# Many-body dynamics of exciton creation in a quantum dot by optical absorption: a quantum quench towards Kondo correlations.

**ABSTRACT** 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.

**0**Bookmarks

**·**

**65**Views

- F. Haupt, S. Smolka, M. Hanl, W. Wüster, J. Miguel-Sanchez, A. Weichselbaum, J. von Delft, A. Imamoglu[Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the effect of many-body interactions on the optical absorption spectrum of a charge-tunable quantum dot coupled to a degenerate electron gas. A constructive Fano interference between an indirect path, associated with an intra-dot exciton generation followed by tunneling, and a direct path, associated with the ionization of a valence-band quantum dot electron, ensures the visibility of the ensuing Fermi-edge singularity despite weak absorption strength. We find good agreement between experiment and renormalization group theory, but only when we generalize the Anderson impurity model to include a static hole and a dynamic dot-electron scattering potential. The latter is a consequence of the significant transient probability amplitude for intra-dot excitation and differentiates our work from earlier studies of X-ray absorption by an ensemble of deep-level states.04/2013; - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**Recent advances in quantum electronics have allowed to engineer hybrid nano-devices comprising on chip a microwave electromagnetic resonator coupled to an artificial atom, a quantum dot. These systems realize novel platforms to explore non-equilibrium quantum impurity physics with light and matter. Coupling the quantum dot system to reservoir leads (source and drain) produces an electronic current as well as dissipation when applying a bias voltage across the system. Focusing on a standard model of biased quantum dot coupled to a photon mode we elucidate the signatures of the electronic correlations in the phase of the transmitted microwave signal. In addition, we illustrate the effect of the electronic degrees of freedom on the photon field, giving rise to non-linearities, damping and dissipation, and discuss how to control these effects by means of gate and bias voltages.Physical Review B 10/2013; 89(19). · 3.66 Impact Factor - SourceAvailable from: export.arxiv.org
##### Article: Edge physics of the quantum spin Hall insulator from a quantum dot excited by optical absorption.

[Show abstract] [Hide abstract]

**ABSTRACT:**The gapless edge modes of the quantum spin Hall insulator form a helical liquid in which the direction of motion along the edge is determined by the spin orientation of the electrons. In order to probe the Luttinger liquid physics of these edge states and their interaction with a magnetic (Kondo) impurity, we consider a setup where the helical liquid is tunnel coupled to a semiconductor quantum dot that is excited by optical absorption, thereby inducing an effective quantum quench of the tunneling. At low energy, the absorption spectrum is dominated by a power-law singularity. The corresponding exponent is directly related to the interaction strength (Luttinger parameter) and can be computed exactly using boundary conformal field theory thanks to the unique nature of the quantum spin Hall edge.Physical Review Letters 04/2014; 112(14):146804. · 7.73 Impact Factor

Page 1

Many-Body Dynamics of Exciton Creation in a Quantum Dot by Optical Absorption:

A Quantum Quench towards Kondo Correlations

Hakan E. Tu ¨reci,1,2M. Hanl,3M. Claassen,2A. Weichselbaum,3T. Hecht,3B. Braunecker,4A. Govorov,5

L. Glazman,6A. Imamoglu,2and 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 ¨t Mu ¨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.106.107402PACS numbers: 78.67.Hc, 78.40.Fy, 78.60.Fi

When a quantum dot (QD) is tunnel coupled to a

Fermionic reservoir (FR) and tuned such that its topmost

occupied level harbors a single electron, it exhibits at low

temperatures the Kondo effect, in which QD and FR are

bound into a spin singlet. It is interesting to ask how Kondo

correlations set in after a quantum quench, i.e., a sudden

change of the QD Hamiltonian, and corresponding predic-

tions have been made in the context of transport experi-

ments [1–4]. Optical transitions in quantum dots [5–7]

offer an alternative arena for probing Kondo quenches:

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 subsequentdynamics 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

experimentally observed, prospects for achieving this

goal improved recently due to two key experimental ad-

vances [9,10].

Here we propose a realistic scenario for an optically

induced quantum quench into a regime of strong Kondo

correlations. A quantum dot tunnel coupled to a FR is

prepared in an uncorrelated initial state [Fig. 1(a)].

