Rabi oscillations within TDDFT: the example of the 2 site Hubbard model

Transcription

1 Rabi oscillations within TDDFT: the example of the 2 site Hubbard model Johanna I. Fuks, H. Appel, Mehdi,I.V. Tokatly, A. Rubio Donostia- San Sebastian University of Basque Country (UPV)

2 Outline Rabi oscillations and effective two-level systems many-body interacting system timescale separation Kohn-Sham system theoretical model for EXX dynamical detuning conclusions

3 Motivation Why Rabi oscillations? because they are non-linear, imply drastic change in the populations and are therefore challenging for a functional to reproduce. Why the 2 site Hubbard Model? because the singlet Hilbert space is 3x3 for the interacting problem and 2x2 for the non-interacting problem, i.e. the two-level assumption in the Kohn-Sham space is exact. We hope to understand the dynamical detuning present in adiabatic-tdddft [?] free of the effects of the Kohn-Sham two-level assumption. simple expressions for the exact adiabatic and the exact time-dependent kohn-sham potentials can be found.

4 atomic units (m e = e = = 1) are used. Rabi oscillations laser field E(t) = E 0 sin(ωt), ω = + δ Rabi frequency : Ω = δ 2 +(d eg E 0 ) 2, d eg =< ψ e j ˆr j ψ g >

5 effective two-level system conditions superposition state observables populations δ <<, Ω(δ = 0) = d eg E 0 << ω, ψ(t) >= a g (t) ψ g > +a e (t) ψ e > a g (t) 2 = n g (t), a e (t) 2 = n e (t) n g (t)+n e (t) = 1 dipole moment (given a symmetric hamiltonian) d(t) = ψ(t) j ˆr j ψ(t)

6 many-body interacting system N-particle system coupled to a laser field in dipole approximation, N Ĥ = Ĥ0 + ˆr j E(t) j=1 After projection onto the 2 2 level space i t ψ(t) >= Ĥ ψ(t) > yields the differential system, i t ( ag (t) a e (t) ) ( = ǫ g d eg E(t) d eg E(t) ǫ e )( ag (t) a e (t) ).

7 many-body interacting system From the time-dependent Schroedinger equation one can derive coupled differential equations for: dipole moment d(t) = d eg ng (t)n e (t) cos(ωt +ϕ(t)) transition current J(t) = 2d eg Im[a g(t)a e (t)] populations n g,e (t) = a g,e (t) 2 + normalization and initial conditions 2 t d(t) = 2 d(t) 2(1 2n e ) d eg 2 E(t)

8 timescale separation Using Rotating Wave Approximation (RWA) we separate fast and slow time scales by writing the dipole moment as, d(t) = d eg [b 1 (t) cos(ωt)+b 2 (t) sin(ωt)] After some algebra..., the population of the excited state! ( ) t 2 n e (t) = δ 2 2 +Ω 0 n e (t)+ 1 2 Ω2 0 n e (t) describes a harmonic oscillator with a restoring force which increases with increasing detuning δ.

9 model system to study Rabi oscillations We focus on a 1D two-particle model system which can be solved exactly: 1D-Helium, whose ground state is a two-particle singlet. Ĥ = ˆT + ˆV ext (x)+ ˆV ext (x )+ ˆV ee (x, x )+(ˆx + ˆx )E(t)

10 2 e singlet in 1D: time evolution many-body system a) δ = 0.08Ω (a) b) δ = 2.2Ω (b) Time (a.u.) Dipole moment d(t) (red line) and populations n e (solid line) and n g (dashed line).

11 2 e singlet in 1D: numerical time propagation of non-interacting KS system in octopus a) EXX δ = (a) b) LDA δ = (b) Time (a.u.)

12 KS hamiltonian and KS two-level expansion The KS hamiltonian has an additional time-dependent term due to V hxc [ρ(r, t)] being a functional of the time-dependent density. Ĥ s = Ĥ0 s + with Ĥ0 s = ˆT s + ˆV ext ( r)+ ˆV hxc [ρ 0 ] ˆV dyn hxc (t)+ˆ r E(t) Effective two-level KS system, φ(t) >= a s g(t) φ g > +a s e(t) φ e > where φ(t) is a one-particle state.

13 non-linear Kohn-Sham differential Equations i t φ( r, t) = Ĥsφ( r, t) time evolution yields, ( ) ( a s i g (t) ǫ s t ae(t) s g +ǫ xc g (t) = dege(t)+f s xc (t) The matrix elements are: ǫ s g,e(t) = φ g,e Ĥs 0 φ g,e ǫ xc g,e(t) = φ g,e ˆV dyn hxc (t) φ g,e F xc (t) = φ g ˆV dyn hxc (t) φ e dege(t)+f s xc (t) ǫ s e +ǫ xc e (t) )( a s g (t) a s e(t) )

14 two-electron singlet in exact exchange (EXX) approximation For simplicity we will focus on EXX approximation because for a 2e singlet it is adiabatic and equal to Hartree-Fock, Vhxc EXX (x, t) = 1 dxˆv ee ( x x )(ρ 0 (x )+δρ(x, t)) 2 and the matrix elements can be conveniently written as: ǫ xc g (t) = λ g n s e(t), ǫ xc e (t) = λ e n s e(t) F xc (t) = g d s eg d s (t). where λ e,g and g are part of the symmetric and antisymmetric contributions to φ(t) dxˆv ee ( x x )δρ(x, t) φ(t).

15 equation of motion for the population of KS excited state n s e(t) For the KS system we define the resonance as the frequency of the linear response ω EXX 0 = s ( s + 2g). Using again the RWA we obtain, ( ) γ t 2 ne(t) s 2 = 2 ns e(t) 2 +Ω 2 s ne(t)+ s 1 2 Ω2 s whith Ω s = d s ege 0 and γ = λ 2g. Unlike n e (t), which corresponds to an harmonic oscillator, n s e(t) corresponds to an anharmonic quartic oscillator, analytic solutions can be found in or it can be solved numerically.

16 EXX: numerical propagation versus theoretical model a) EXX octopus (a) b)exx analytic model (b) Time (a.u.) using RWA neglects terms that lead to an error of about 10%.

17 dynamical detuning For any adiabatic functional the potential will change due to the changing density and the system is driven out of resonance. ( ) t 2 n e (t) = δ 2 +Ω 2 0 n e (t)+ 1 2 Ω2 0 ( ) γ t 2 ne(t) s 2 = 2 ns e(t) 2 +Ω 2 s ne(t)+ s 1 2 Ω2 s Resonant Rabi oscilllations are not captured by the adiabatic approximation because the instantaneous dependence on the state of the system introduces a time-dependent detuning.

18 Conclusions Dynamical detuning is not limited to adiabatic TDDFT but is generic for all mean-field theories, e.g. HF or all hybrids, when the effective potential depends instantaneously on the state of the system. Only the inclusion of an appropriate memory dependence can correct the fictitious time-dependence of the resonant frequency. Adiabatic functionals will fail similarly in the description of all processes involving a change in the population of states. J.I. Fuks, N. Helbig, I.V. Tokatly, A. Rubio, Non-linear phenomena in time-dependent density-functional theory: What Rabi physics can teach us, Physical Review B 84, (2011)

19 Thanks!