Nonequilibrium properties of strongly correlated artificial atoms - a Green's functions approach
NNonequilibrium properties of strongly correlatedartificial atoms—a Green’s functions approach
K Balzer and M Bonitz
Institut f¨ur Theoretische Physik und Astrophysik,Christian-Albrechts-Universit¨at Kiel, Leibnizstrasse 15, 24098 Kiel, GermanyE-mail: [email protected]
Abstract.
A nonequilibrium Green’s functions (NEGF) approach for spatiallyinhomogeneous, strongly correlated artificial atoms is presented and applied tocompute the time-dependent properties while starting from a (correlated) initialfew-electron state at finite temperatures. In the regime of moderate to strongcoupling, we consider the Kohn mode of a three-electron system in a parabolicconfinement excited by a short pulsed classical laser field treated in dipoleapproximation. In particular, we numerically confirm that this mode is preservedwithin a conserving (e.g. Hartree-Fock or second Born) theory .PACS numbers: 05.30.-d, 73.21.-b
Keywords : artificial atoms, nonequilibrium Green’s functions, nonequilibriumbehavior, collective excitations
1. Introduction ’Artificial atoms’ (AA) are inhomogeneous quantum few-particle systems confined in atrapping potential and show bound (discrete) electronic states, as they are occurring inreal atoms[1]. Most artificial atoms are realized in an (isotropic) parabolic confinementand quantum dots are a synonym convention for these systems, e.g. Refs [2, 3].But AAs are also formed by ions in Penning and Paul traps, charge carriers insemiconductor heterostructures (quantum wells), or electrons in metal clusters. AAsin one- (1D) and two-dimensional (2D) entrapment show interesting properties farfrom ideal Fermi-gas behavior including ring and shell structures—see the 2D groundstate configurations for N = 2 , , . . . , a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t onequilibrium properties of strongly correlated artificial atoms E/E ∗ = 5 . a ) N = 2 . . E/E ∗ = 12 . b ) N = 3 . . E/E ∗ = 21 . c ) N = 4 . . E/E ∗ = 33 . d ) N = 5 . . E/E ∗ = 46 . e ) N = 6 . . E/E ∗ = 61 . f ) N = 7 . . Figure 1.
Examples for the one-electron ground state (GS) density n GS N ( r ) of N = 2 , , . . . , β = 100 and λ = 5 . a ) to ( f ) display the result of (symmetry broken) unrestricted Hartree-Fock calculations. The side length of the density plots measures 8 x ∗ . which is then propagated in time according to the Keldysh/Kadanoff-Baym equations.In Sec. 5, we present results for a three-electron AA in a 1D and 2D trap geometry.
2. Model
The d -dimensional N -electron Hamiltonian of the artificial atom in a time-dependentlaser field E ( t ) described in dipole approximation readsˆ H ( t ) = N (cid:88) i =1 (cid:18) − (cid:126) m ∗ e ∇ i + m ∗ e ω r i + e E ( t ) r i (cid:19) + N (cid:88) i 1, theartificial atom behaves similar to an ideal Fermi gas. For λ ∼ 1, the equilibriumstate of the AA is Fermi liquid-like, whereas in the limit λ → ∞ , it is x ∗ (cid:29) a B , and quantum effects vanish in favor of classical interaction dominated chargecarriers [10]. For moderate coupling ( λ (cid:38) 1) the AAs typically show spatially welllocalized carrier density including Wigner molecule (Wigner crystal-like) behavior[4],see Fig. 1. Moreover, if the AA is not in its ground state (GS), one has to take intoaccount thermodynamic fluctuations due to a surrounding heat bath of dimensionlesstemperature β − = k B T /E ∗ . Below, all presented results are related to the system ofunits { x ∗ , E ∗ } and in the definition of the NEGFs we take (cid:126) = 1. 