Optical absorption of a photon creates an exciton, thereby

inducing a quantum quench to a state conducive to Kondo

correlations [Fig. 1(b)]. The subsequent emergence of spin

correlations between the QD-electron and the FR, leading

to a screened spin singlet, is imprinted on the optical abso-

rption line shape [Fig. 1(c)]: its high-, intermediate-, and

low-detuning behaviors are governed by the three fixed

points of the single-impurity Anderson model (AM)

[Fig. 1(d)]. We present detailed numerical and analytical

results for the line shape as a function of temperature and

magnetic field. At zero temperature we predict a tunable

Andersonorthogonalitycatastrophe, since thedifference in

initial and final ground state phase shifts of FR electrons

FIG. 1 (color online).

to a FR and (a) assumed empty at t ¼ 0, (b) is filled at t ¼ 0þ

when photon absorption produces a neutral exciton, leading to

Kondo correlations between QD and FR for t ! 1. (c) Starting

from an empty QD state jGii(for T ¼ 0), the absorption rate at

frequency !L¼ !thþ ? (with detuning ? from the threshold

!th¼ Ef

energy ?. (d) Cartoons illustrating the nature of the free orbital

(FO), local moment (LM) and strong-coupling (SC) fixed points

of the Anderson impurity model, which are dominated by charge

fluctuations, spin fluctuations (indicated by dashed arrows) and a

screened spin singlet, respectively.

A localized QD e level, tunnel coupled

G? Ei

G) probes the spectrum of Hfat excitation

PRL 106, 107402 (2011)

PHYSICAL REVIEWLETTERS

week ending

11 MARCH 2011

0031-9007=11=106(10)=107402(4)107402-1

? 2011 American Physical Society

Page 2

[indicated by wavy lines in Fig. 1(d)] can be tuned by

magnetic field and gate voltage via their effects on the

level occupancy.

Model.—We consider a QD, tunnel coupled to a FR,

whose charge state is controllable via an external gate

voltage Vgapplied between a top Schottky gate and the

FR [see Fig. 1(a) and 1(b)]. In a gate voltage regime for

which the QD is initially uncharged, a circularly polarized

light beam (polarization ?) at a suitably chosen frequency

!Lpropagating along the z axis of the heterostructure

will create a so-called neutral exciton [11] (X0), a bound

electron-hole pair with well-defined spins ? and ? ? ¼ ??

(2 fþ;?g) in the lowest available localized s orbitals

of the QD’s conduction- and valence bands (to be called

e and h levels, with creation operators ey

spectively). The QD-light interaction is described by

HL/ ðey

and after absorption by the initial and final Hamiltonian

Hi=f¼ Hi=f

Ha

?

? and hy

? ?, re-

?hy

? ?e?i!Ltþ H:c:Þ. We model the system before

e

þ Hcþ Ht, where

e¼

X

"a

e?ne?þ Une"ne#þ ?af"h ? ?

ða ¼ i;fÞ

(1)

describes the QD, with Coulomb cost U for double

occupancy of the e level, ne?¼ ey

"h ? ?(> 0,on the order of theband gap).The elevel’s initial

andfinalenergiesbefore

"a

between the newly created electron-hole pair, which

pulls the final e level downward, "a

[Fig. 1(b)]. This stabilizes the excited electron against

decay into the FR, provided that "f

Fermi energy "F¼ 0. Since Hf? Hi, absorption imple-

ments a quantum quench, which, as elaborated below,

can be tuned by electric and magnetic fields. The term

Hc¼P

constant density of states ? per spin, while Ht¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi

tunnel coupling to the e level, giving it a width ?. A

magnetic field B along the growth-direction of the hete-

rostructure (Faraday configuration) causes a Zeeman

splitting, "e?¼ "eþ1

Zeeman splitting of FR states can be neglected for our

purposes [12]). The electron-hole pair created by photon

absorption will additionally experience a weak but highly

anisotropic intradot exchange interaction [12]. Its effects

can be fully compensated by applying a magnetic field

fine-tuned to a value, say B?

eh, that restores degeneracy of

the e level’s two spin configurations [12]. Hence-

forth, B is understood to be measured relative to B?

