week ending PRL 106, 107402 (2011) PHYSICAL REVIEW LETTERS 11 MARCH 2011
Many-Body Dynamics of Exciton Creation in a Quantum Dot by Optical Absorption: A Quantum Quench towards Kondo Correlations
Hakan E. Tu¨reci,1,2 M. Hanl,3 M. Claassen,2 A. Weichselbaum,3 T. Hecht,3 B. Braunecker,4 A. Govorov,5 L. Glazman,6 A. Imamoglu,2 and J. von Delft3 1Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA 2Institute for Quantum Electronics, ETH-Zu¨rich, CH-8093 Zu¨rich, Switzerland 3Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universita¨tMu¨nchen, D-80333 Mu¨nchen, Germany 4Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland 5Department of Physics and Astronomy, Ohio University, Athens, Ohio 45701, USA 6Sloane Physics Laboratory, Yale University, New Haven, Connecticut 06520, USA (Received 14 May 2010; published 9 March 2011) We study a quantum quench for a semiconductor quantum dot coupled to a Fermionic reservoir, induced by the sudden creation of an exciton via optical absorption. The subsequent emergence of correlations between spin degrees of freedom of dot and reservoir, culminating in the Kondo effect, can be read off from the absorption line shape and understood in terms of the three fixed points of the single- impurity Anderson model. At low temperatures the line shape is dominated by a power-law singularity, with an exponent that depends on gate voltage and, in a universal, asymmetric fashion, on magnetic field, indicative of a tunable Anderson orthogonality catastrophe.
DOI: 10.1103/PhysRevLett.106.107402 PACS numbers: 78.67.Hc, 78.40.Fy, 78.60.Fi
When a quantum dot (QD) is tunnel coupled to a correlations between the QD-electron and the FR, leading Fermionic reservoir (FR) and tuned such that its topmost to a screened spin singlet, is imprinted on the optical abso- occupied level harbors a single electron, it exhibits at low rption line shape [Fig. 1(c)]: its high-, intermediate-, and temperatures the Kondo effect, in which QD and FR are low-detuning behaviors are governed by the three fixed bound into a spin singlet. It is interesting to ask how Kondo points of the single-impurity Anderson model (AM) correlations set in after a quantum quench, i.e., a sudden [Fig. 1(d)]. We present detailed numerical and analytical change of the QD Hamiltonian, and corresponding predic- results for the line shape as a function of temperature and tions have been made in the context of transport experi- magnetic field. At zero temperature we predict a tunable ments [1–4]. Optical transitions in quantum dots [5–7] Anderson orthogonality catastrophe, since the difference in offer an alternative arena for probing Kondo quenches: initial and final ground state phase shifts of FR electrons the creation of a bound electron-hole pair—an exciton— via photon absorption implies a sudden change in the local charge configuration. This induces a sudden switch-on of both a strong electron-hole attraction [6–8] and an ex- change interaction between the bound electron and the FR. The subsequent dynamics is governed by energy scales that become ever lower with increasing time, leaving tell- tale signatures in the absorption and emission line shapes. For example, at low temperatures and small detunings relative to the threshold, the line shape has been predicted to show a gate-tunable power-law singularity [8]. Though optical signatures of Kondo correlations have not yet been FIG. 1 (color online). A localized QD e level, tunnel coupled experimentally observed, prospects for achieving this to a FR and (a) assumed empty at t ¼ 0, (b) is filled at t ¼ 0þ goal improved recently due to two key experimental ad- when photon absorption produces a neutral exciton, leading to vances [9,10]. Kondo correlations between QD and FR for t !1. (c) Starting 0 Here we propose a realistic scenario for an optically from an empty QD state jGii (for T ¼ ), the absorption rate at frequency !L ¼ !th þ (with detuning from the threshold induced quantum quench into a regime of strong Kondo f i f !th ¼ E E ) probes the spectrum of H at excitation correlations. A quantum dot tunnel coupled to a FR is G G energy . (d) Cartoons illustrating the nature of the free orbital prepared in an uncorrelated initial state [Fig. 1(a)]. (FO), local moment (LM) and strong-coupling (SC) fixed points Optical absorption of a photon creates an exciton, thereby of the Anderson impurity model, which are dominated by charge inducing a quantum quench to a state conducive to Kondo fluctuations, spin fluctuations (indicated by dashed arrows) and a correlations [Fig. 1(b)]. The subsequent emergence of spin screened spin singlet, respectively.