3. Preparation of equilibrium states Introducing electron annihilation (creation) operators ψ ( † ) ( r t ) acting in theHeisenberg picture at a space-time point 1 = ( r , t ), the second-quantized form of(3) reads H λ ( t ) = (cid:90) d d r ˆ ψ † (1) h (1) ˆ ψ (1) (4)+ (cid:90) (cid:90) d d r d d ¯ r ˆ ψ † (1) ˆ ψ † (¯1) w ( r − r ¯1 ) ˆ ψ (¯1) ˆ ψ (1) , with the one-electron energy h (1) = ( −∇ r + r ) / γ ( t ) r and the interaction w ( r − r ¯1 ) = λ | r − r ¯1 | − . In the following, we study Hamiltonian (4) at finitetemperatures β − by means of the one-particle nonequilibrium Green’s function G (1 , ¯1), which is defined on the Schwinger/Keldysh contour C (see e.g. Refs. [5, 6]) as G (1 , ¯1) = − i (cid:104) T C ˆ ψ (1) ˆ ψ † (¯1) (cid:105) , (5)where T C denotes time-ordering on C . G (1 , ¯1) obeys the two-time Keldysh/Kadanoff-Baym equation (KBE)[7][ i∂ t − h (1) ] G (1 , ¯1) = δ C (1 − ¯1) − i (cid:90) C d2 W (1 − G (1 , 2; ¯1 , + ) , (6)and its adjoint, where W (1 − ¯1) = δ C ( t − t ¯1 ) w ( r − r ¯1 ) acts instantaneously withcontour delta function δ C , and 2 + indicates the time limit t → t + 0. Here, thetwo-particle Green’s function, G (1 , 2; ¯1 , ¯2) = ( − i ) (cid:104) T C ˆ ψ † (1) ˆ ψ † (¯1) ˆ ψ † (¯2) ˆ ψ † (2) (cid:105) , (7)appears as a generalization of the two-particle density matrix. In terms of G , we can formulate all relevant many-body approximations: For instance,substituting G (1 , 2; ¯1 , ¯2) → G (1 , ¯1) G (2 , ¯2) − G (1 , ¯2) G (2 , ¯1) yields the Hartree-Fock(HF) approximation. Second and higher order approximations (e.g. second Born orGW), can be systematically obtained by diagram expansions known from ground statemany-body theory and are valid for both equilibrium and nonequilibrium situations. onequilibrium properties of strongly correlated artificial atoms γ ( t ) ≡ −∞ < t ≤ t , the AA stays inthermodynamic equilibrium until at a time t > t γ ( t ) becomes nonzero. Withoutloss of generality we thereby can take t = 0. Specifying the time-independent single-electron part of Eq. (4) as h ( r ) = h (1) | γ ( t )=0 , the KBE then reduce, for t , ¯1 ≤ − ∂ τ − h ( r ) ] G M ( r , r ¯1 ; τ ) (8)= δ ( τ ) + (cid:90) d ¯ r (cid:90) β d¯ τ Σ Mλ ( r , ¯ r ; τ − ¯ τ ) G M (¯ r , r ¯1 ; ¯ τ ) . Here, the Matsubara Green’s function being defined as G M (1 , ¯1) = G M ( r , r ¯1 ; τ ) = G ( r − iτ , r ¯1 − iτ ¯1 ), with τ = τ − τ ¯1 ∈ [ − β, β ], characterizes the equilibrium(initial) state of the AA. Further, on the right hand side, we have introduced theself-energy Σ Mλ (1 , ¯1) = Σ Mλ ( r , r ¯1 ; τ ) according to − iW (1 − G (1 , 2; ¯1 , + ) | t , ¯1 ≤ =Σ[ G ](1 , G (2 , ¯1). A conserving many-body approximation[7], i.e. an approximationfor Σ that preserves density (continuity equation), total energy and momentum, cannow be formulated in terms of a functional Φ such that Σ(1 , ¯1) = δ Φ[ G ] /δG (¯1 , β − , most of the equilibrium properties of the AA system(1), e.g. total energy, one-particle density and energy spectrum, are contained in G M , see the formulas in Sec. 4.2 and take the limit t → 0. For the numericaltechniques applicable in solving Eq. (8) in matrix form see e.g. Refs. [3, 8, 9]. In HFapproximation, the self-consistent solution can be written as G M ( r , r ¯1 ; τ ) = n b − (cid:88) m =0 φ ∗ m ( r ) φ m ( r ¯1 ) g Mmm ( τ ) , (9) g Mmm ( τ ) = f β ( (cid:15) m − µ ) e − τ ( (cid:15) m − µ ) = e − τ ( (cid:15) m − µ ) / ( e β ( (cid:15) m − µ ) + 1) , with interaction renormalized (effective single-electron) HF orbitals φ m ( r )[10],quantum numbers m = 0 , . . . , n b − 1, discrete energies (cid:15) m , and a chemical potential µ .Beyond HF level, G M will be no longer diagonal in the functions φ m , and the respectiveoccupation probabilities will deviate from a Fermi-Dirac distribution f β ( (cid:15) m − µ ) dueto additional electron scattering processes. In particular, the inclusion of electron-electron correlations leads to orbital-dependent energy shifts and broadening in theHF spectrum a ( ω ) = (cid:80) m δ ( ω − (cid:15) m ), see Ref. [3]. 