j ext]; t) 2ω (λσ z ([v ext, j ext]; t) + j ext(t))

23 Using functional variable-transformation Expressed using external potentials: 2 2 σz ([vext, jext]; t) = 4tx {tx σz ([vext, jext]; t) t +v ext(t)σ x ([v ext, j ext]; t) + λ pσ x ([v ext, j ext]; t)} 2 2 t p([vext, jext]; t) = ω2 p([v ext, j ext]; t) 2ω (λσ z ([v ext, j ext]; t) + j ext(t)) Now, we use the 1:1 correspondence σ z (t) 1:1 v ext(t) and p(t) 1:1 j ext(t) and we can formulate the problem as follows: Expressed using densities: 2 2 σz (t) = 4tx {tx σz (t) + vext(t)σx ([σz (t), p(t)]; t) + λ pσx ([σz (t), p(t)]; t)} t 2 2 t p(t) = ω2 p(t) 2ω (λσ z (t) + j ext(t)) Hence, instead of solving for the (numerically expensive) wavefunction, solve non-linear coupled evolution equations.

24 Convergence of fixed point iteration

25 Mean-field - stronger-coupling limit - Rabi oscillation t x =, ω = 1, v ext(t) = j ext(t) = 0. Initial state: Ψ 0 = Φ 0 = 1 0 v KS (t) [a.u.] σ z (t) [a.u.] λ = 0.1 (a) (b) exact mean-field t [a.u.] j KS (t) [a.u.] A(t) [a.u.] (c) (d) t [a.u.]

26 Mean-field - weak-coupling limit - coherent states t x =, ω = 1, v ext(t) = j ext(t) = 0. Initial state: Ψ 0 = Φ 0 = e α, α = f n(α) n, with f n(α) = αn exp ( 1 (n!) 2 α 2) n=0 v KS (t) [a.u.] σ z (t) [a.u.] λ = 0.01 <n > = (a) (b) 0.0 exact mean-field t [a.u.] j KS (t) [a.u.] A(t) [a.u.] (c) (d) t [a.u.]

27 How to construct approximations? Optimized effective Potential (OEP) Similar derivation as in normal DFT: (Review: S. Kümmel and L. Kronik, Rev. Mod. Phys. 80, 3 (2008). C. Pellegrini et. al. (2014) in preparation. static OEP equation a δ λ 2 ω + 4 a a 2 + T 2 (ω + 2 ) 2 = 0 a 2 + T 2 TDOEP equation (Volterra equation first kind) t +ωλ 2 R 0 t t 2 dt V xc(t )I{d 12 (t )d 21 (t)} = ωλ 2 R dt c(t, t )d 12 (t) dt c(t, t)d 21 (t )e iω(t t ) + 2ωλ2 t 2W + ω I dt c(t, t)d 12 (0)e iωt 0 0 [ n(t) R {d ] 12(t)} d 12 (0) t dt ωλ 2 d 2 12 (0) 1 (2W + ω) 2

28 OEP - weak-coupling limit - Sudden switch t x = 0.7, ω = 1, v ext(t) = 0.2, j ext(t) = 0. σ z (λ) [a.u.] (a) exact static OEP static mean-field E(λ) [a.u.] (b) λ [a.u.]

29 OEP - weak-coupling limit - Sudden switch t x = 0.7, ω = 1, v ext(t) = 0.2, j ext(t) = 0. σ z (t) [a.u.] v KS (t) [a.u.] λ = 0.01 exact TDOEP mean-field 0.8 (a) (b) time in a.u σ z (ω) [a.u.] t [a.u.] 6 (c) ω [a.u.]

30 OEP - stronger-coupling limit - Sudden switch t x = 0.7, ω = 1, v ext(t) = 0.2, j ext(t) = 0. σ z (t) [a.u.] v KS (t) [a.u.] σ z (ω) [a.u.] λ = exact TDOEP mean-field 0.8 (a) (b) 0.1 time in a.u t [a.u.] 6 (c) ω [a.u.]