0031-9007=11=106(10)=107402(4) 107402-1 Ó 2011 American Physical Society week ending PRL 106, 107402 (2011) PHYSICAL REVIEW LETTERS 11 MARCH 2011
[indicated by wavy lines in Fig. 1(d)] can be tuned by Figs. 1(a) and 1(b), where the e level is essentially empty magnetic field and gate voltage via their effects on the in the initial state and singly occupied in the ground state of level occupancy. i 0 f 1 a theP final Hamiltonian, ne ’ and ne ’ . (Here ne ¼ Model.— a a We consider a QD, tunnel coupled to a FR, ne , and ne ¼hne ia is the thermal average of ne whose charge state is controllable via an external gate a i with respect to H .) This requires "e , and U þ voltage Vg applied between a top Schottky gate and the & f & "e pffiffiffiffiffiffiffiffiffiffiffiffiffi. The Kondo temperature accociated with FR [see Fig. 1(a) and 1(b)]. In a gate voltage regime for f f 2 f f ¼ 2 j"e ð"e þUÞj=ð U Þ ¼ 2 which the QD is initially uncharged, a circularly polarized H is TK U= e .If"e U= , f 1 f light beam (polarization ) at a suitably chosen frequency so that ne ¼ , then H represents the symmetric ! propagating along the z axis of the heterostructure excitonic Anderson model, to be denoted by writing L f will create a so-called neutral exciton [11](X0), a bound H ¼ SEAM. electron-hole pair with well-defined spins and ¼ Absorption line shape.—Absorption sets in once !L (2fþ; g) in the lowest available localized s orbitals exceeds a threshold frequency, !th. The line shape at of the QD’s conduction- and valence bands (to be called temperature T and detuning ¼ !L !th is, by the e and h levels, with creation operators ey and hy , re- golden rule, proportional to the spectral function (see [13]) X spectively). The QD-light interaction is described by 2 i 0 y 2 f i y y A ð Þ¼ mj hm je jmiij ð!L E 0 þ EmÞ: (2) i!Lt H c f m HL /ðe h e þ : :Þ. We model the system before mm0 and after absorption by the initial and final Hamiltonian a a i=f i=f Here jmia and Em are exact eigenstates and energies of H , H ¼ He þ Hc þ Ht, where i i Em=T i X depicted schematically in Fig. 1(c), and m ¼ e =Z a a the initial Boltzmann weights. The threshold frequency He ¼ "e ne þ Une"ne# þ af"h ða ¼ i; fÞ (1) f i a evidently is !th ¼ EG EG (EG is the ground state energy a f of H ), which is on the order of " þ " (up to correc- describes the QD, with Coulomb cost U for double e h y tions due to tunneling and correlations). occupancy of the e level, ne ¼ e e , and hole energy ð Þ 0 We calculated A using the Numerical "h (> , on the order of the band gap). The e level’s initial Renormalization Group (NRG) [14], generalizing the ap- and final energies before and after absorption, proach of Refs. [8,15]toT 0 by following Ref. [16]. The a 0 "e (a ¼ i, f), differ by the Coulomb attraction Uehð> Þ inset of Fig. 2 shows a typical result: As temperature is between the newly created electron-hole pair, which a gradually reduced, an initially rather symmetric line shape pulls the final e level downward, "e ¼ "e afUeh becomes highly asymmetric, dramatically increasing in [Fig. 1(b)]. This stabilizes the excited electron against peak height as T ! 0.AtT ¼ 0, the line shape displays f decay into the FR, provided that "e lies below the FR’s a threshold, vanishing for <0 and diverging as tends to 0 f i Fermi energy "F ¼ . Since H H , absorption imple- 0 from above. Figure 2 analyzes this divergence on a log- ments a quantum quench, which, as elaborated below, log plot, for the case that T, which cuts off the divergence, can beP tuned by electric and magnetic fields. The term is smaller than all other relevant energy scales. Three y Hc ¼ k "k ck ck represents a noninteracting condu- distinct functional forms emerge in the regimes of ‘‘large’’, ction band (the FR) with half-width D ¼ 1=ð2 Þ and ‘‘intermediate’’ or ‘‘small’’ detuning, labeled (for reasons discussed below) FO, LM and SC, respectively, (given here pconstantffiffiffiffiffiffiffiffiffiffiffiffiffiP