?e?, and hole energy

andafter absorption,

e?(a ¼ i, f), differ by the Coulomb attraction Uehð>0Þ

e?¼ "e?? ?afUeh

e? lies below the FR’s

k?"k?cy

k?ck?represents a noninteracting condu-

ction band (the FR) with half-width D ¼ 1=ð2?Þ and

p

?=??

P

?ðey

?c?þ H:c:Þ, with c?¼P

kck?, describes its

2?geB, "h?¼ "hþ3

2?ghB (the

eh.

We set ?B¼ @ ¼ kB¼ 1, give energies in units of

D ¼ 1 throughout, and assume T, B ? ? ? U, Ueh?

D ? "h ? ?. The electron-hole recombination rate is as-

sumed to be negligibly small compared to all other

energy scales. We focus on the case, illustrated in

Figs. 1(a) and 1(b), where the e level is essentially empty

intheinitial state and singly occupiedinthegroundstate of

the final Hamiltonian, ? nie’ 0 and ? nf

P

with respect to Ha.) This requires "ie?? ?, and ?U þ

? & "f

Hfis TK¼

so that

? nf

excitonic Anderson model, to be denoted by writing

Hf¼ SEAM.

Absorption line shape.—Absorption sets in once !L

exceeds a threshold frequency, !th. The line shape at

temperature T and detuning ? ¼ !L? !th is, by the

golden rule, proportional to the spectral function (see [13])

A?ð?Þ ¼ 2?X

Here jmiaand Ea

depicted schematically in Fig. 1(c), and ?im¼ e?Eim=T=Zi

the initial Boltzmann weights. The threshold frequency

evidently is !th¼ Ef

of Ha), which is on the order of "f

tions due to tunneling and correlations).

We calculated

A?ð?Þ

Renormalization Group (NRG) [14], generalizing the ap-

proachof Refs. [8,15] to T ? 0 by following Ref. [16]. The

inset of Fig. 2 shows a typical result: As temperature is

gradually reduced, an initially rather symmetric line shape

becomes highly asymmetric, dramatically increasing in

peak height as T ! 0. At T ¼ 0, the line shape displays

a threshold, vanishing for ? < 0 and diverging as ? tends to

0 from above. Figure 2 analyzes this divergence on a log-

log plot, for the case that T, which cuts off the divergence,

is smaller than all other relevant energy scales. Three

distinct functional forms emerge in the regimes of ‘‘large’’,

‘‘intermediate’’ or ‘‘small’’ detuning, labeled (for reasons

discussed below) FO, LM and SC, respectively, (given here

for Hf¼ SEAM):

j"f

TK& ? & j"f

T & ? & TK:

ASC

e ’ 1. (Here ? na

e¼

?? na

e?, and ? na

e?¼ hne?iais the thermal average of ne?

e?& ??. The Kondo temperature accociated with

ffiffiffiffiffiffiffiffiffiffiffiffiffi

represents the symmetric

?U=2

p

e??j"f

eð"f

eþUÞj=ð2U?Þ. If "f

e?¼ ?U=2,

e ¼ 1, then Hf

mm0?imjfhm0jey

?jmiij2?ð!L? Ef

m0 þ EimÞ:

(2)

mare exact eigenstates and energies of Ha,

G? Ei

G(Ea

Gis the ground state energy

e?þ "h ? ?(up to correc-

using theNumerical

e?j & ? & D:

AFO

?ð?Þ ¼4?

?ð?Þ ¼3?