4. Time-propagation of initial states When for t > γ ( t ) (cid:54) = 0, the quantum state of theAA evolves in real time according to the KBE, Eq. (6) and its adjoint. Thereby, beingcomputed from the Dyson equation (8) in a self-consistent manner, the MatsubaraGreen’s function serves as initial (Kubo-Martin-Schwinger) condition for the time-propagation. In particular, for t = 0, one has G ( r − iτ , r ¯1 − iτ ¯1 ) = i [ G M ( r , r ¯1 ; τ ) − G M ( r , r ¯1 ; − τ ¯1 )] . (10)Beyond mean field level, all relevant initial correlations are taken into account via G M and, consequently, evolve in time, leading to a correlated N -particle dynamics. onequilibrium properties of strongly correlated artificial atoms The expansion of the NEGF in terms of a HF basis, see Eq. (9), advises us also tosolve the real-time KBE in matrix form[8, 11]. This means that we generally consider G (1 , ¯1) = n b − (cid:88) m,n =0 φ ∗ m ( r ) φ n ( r ¯1 ) g mn ( t , t ¯1 ) , (11)with time arguments t , t ¯1 on the contour C , the coefficient matrix g mn ( t , t ¯1 ) = θ ( t , t ¯1 ) g >mn ( t , t ¯1 ) − θ ( t ¯1 , t ) g 00 000 0 . . . . . . . . . . . . µ nd Born2 nd BornHFHFidealideal a ) b ) b ) Figure 2. Thermodynamic properties of N = 3 charge carriers in the 1D AAwith λ = 2 . β = 2. a ) HF distribution function f β ( (cid:15) m − µ ) (solid curve)and renormalization at the second Born level (dashed curve). The triangle marksthe position of the chemical potential in HF approximation. Further, the idealenergies (cid:15) m = m + 1 / λ ≡ µ . b ) Density profile (cid:104) ˆ n (cid:105) ( x ) for the ideal system as well as for the HFand second Born approximation. (cid:104) ˆ n m (cid:105) ( t ) = g 5. Numerical results In this section, we study the dynamical properties of a three-electron AA, when,initially in thermodynamic equilibrium, the system is excited by a single few-cyclelaser pulse described in dipole approximation, cf. Eq. (3). More precisely, the field islinearly polarized in x -direction and has the time-dependence γ ( t ) = E e − ( t − t l ) / (2 τ l ) cos( ω l ( t − t l )) , (20) onequilibrium properties of strongly correlated artificial atoms τ l = 10 τ l = 2 τ l = 5 n n n n n n n n n n n n n n n n n n o cc up a t i o np r o b a b ili t y h ˆ n m i ( t ) time t [ ω − ] time t [ ω − ]time t [ ω − ] 0 0000 . . . . 81 10 1010 20 2020 30 3030 40 4040 50 5050 60 6060 70 a ) c ) b ) pu l s e . Figure 3. Nonequilibrium behavior of the N = 3 AA system (1D) withparameters λ = 2 . β = 2. a ) to c ) show the mean-field dynamics of theHF orbital occupation probabilities (cid:104) n i (cid:105) ( t ) at a near-resonant laser frequency ω l = 1 . ω and three different pulse durations τ l . The corresponding pulseshapes γ ( t ) are indicated above the figures. where E = E l / √ π denotes the amplitude of the electric field, the Gaussian envelope iscentred at t l , the pulse duration (variance) is given by τ l , and the oscillation frequencyis ω l , cf. Figs. 3 a )- c ).As the response characteristics Ξ( ω l ) of the quantum system we define the amountof energy that has been absorbed from the laser field for a fixed frequency ω l , i.e.Ξ( ω l ) = (cid:104) ˆ E tot (cid:105) ω l ( t → ∞ ) − (cid:104) ˆ E tot (cid:105) (0) . (21)This quantity together with the time-dependent occupation probabilities (cid:104) ˆ n m (cid:105) ( t )allows for the determination of (off)resonant nonequilibrium behavior (and nonlineareffects), see Secs. 5.1 and 5.2.The nonequilibrium behavior of the quantum system (1) is theoretically wellknown: Driven by the laser field, the AA exactly responds according to the excitationof the centre of mass (Kohn or sloshing) mode. This is obtained from the Kohntheorem, and its generalization to the case of an additional external dipole field, seeRef. [13] and references therein. Its statement is that, independent of dimensionality,the centre of mass coordinate R ( t ) = N − (cid:80) i r i ( t ) of a parabolically confined,interacting electron system performs (equivalently to a single particle in the AA)the motion of a forced harmonic oscillator, ¨ | R | + ω | R | = N eE ( t ) /m ∗ e . Furthermore,this effect is accompanied by a rigid translation of the density profile (cid:104) ˆ n (cid:105) ( r ), since theparticle interaction appears only in the relative Hamiltonian and [ H c . m . , H rel . ] = 0.The key point in the present study is, however, that the Kohn theorem also holdswhen the interaction is treated approximately, as long as density, total energy andmomentum are preserved (conserving approximation), and also applies to zero and onequilibrium properties of strongly correlated artificial atoms τ l = 10 τ l = 2 τ l = 5 h ˆ E tot i ( t ) h ˆ E tot i ( t ) h ˆ E tot i ( t ) h ˆ E single i ( t ) h ˆ E single i ( t ) h ˆ E single i ( t ) h ˆ E kin i ( t ) h ˆ E kin i ( t ) h ˆ E kin i ( t ) h ˆ E pot i ( t ) h ˆ E pot i ( t ) h ˆ E pot i ( t ) h ˆ E HF i ( t ) h ˆ E HF i ( t ) h ˆ E HF i ( t ) e n e r g y c o n t r i bu t i o n s h ˆ E i ( t ) time t [ ω − ] time t [ ω − ]time t [ ω − ] 0 0002468 10 101010 20 2020 30 3030 40 4040 50 5050 60 6060 70 a ) c ) b ) pu l s e . Figure 4. For the laser irradiated artificial atom [ ω l = 1 . ω ] also considered inFig. 