density of states perP spin, while Ht ¼ y H c for Hf ¼ SEAM): = ðe c þ : :Þ, with c ¼ kck , describes its tunnel coupling to the e level, giving it a width .A 4 j"f j & & D: AFOð Þ¼ ð j"f jÞ; (3a) magnetic field B along the growth-direction of the hete- e 2 e rostructure (Faraday configuration) causes a Zeeman 3 1 3 f LM 2 splitting, " ¼ " þ g B, " ¼ " þ g B (the T & & j"e j: A ð Þ¼ ln ð =T Þ; (3b) e e 2 e h h 2 h K 4 K Zeeman splitting of FR states can be neglected for our SC 1 T & & T : A ð Þ/T ð =T Þ : (3c) purposes [12]). The electron-hole pair created by photon K K K absorption will additionally experience a weak but highly anisotropic intradot exchange interaction [12]. Its effects The remarkable series of crossovers found above are can be fully compensated by applying a magnetic field symptomatic of three different regimes of charge and spin dynamics. They can be understood analytically using fine-tuned to a value, say Beh, that restores degeneracy of the e level’s two spin configurations [12]. Hence- fixed-point perturbation theory (FPPT). To this end, note that at T ¼ 0 the absorption line shape can be written as forth, B is understood to be measured relative to Beh. ¼ @ ¼ ¼ 1 Z 1 We set B kB , give energies in units of i f y 2Re it þ iH t iH t 1 A ð Þ¼ dte ihGje e e e jGii; (4) D ¼ throughout, and assume T, B U, Ueh 0 D "h . The electron-hole recombination rate is as- a a a 0 sumed to be negligibly small compared to all other where H ¼ H EG and þ ¼ þ i þ. Thus it directly y energy scales. We focus on the case, illustrated in probes the postquench dynamics, with initial state e jGii, 107402-2 week ending PRL 106, 107402 (2011) PHYSICAL REVIEW LETTERS 11 MARCH 2011 FO 4 SC LM FO For r ¼ and LM, A ð Þ can be calculated using 10 0 0 i i perturbation theory in Hr.ForT ¼ , note that Γ = Γ ε = 0.75U −η Γi e ν σ = 0 εf = −0.5U 1 e y A ð Þ¼ 2ImhGje e jGii; (5) Γ = 0.03U i f 0 T þ H 10 T = 3.3⋅10−10Γ )
K −6 0 T = 5.9⋅10 Γ f K set H ! Hr þ Hr and expand the resolvent in powers of (T
σ T 0 K 3π/4 U = 0.1D Hr. One readily finds (see [13]) 2 B = 0
) / A 1 ν ν −4 10 T / T ln ( /T ) ν 10 T=0 K K 2 1 ( )] r 0 0 y K σ 0 0.01 Im 10 A ð Þ’ 2 ihGje Hr Hre jGii; (6)
(T A 0.1 σ −1 f þ Hr 10 1 |ε | 10 eσ ) / [A −2 ν ( 10 100 which reveals the relevant physics: Large detuning −8 σ A −3 10 10 4Γ (r ¼ FO) is described by the spectral function of the 0 50 100 2 ν / T ν y K operator Hte ; the absorption process can thus be under- −4 0 4 8 10 10 10 10 stood as a two-step process consisting of a virtual excita- ν / T tion of the QD resonance, followed by a tunneling event to K a final free-electron state above the Fermi level. In contrast, FIG. 2 (color online). Log-log plot of the absorption line shape intermediate detuning (r ¼ LM) is described by the 0 f SEAM y A ð Þ for T TK, B ¼ and H ¼ (for which spectral function of s~ s~ e , i.e., it probes spin flu- 1 c e ¼ 2 ), showing three distinct functional forms for high, ctuations. Evaluating these spectral functions is ele- intermediate and small detuning, labeled FO, LM, and SC, mentary since HFO and HLM involve only free fermions. respectively, according to the corresponding fixed points of the f 1 For B ¼ 0 and j"e j¼2 U, we readily recover Eqs. (3a) Anderson model. Arrows indicate the crossover scales T, TK and f and (3b) (see [13]), which quantitatively agree with the j"e j. Fixed-point perturbation theory [FPPT, red dashed lines, NRG results of Fig. 2.