?ð?Þ / T?1

?2?ð? ? j"f

e?jÞ;

(3a)

e?j: ALM

4?ln?2ð?=TKÞ;

Kð?=TKÞ???:

(3b)

(3c)

The remarkable series of crossovers found above are

symptomatic of three different regimes of charge and

spin dynamics. They can be understood analytically using

fixed-point perturbation theory (FPPT). To this end, note

that at T ¼ 0 the absorption line shape can be written as

Z1

where? Ha¼ Ha? Ea

probes the postquench dynamics, with initial state ey

A?ð?Þ ¼ 2Re

0

dteit?þihGjei?Hite?e?i?Hftey

?jGii;

(4)

Gand ?þ¼ ? þ i0þ. Thus it directly

?jGii,

PRL 106, 107402 (2011)

PHYSICALREVIEWLETTERS

week ending

11 MARCH 2011

107402-2

Page 3

of a photogenerated e-electron coupled to a FR. Evidently,

large, intermediate or small detuning, corresponding to

ever longer time scales after absorption, probes excitations

at successively smaller energy scales [see Fig. 1(c)], for

which ?Hfcan be represented by expansions H?

around the three well-known fixed points [14] of the AM:

the free orbital, local moment and strong-coupling fixed

points (r ¼ FO, LM, SC), characterized by charge fluctua-

tions, spin fluctuations and a screened spin singlet, respec-

tively, as illustrated in Fig. 1(d).

Large and intermediate detuning—perturbative re-

gime.—For large detuning, probing the time interval

t & 1=j"f

pears as a free, filled orbital perturbed by charge fluctua-

tions, described by [14] the fixed-point Hamiltonian

H?

H0

1=j"f

have frozen out, resulting in a stable local moment; how-

ever, virtual charge fluctuations still cause the local mo-

ment to undergo spin fluctuations, which are not yet

screened. This is described by [14] H?

and the RG-relevant perturbation H0

~ sj¼1

the e level and conduction band, respectively, (~ ? are Pauli

matrices), and Jð?Þ ¼ ln?1ð?=TKÞ is an effective, scale-

dependent dimensionless exchange constant.

rþ H0r

e?j immediately after absorption, the e level ap-

FO¼ Hcþ Hf

FO¼ Ht. Intermediate detuning probes the times

e?j & t & 1=TKfor which real charge fluctuations

eþ const and the relevant perturbation

LM¼ Hcþ const

LM¼Jð?Þ

?~ se? ~ sc. Here

2

P

??0jy

?~ ???0j?0 (for j ¼ e, c) are spin operators for

For r ¼ FO and LM, A?ð?Þ can be calculated using

perturbation theory in H0r. For T ¼ 0, note that

A?ð?Þ ¼ ?2ImihGje?

1

?þ?? Hfey

?jGii;

(5)

set? Hf! H?

H0r. One readily finds (see [13])

rþ H0rand expand the resolvent in powers of

Ar?ð?Þ ’ ?2

?2ImihGje?H0r

1

?þ? H?

r

H0rey

?jGii;

(6)

which reveals the relevant physics: Large detuning

(r ¼ FO) is described by the spectral function of the

operator Htey

stood as a two-step process consisting of a virtual excita-

tion of the QD resonance, followed by a tunneling event to

afinal free-electronstate above theFermi level.In contrast,

intermediate detuning (r ¼ LM) is described by the

spectral function of ~ sc? ~ seey

ctuations. Evaluating these spectral functions is ele-

mentary since H?

LMinvolve only free fermions.

For B ¼ 0 and j"f

and (3b) (see [13]), which quantitatively agree with the

NRG results of Fig. 2.—Though the latter was calculated

for Hf¼ SEAM, Eq. (3b) holds more generally as long as

Hfremains in the LM regime, with ? nf

depends on "f

e?, U and ? only through their influence on

TK, and hence is a universal function of ? and TK.

The FPPT strategy for calculating FO and LM line

shapes can readily be generalized to finite temperatures

[12], using the methods of Ref. [17] (Section III.A) for

finding the finite-T dynamic magnetic susceptibility [13].

For j?j ? j"f

? ð?Þ ¼3?

4

1 ? e??=T

where ?Korð?;TÞ ¼ ?T=ln2½maxðj?j;TÞ=TK? is the scale-

dependent Korringa relaxation rate [17]. It is smaller than

T by a large logarithmic factor, implying a narrower and

higher absorption peak than for thermal broadening.

