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A Quantum Quench Towards Kondo Correlations

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A Quantum Quench Towards Kondo Correlations 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 À 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 ¼hneia is the thermal average of ne whose charge state is controllable via an external gate a i with respect to H .) This requires "e À, and ÀU þ voltage Vg applied between a top Schottky gate and the À & f & À "e Àpffiffiffiffiffiffiffiffiffiffiffiffiffi. The Kondo temperature accociated with FR [see Fig. 1(a) and 1(b)]. In a gate voltage regime for f f 2 À f f ¼ À 2 Àj"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 À !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 AðÞ¼ mj hm jejmiij ð!L À E 0 þ EmÞ: (2) Ài!Lt H c f m HL /ðeh 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 ÀEm=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 "ene þ Une"ne# þ af"h ða ¼ i; fÞ (1) f i a evidently is !th ¼ EG À 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 ee, 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 Þ 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 À 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 Þ 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"kckck represents a noninteracting condu- distinct functional forms emerge in the regimes of ‘‘large’’, ction band (the FR) with half-width D ¼ 1=ð2Þ 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 À y H c for Hf ¼ SEAM): = ðec þ : :Þ, with c ¼ kck, describes its tunnel coupling to the e level, giving it a width À.A 4 j"f j & & D: AFOðÞ¼ 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 À2 splitting, " ¼ " þ g B, " ¼ " þ g B (the T & & j"ej: A ðÞ¼ 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 ðÞ/TÀ ð=T ÞÀ : (3c) purposes [12]).
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