—Though the latter was calculated from Eq. (3)] and NRG (thick blue line for i 0; thin blue line f ¼ SEAM for i ¼ 0) agree well. Inset: A ð Þ for five temperatures in for H , Eq. (3b) holds more generally as long as 0 f f LM semilog scale, obtained from FPPT for i ¼ [dashed lines, H remains in the LM regime, with n e ’ 1; then A ð Þ from Eq. (7)] and NRG (solid lines). f depends on "e , U and only through their influence on TK, and hence is a universal function of and TK. of a photogenerated e-electron coupled to a FR. Evidently, The FPPT strategy for calculating FO and LM line large, intermediate or small detuning, corresponding to shapes can readily be generalized to finite temperatures ever longer time scales after absorption, probes excitations [12], using the methods of Ref. [17] (Section III.A) for at successively smaller energy scales [see Fig. 1(c)], for finding the finite-T dynamic magnetic susceptibility [13]. f 0 f max which H can be represented by expansions Hr þ Hr For j j j"e j and ½j j;T TK, we obtain around the three well-known fixed points [14] of the AM: 3 =T Korð ; TÞ= the free orbital, local moment and strong-coupling fixed ALMð Þ¼ ; (7) 4 1 =T 2 þ 2 ð Þ points (r ¼ FO, LM, SC), characterized by charge fluctua- e Kor ; T 2 tions, spin fluctuations and a screened spin singlet, respec- where Korð ; TÞ¼ T=ln ½maxðj j;TÞ=TK is the scale- tively, as illustrated in Fig. 1(d). dependent Korringa relaxation rate [17]. It is smaller than Large and intermediate detuning—perturbative re- T by a large logarithmic factor, implying a narrower and gime.—For large detuning, probing the time interval higher absorption peak than for thermal broadening. f t & 1=j"e j immediately after absorption, the e level ap- Small detuning and Kondo-edge singularity—strong- pears as a free, filled orbital perturbed by charge fluctua- coupling regime.—As is lowered through the bottom of tions, described by [14] the fixed-point Hamiltonian the LM regime, Jð Þ increases through unity into the f strong-coupling regime, and A ð Þ monotonically crosses HFO ¼ Hc þ He þ const and the relevant perturbation 0 over to the SC regime. It was first studied for the present HFO ¼ Ht. Intermediate detuning probes the times f model (for B ¼ 0Þ in Ref. [8], which found a power-law 1=j" j & t & 1=T for which real charge fluctuations e K line shape of the form (3c), characteristic of a Fermi edge have frozen out, resulting in a stable local moment; how- singularity, with an exponent that followed Hopfield’s ever, virtual charge fluctuations still cause the local mo- rule [18]. The power-law behavior reflects Anderson ment to undergo spin fluctuations, which are not yet orthogonality [19,20]: it arises because the final ground screened. This is described by [14] HLM ¼ Hc þ const state jGfi that is reached in the long-time limit is charac- and the RG-relevant perturbation H0 ¼ Jð Þ s~ s~ . Here P LM e c terized by a screened singlet. The singlet ground state 1 y s~j ¼ 2 0 j ~ 0 j 0 (for j ¼ e, c) are spin operators for induces different phase shifts [as indicated in Fig. 1(d) by the e level and conduction band, respectively, ( ~ are Pauli wavy lines] for FR electrons than the unscreened initial ln 1 y matrices), and Jð Þ¼ ð =TKÞ is an effective, scale- state just after photon absorption, e jGii, and hence is dependent dimensionless exchange constant. orthogonal to the latter. It is straightforward to generalize 107402-3 week ending PRL 106, 107402 (2011) PHYSICAL REVIEW LETTERS 11 MARCH 2011
charge: ne;lower ! 1 while ne;upper ! 0. As a result, Anderson orthogonality [19] is completely absent 0 0 ( ne 0 ¼ ) for photo-excitation into the lower level, since subsequently the e-level spin need not adjust at all. In 0 1 contrast, it is maximal ( ne 0 ¼ ) for photo-excitation into the upper level, since subsequently the e-level spin has to create a spin-flip electron-hole pair excitation in the FR to reach its longtime value. It follows, remarkably, that a magnetic field tunes the strength of Anderson orthogonal- ity, implying a dramatic asymmetry for the evolution of the line shape A ð Þ/ with increasing jBj [Fig. 3(a)]. Conclusions.