Small detuning and Kondo-edge singularity—strong-

coupling regime.—As ? is lowered through the bottom of

the LM regime, Jð?Þ increases through unity into the

strong-coupling regime, and A?ð?Þ monotonically crosses

over to the SC regime. It was first studied for the present

model (for B ¼ 0Þ in Ref. [8], which found a power-law

line shape of the form (3c), characteristic of a Fermi edge

singularity, with an exponent ? that followed Hopfield’s

rule [18]. The power-law behavior reflects Anderson

orthogonality [19,20]: it arises because the final ground

state jGfi that is reached in the long-time limit is charac-

terized by a screened singlet. The singlet ground state

induces different phase shifts [as indicated in Fig. 1(d) by

wavy lines] for FR electrons than the unscreened initial

state just after photon absorption, ey

orthogonal to the latter. It is straightforward to generalize

?; the absorption process can thus be under-

?, i.e., it probes spin flu-

FOand H?

e?j ¼1

2U, we readily recover Eqs. (3a)

e ’ 1; then ALM

?ð?Þ

e?j and max½j?j;T? ? TK, we obtain

?=T

ALM

?Korð?;TÞ=?

?2þ ?2

Korð?;TÞ;

(7)

?jGii, and hence is

10

−4

10

0

10

4

10

8

10

−8

10

−4

10

0

10

4

SCLMFO

3π/4

ν ln2(ν/TK)

ν−ησ

4Γ

ν2

|εf

eσ|

T

TK

Γi = Γ

Γi = 0

ν / TK

Aσ(ν) / Aσ(TK)

050 100

10

−3

10

−2

10

−1

10

0

10

1

ν / TK

Aσ(ν) / [Aσ(TK)]T=0

T / TK

0.01

0.1

1

10

100

εe

εe

Γ = 0.03U

T = 3.3⋅10−10Γ

TK = 5.9⋅10−6Γ

U = 0.1D

B = 0

i = 0.75U

f = −0.5U

FIG. 2 (color online).

A?ð?Þ for T ? TK, B ¼ 0 and Hf¼ SEAM (for which

??¼1

intermediate and small detuning, labeled FO, LM, and SC,

respectively, according to the corresponding fixed points of the

Anderson model. Arrows indicate the crossover scales T, TKand

j"f

from Eq. (3)] and NRG (thick blue line for ?i? 0; thin blue line

for ?i¼ 0) agree well. Inset: A?ð?Þ for five temperatures in

semilog scale, obtained from FPPT for ?i¼ 0 [dashed lines,

from Eq. (7)] and NRG (solid lines).

Log-log plot of the absorption line shape

2), showing three distinct functional forms for high,

e?j. Fixed-point perturbation theory [FPPT, red dashed lines,

PRL 106, 107402 (2011)

PHYSICAL REVIEWLETTERS

week ending

11 MARCH 2011

107402-3

Page 4

the arguments of Refs. [8,18] to the case of B ? 0

(see [13]). One readily finds the generalized Hopfield rule

??¼ 1 ?X

?n0

units of e, that flows from the scattering site to infinity

when ey

is the local occupation difference between jGfi and jGii.

According to Eq. (8), ??can be tuned not only via gate

voltage but also via magnetic field, since both modify "a

and hence ?n0

e?0. This tunability can be exploited to study

universal aspects of Anderson orthogonality physics. In

particular, if the system is tuned such that ? nie¼ 0 and

? nf

2mf

ð? nf

ponents ??then are universal functions of geB=TK, with

simple limits for small and large fields [see Fig. 3(b)]:

??!1

jgeBj ? TK. Here the subscript ‘‘lower’’ or ‘‘upper’’ dis-

tinguishes whether the spin-? electron is photoexcited into

the lower or upper of the Zeeman-split pair (?geB < 0

or >0, respectively). The sign difference ?1 for ??arises

since these cases yield fully asymmetric changes in local

?0ð?n0

e?0Þ2;?n0

e?0 ¼ ???0 ? ?ne?0; (8)

e?0 is the displaced charge of electrons with spin ?0, in

?jGii is changed to jGfi, and ?ne?0 ¼ ? nf

e?0 ? ? ni

e?0

e?