3, a ) to c ) show the mean-field dynamics of the relevant energy contributionsin dependence of the pulse duration. For τ l = 10, the AA returns after theexcitation close to its initial state (off-resonant situation). Whereas, in Figs. b )and c ) the spectral width of the laser frequency is essentially increased comparedto a ), leading to the AA remaining in an excited state of the Kohn mode (resonantcase). Potential and kinetic energy thereby oscillate out of phase with exactlydouble confinement frequency ω , while the HF energy stays constant. finite temperatures—for the proof see Ref. [13].The following mean-field results for 1D and 2D have been obtained from NEGFcalculations with up to n b = 40 HF orbitals. The main limitations of the approachare thereby (i) the basis size, which sets the dimension of the time-evolution matrix U ( t ) = exp( − i [ h ( t ) + Σ HF ( t )] t ) to be computed in each time-step (diagonalizing h + Σ HF ), and (ii) the two-electron integrals w ij,kl , Eq. (15), that generally requirelarge memory resources [scaling with n b ] and need to be processed very frequentlyin the self-energy expression Σ ( t , t ¯1 ). With more than 50000 time-steps neededto achieve convergence, this results in computing times of typically several hourson a single machine. For the correlated time-evolution of the AA in second Bornapproximation, the propagation must be carried out in the whole two-time plane( t , t ¯1 ). This is an even more intricate task as one needs to compute all the higherorder collision integrals on the r.h.s. of the KBE. However, these calculations arecurrently near completion—examples are to be found in Ref. [8] and for applicationson real atoms and small molecules see Refs. [9, 11, 12]. For the 1D AA calculations in equilibrium and nonequilibrium, we, in Eq. (15) havereplaced the pure Coulomb interaction w ( x − ¯ x ) by λ [( x − ¯ x ) + α ] − / with α = 0 . onequilibrium properties of strongly correlated artificial atoms τ l = 10 τ l = 2 τ l = 5 r e s p o n s ec h a r a c t e r i s t i c s Ξ ( ω l ) laser frequency ω l [ ω ]00 . . . . . E l = 0 . Figure 5. Response characteristics Ξ( ω l ), Eq. (21), for the pulsed laser excitationof N = 3 electrons in the 1D artificial atom. The system parameters are asin Fig. 2. The AA shows resonance behavior at the confinement frequencyonly, i.e. for ω l = ω , and responds via the c.m. motion (Kohn mode)—rigidtranslation of the whole density. With increasing pulse durations τ l (sharpenedlaser frequency) the resonance curves become more and more peaked. a regularization parameter. This is necessary to make the integrals w mn,kl finite and,in a physical interpretation, allows for a small transversal spread of the one-electronwave functions[2]. In 2D, we used α ≡ β = 2. With λ = 2 . f β ( (cid:15) m − µ ) of the equilibrium (initial) state, including itscollisional renormalization in second Born approximation, see Fig. 2 a ). The one-electron density (cid:104) ˆ n (cid:105) ( x ) is displayed in Fig. 2 b ). Compared to the HF result (solidcurve, (cid:104) ˆ E HFtot (cid:105) = 8 . (cid:104) ˆ E (cid:105) = 8 . . G M ( r , r ¯1 ; τ ), the AA was now propagatedin time under the presence of a laser field (centred at t l = 25) with amplitude E l = 0 . ω l = 1 . ω . What happens to the orbital occupations and the energiesfor different pulse durations τ l is shown in Figs. 3 and 4. In all cases, gradually, theHF orbitals m < m ≥ N ( t ). Oscillations of the increased total, kinetic andpotential energy and (cid:104) ˆ n m (cid:105) ( t ) thereby occur with twice the confinement frequency.In Figs. 3 a ) and 4 a ), respectively, the laser excitation is such that the N -particledynamics is decelerated and almost freezed after the pulse has passed. Consequently,we nearly recover the initial state characterized by (cid:104) ˆ n m (cid:105) (0) and (cid:104) ˆ E tot (cid:105) (0). Also, for onequilibrium properties of strongly correlated artificial atoms ǫ = 3 . ǫ = 4 . ǫ = 4 . ǫ = 5 . ǫ = 5 . ǫ = 6 . b ) a ) c ) d ) e ) f ) g ) x [ x ∗ ] y [ x ∗ ] h ˆ n i ( r ) − − − − 30 00 0 00 . . Figure 6. Thermodynamic initial state of the 2D artificial atom ( N = 3) at λ = 2 . β = 2 (in HF approximation). a ) Single-electron density profile (cid:104) ˆ n (cid:105) ( r ) which is rotationally symmetric. Figs. b ) to g ): Energetically lowestspatial, unrestricted HF states φ m ( r ) with orbital energies (cid:15) , . . . , (cid:15) where (cid:15) and (cid:15) as well as (cid:15) and (cid:15) are degenerate). The arrows in c ) and d ) mark thedirection of polarization of the laser field γ ( t ), Eq. (20). n n n n n n n n n occupation probability h ˆ n m i ( t ) t i m e t [ ω − ] . . 01 110203040506070 pulse τ l = 5 Figure 7. Mean-field dynamics of the HF orbital occupation probabilities (cid:104) ˆ n m (cid:105) ( t ) = n m for three charge carriers in a 2D artificial atom with couplingparameter λ = 2 . β = 2. The laser frequency is againnear-resonant, ω l = 1 . ω , and the pulse duration is τ l = 5. In the initial statethe occupation numbers n and n , n and n as well as n and n are practicallypairwise degenerate, compare with the energy spectrum displayed in Fig. 6. different pulse durations, i.e. different spectral profiles of the laser, the maximumlaser energy absorption is observed at the confinement frequency ω , cf. the responsefunction Ξ( ω l ) in Fig. 5. In addition, the single resonance-peak at ω l = ω sharpenswith the increase of τ l . Consider now the spatial dynamics. We observe in all cases,that the center of mass of the AA R ( t ) performs a harmonic oscillation with frequency ω while the whole density profile itself is translated rigidly. Accompanying this fact, E HF is constant in time, see Fig. 4. Thus, we numerically confirm that the Kohntheorem is satisfied. onequilibrium properties of strongly correlated artificial atoms For the three-electron AA in 2D, we have chosen the same system parameters, λ = 2 . β = 2—however, no regularization parameter α was needed in thetwo-electron integrals w mn,kl . The (almost) rotationally symmetric density profilefor the equilibrium state is shown in Fig. 6 and indicates a ring-like structure. It isinstructive to note, that the unrestricted HF solution of the Dyson equation leads toorbitals φ m ( r ), that are in general arbitrarily oriented in space, compare Fig. 6 b )to g ). Together with the energetically degenerate states m = 1 and 2 ( m = 3 and4, etc.), this has the following consequence on the dipole excitation: As degenerateorbitals can be differently oriented relative to the laser field, in the time-evolution ofthe artificial atom this degeneracy is lifted. In the present case, the orbitals m = 1and m = 2 (see Fig. 6 c ) and d )) are almost aligned with the diagonals (dotted lines),nevertheless the small deviations are sufficient to clearly influence the evolution of theoccupation numbers (cid:104) ˆ n m (cid:105) ( t ), see (cid:104) ˆ n (cid:105) and (cid:104) ˆ n (cid:105) as well as (cid:104) ˆ n (cid:105) and (cid:104) ˆ n (cid:105) in Fig. 7. 6. Conclusion and outlook We have presented an analysis of femtosecond relaxation of few-particle quantumdots during and after a short laser pulse. The method of NEGF wa shown to beefficient to describe the dynamics even in the range of strong Coulomb correlations.Numerically, the C.m. mode excitation can serve as a very sensitive test for theNEGF calculation [and any other numerical code] involving quantum many-bodyapproximations Ref. [13]. Acknowledgements We acknowledge stimulating discussions with R. van Leeuwen, A. Filinov and S. 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