—We have shown that optical absorption in a single quantum dot can implement a quantum quench in an experimentally accessible solid-state system that allows the emergence of Kondo correlations and Anderson or- thogonality to be studied in a tunable setting. A. I. and H. E. T. acknowledge support from the Swiss FIG. 3 (color online). Asymmetric magnetic-field dependence of the line shape for Hf ¼ SEAM and T ¼ 0. (a) Depending on NSF under Grant No. 200021-121757. H. E. T. acknowl- whether the electron is photoexcited into the lower or upper of edges support from the Swiss NSF under Grant No. PP00P2-123519/1. B. B. acknowledges support from the Zeeman-split e levels ( geB<0 or >0, solid or dashed lines, respectively), increasing jBj causes the near-threshold the Swiss NSF and NCCR Nanoscience (Basel). J. v. D. divergence, A ð Þ/ , to be either strengthened, or sup- acknowledges support from the DFG (SFB631, pressed via the appearance of a peak at ’ geB, respectively. SFB-TR12, De730/3-2, De730/4-1), the Cluster of (The peak’s position is shown by the red line in the geB- Excellence NIM and in part from NSF under Grant plane.) (b) Universal dependence on geB=TK of the local mo- No. PHY05-51164. A. I. acknowledges support from an f ment me (dash-dotted line), and the corresponding prediction of ERC Advanced Investigator Grant, and L. G. from NSF Hopfield’s rule, Eq. (8), for the infrared exponents lower (solid Grant No. DMR-0906498. line) and upper (dashed line) for ¼þ. Symbols: þ values extracted from the near-threshold þ divergence of Aþð Þ. Symbols and lines agree to within 1%. 83 0 [1] P. Nordlander et al., Phys. Rev. Lett. , 808 (1999). the arguments of Refs. [8,18] to the case of B [2] M. Plihal, D. C. Langreth, and P. Nordlander, Phys. Rev. B (see [13]). One readily finds the generalized Hopfield rule X 71, 165321 (2005). 1 0 2 0 [3] F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 ¼ ð ne 0 Þ ; ne 0 ¼ 0 ne 0 ; (8) 0 (2005). [4] D. Lobaskin and S. Kehrein, Phys. Rev. B 71, 193303 0 0 ne 0 is the displaced charge of electrons with spin ,in (2005). units of e, that flows from the scattering site to infinity [5] T. V. Shahbazyan, I. E. Perakis, and M. E. Raikh, Phys. y f i Rev. Lett. 84, 5896 (2000). when e jG i is changed to jG i, and n 0 ¼ n 0 n 0 i f e e e [6] K. Kikoin and Y. Avishai, Phys. Rev. B 62, 4647 (2000). j i j i is the local occupation difference between Gf and Gi . [7] A. O. Govorov, K. Karrai, and R. J. Warburton, Phys. Rev. According to Eq. (8), can be tuned not only via gate B 67, 241307 (2003). a voltage but also via magnetic field, since both modify "e [8] R. W. Helmes et al., Phys. Rev. B 72, 125301 (2005). 0 94 and hence ne 0 . This tunability can be exploited to study [9] J. M. Smith et al., Phys. Rev. Lett. , 197402 (2005). universal aspects of Anderson orthogonality physics. In [10] P. A. Dalgarno et al., Phys. Rev. Lett. 100, 176801 (2008). i ¨ et al. 93 particular, if the system is tuned such that n e ¼ 0 and [11] A. Hogele , Phys. Rev. Lett. , 217401 (2004). n f ¼ 1 at B ¼ 0, Eq. (8) can be expressed as ¼ 1 þ [12] H. Tureci et al. (to be published). e 2 [13] See supplemental material at http://link.aps.org/ 2 f 2 f 2 f 1 me ðme Þ , where the final magnetization me ¼ 2 supplemental/10.1103/PhysRevLett.106.107402. f f [14] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, ðneþ ne ) is a universal function of geB=TK. The ex- ponents then are universal functions of g B=T , with Phys. Rev. B 21, 1003 (1980). e K 55 simple limits for small and large fields [see Fig. 3(b)]: [15] T. A. Costi, Phys. Rev. B , 3003 (1997). 1 [16] A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99, ! for jg Bj T , while lower upper ! 1 for 2 e K = 076402 (2007). jgeBj TK. Here the subscript ‘‘lower’’ or ‘‘upper’’ dis- [17] M. Garst et al., Phys. Rev. B 72, 205125 (2005). tinguishes whether the spin- electron is photoexcited into [18] J. Hopfield, Comments Solid State Phys. 2, 40 (1969). 0 the lower or upper of the Zeeman-split pair ( geB< [19] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967). or >0, respectively). The sign difference 1 for arises [20] P. Nozie`res and C. T. De Dominicis, Phys. Rev. 178, 1097 since these cases yield fully asymmetric changes in local (1969). 107402-4