e ¼ 1 at B ¼ 0, Eq. (8) can be expressed as ??¼1

e? ? 2ðmf

eþ? ? nf

2þ

2?

eÞ2, where the final magnetization mf

e?) is a universal function of geB=TK. The ex-

e ¼1

2for jgeBj ? TK, while ?lower=upper! ?1 for

charge: ?ne;lower! 1 while ?ne;upper! 0. As a result,

Andersonorthogonality[19]

(?n0

subsequently the e-level spin need not adjust at all. In

contrast, it is maximal (?n0

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

NSF under Grant No. 200021-121757. H.E.T. acknowl-

edges support from the Swiss NSF under Grant

No. PP00P2-123519/1. B.B. acknowledges support from

the Swiss NSF and NCCR Nanoscience (Basel). J.v.D.

acknowledgessupportfrom

SFB-TR12, De730/3-2, De730/4-1), the Cluster of

Excellence NIM and in part from NSF under Grant

No. PHY05-51164. A.I. acknowledges support from an

ERC Advanced Investigator Grant, and L.G. from NSF

Grant No. DMR-0906498.

iscompletelyabsent

e?0 ¼ 0) for photo-excitation into the lower level, since

e?0 ¼ 1) for photo-excitation

theDFG(SFB631,

[1] P. Nordlander et al., Phys. Rev. Lett. 83, 808 (1999).

[2] M. Plihal, D.C. Langreth, and P. Nordlander, Phys. Rev. B

71, 165321 (2005).

[3] F.B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801

(2005).

[4] D. Lobaskin and S. Kehrein, Phys. Rev. B 71, 193303

(2005).

[5] T.V. Shahbazyan, I.E. Perakis, and M.E. Raikh, Phys.

Rev. Lett. 84, 5896 (2000).

[6] K. Kikoin and Y. Avishai, Phys. Rev. B 62, 4647 (2000).

[7] A.O. Govorov, K. Karrai, and R.J. Warburton, Phys. Rev.

B 67, 241307 (2003).

[8] R.W. Helmes et al., Phys. Rev. B 72, 125301 (2005).

[9] J.M. Smith et al., Phys. Rev. Lett. 94, 197402 (2005).

[10] P.A. Dalgarno et al., Phys. Rev. Lett. 100, 176801 (2008).

[11] A. Ho ¨gele et al., Phys. Rev. Lett. 93, 217401 (2004).

[12] H. Tureci et al. (to be published).

[13] Seesupplementalmaterial

supplemental/10.1103/PhysRevLett.106.107402.

[14] H.R. Krishna-murthy, J.W. Wilkins, and K.G. Wilson,

Phys. Rev. B 21, 1003 (1980).

[15] T.A. Costi, Phys. Rev. B 55, 3003 (1997).

[16] A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99,

076402 (2007).

[17] M. Garst et al., Phys. Rev. B 72, 205125 (2005).

[18] J. Hopfield, Comments Solid State Phys. 2, 40 (1969).

[19] P.W. Anderson, Phys. Rev. Lett. 18, 1049 (1967).

[20] P. Nozie `res and C.T. De Dominicis, Phys. Rev. 178, 1097

(1969).

athttp://link.aps.org/

FIG. 3 (color online).

of the line shape for Hf¼ SEAM and T ¼ 0. (a) Depending on

whether the electron is photoexcited into the lower or upper of

the Zeeman-split e levels (?geB < 0 or >0, solid or dashed

lines, respectively), increasing jBj causes the near-threshold

divergence, A?ð?Þ / ????, to be either strengthened, or sup-

pressed via the appearance of a peak at ? ’ ?geB, respectively.

(The peak’s position is shown by the red line in the ?geB-?

plane.) (b) Universal dependence on geB=TKof the local mo-

ment mf

e (dash-dotted line), and the corresponding prediction of

Hopfield’s rule, Eq. (8), for the infrared exponents ?lower(solid

line) and ?upper(dashed line) for ? ¼ þ. Symbols: ?þvalues

extracted from the near-threshold ???þdivergence of Aþð?Þ.

Symbols and lines agree to within 1%.

Asymmetric magnetic-field dependence

PRL 106, 107402 (2011)

PHYSICALREVIEWLETTERS

week ending

11 MARCH 2011

107402-4

Page 5

EPAPS: Supplementary information

for “Many-Body Dynamics of Exciton Creation in a Quantum

Dot by Optical Absorption:

A Quantum Quench towards Kondo Correlations”

Hakan E. T¨ ureci1,2, M. Hanl3, M. Claassen2, A. Weichselbaum3, T. Hecht3,

B. Braunecker4, A. Govorov5, L. Glazman6, A. Imamoglu2& J. von Delft3

1Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA

2Institute for Quantum Electronics, ETH-Z¨ urich, CH-8093 Z¨ urich, Switzerland

3Arnold Sommerfeld Center for Theoretical Physics,

Ludwig-Maximilians-Universit¨ at M¨ unchen, D-80333 M¨ unchen, 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, CT 06520, USA

We provide below some intermediate steps for the derivation of the main equations of the

main text. For clarity, information contained in the main text is typeset in blue.

Spectral function

The origin of Eq. (2) for the absorption rate can be understood as follows [1]. We begin

with a Hamiltonian slightly more general than those of the main text, in that it includes the

hole degree of freedom: H = Heh+ Hc+ Ht, where Hcand Htare given in the main text,

and

Heh=

σ

?

(εeσneσ+ εhσnhσ) + Une↑ne↓−

?

σσ?

Uehneσnhσ?

(S1)

describes the QD, with e-level charging energy U(> 0), e-h Coulomb attraction Ueh(> 0),

neσ= e†

The QD-light interaction is described by HL∝ (e†

ωLexceeds a threshold frequency, say ωth. According to Fermi’s golden rule, treating HLas

harmonic perturbation, the absorption lineshape at temperature T and detuning ν = ωL−ωth

is proportional to

?

where the ρmare Boltzmann weights. Noting that the dynamics of the optically created hole

is trivial, [nhσ,H] = 0, we can write all initial states |m? as |m?i⊗ |0?hand all final states

|m?? in the form |m??f⊗ |¯ σ?h, thus arriving at Eq. (2) from the main text:

?

Here |m?a(a = i,f) are the many-body eigenstates of the effective initial and final Hamil-

tonians, Hi=h?0|H|0?hand Hf=h?¯ σ|H|¯ σ?h, that only include the QD and FR electronic

σeσ, nhσ= h†

σhσ. The hole energy εh¯ σ(> 0) is on the order of the band gap.

σh†

¯ σe−iωLt+h.c.). Absorption sets in once

Aσ(ν) =2π

mm?

ρm|?m?|e†

σh†

¯ σ|m?|2δ(ωL− Em? + Em),(S2)

Aσ(ν)=2π

mm?

ρi

m|f?m?|e†

σ|m?i|2δ(ωL− Ef

m? + Ei

m). (S3)

1

Page 6

degrees of freedom. They are given by Hi/f= Hi/f

?

To bring the spectral function into a resolvent form suitable for fixed point perturbation

theory (FPPT), we use Dirac’s identity

e

+ Hc+ Ht, with Hi/f

e

given by Eq. (1):

Ha

e=

σ

εa

eσneσ+ Une↑ne↓+ δafεh¯ σ

(a = i,f),withεa

eσ= εeσ− δafUeh.(S4)

δ(ωL− Ef

m? + Ei

m) = −1

πIm

?

1

ωL− Ef

m? + Eim+ i0+

?

,(S5)

to rewrite Eq. (S3) as follows:

Aσ(ν)=

−2Im

??

??

mm?

ρi

m i?m|eσ

1

ωL− Ef

m? + Eim+ i0+|m??f f?m?|e†

σ|m?i

?

.(S6)

=

−2Im

m

ρi

m i?m|eσ

1

ωL− Hf+ Eim+ i0+e†

m? by Hfand used?

σ|m?i

?

.(S7)

For the last step, we replaced Ef

m?|m??f f?m?| = 1.

Zero temperature:

At T = 0, the initial density matrix is |G?i i?G|. With the definitions introduced in the

text,¯Ha= Ha− Ea

the main text:

G, ν = ωL− Ef

G+ Ei

Gand ν+= ν + i0+, Eq. (S7) reduces to Eq. (5) of

Aσ(ν) = −2Imi?G|eσ

1

ν+−¯Hfe†

σ|G?i.(S8)

For the parameters considered in the main text, the ground state of Hican be approx-

imated by a free Fermi sea: |G?i??

1

ν+−¯Hf=

with the T-matrix given by

εkσ<εFc†

kσ|Vac?. Next comes the key step of FPPT:

rof the resolvent replace¯Hf→ H∗

r+ H?

rand do a perturbation expansion in H?

1

ν+− H∗

r

+

1

ν+− H∗

r

?Tr

1

ν+− H∗

r

(S9)

?Tr= H?

r+ H?

r

1

ν+− H∗

r

H?

r+ ···

(S10)

According to the main text, the free orbital (FO) regime is described by

H∗

FO

=Hc+ Hf

e− EFO=

?

kσ

εkσc†

kσckσ+

?

σ

εf

eσneσ+ Une↑ne↓− EFO, (S11a)

perturbed by

H?

FO

=Ht=

?

Γ/πρ

?

σ

(e†

σcσ+ h.c.) ,(S11b)

2

Page 7

and the local moment (LM) regime by

H∗

LM

=Hc− ELM=

?

kσ

εkσc†

kσckσ− ELM,(S12a)

perturbed by

H?

LM

=

J(ν)

ρ

? se·? sc,(S12b)

with e-level and conduction band spin operators given by ? sj=1

The subtracted constants EFOand ELMcorrespond to the subtraction of Ef

of¯Hf= Hf−Ef

state of H∗

|σ? ≡ e†

Now, when inserted into Eq. (S8), the first term in Eq. (S9) gives a δ(ν) not relevant

for the regime ν ? TKthat we are focusing on. The second term gives −2Im?σ|?Tr|σ?/ν2=

to lowest non-zero order, we obtain Eq. (6) of the main text, namely

2

?

σσ?j†

σ? τσσ?jσ? (for j = e,c).

Gin the definition

G→ H∗

r+H?

r(see the main text after Eq. (4)); they ensure that the ground

rhas eigenvalue 0. For both r = FO and LM, this ground state is given by

σ|Gi?.

Im[T(1)

r

+ T(2)

r

+ ···]/ν2. For both r = FO and LM, T(1)

r

= ?σ|H?

r|σ? = 0 (for B = 0). Thus,

Ar

σ(ν) = −2Im[T(2)

r

]/ν2,

T(2)

r

= ?σ|H?

r

1

ν+− H∗

r

H?

r|σ?.(S13)

T(2)

of uncorrelated free-fermion states, all of which are eigenstates of

regime, we obtain

r

can be evaluated straightforwardly using Wick’s theorem, since H?

r|σ? produces a sum

r. For the free orbital

1

ν+−H∗

T(2)

FO=

??

Γ

πρ

Γ/πρ

?2 ?

(1 − f(εk))

ν+− (εk− εfe)+

ks,k?s?

?σ|

?

c†

kses+ h.c.

?

1

ν+− H∗

f(εk)

FO

?

c†

k?s?es? + h.c.

?

|σ?

(S14)

=

?

k

?

ν+− (−εk+ εfe+ U)

?

,(S15)

where f(ε) = θ(−ε) stands for the Fermi function at zero temperature. Inserting this into

Eq. (S13) we obtain (for D ? |ν|):

σ(ν) =2Γ

ν2

AFO

?

θ(ν − |εf

eσ|) + θ(ν − (εf

eσ+ U)

?

.(S16)

For the case of Hf=SEAM (with εf

Eq. (3a):

e= −U/2) considered in the main text, this reduces to

AFO

σ(ν) =4Γ

ν2θ(ν − |εf

eσ|) .(S17)

The calculation for the local moment regime, TK? |ν| ? min[|εf

eσ|,εf

eσ+U], is analogous.

3

Page 8

Writing ? se·? sc=1

2(sσ

es¯ σ

?2

c+ s¯ σ

esσ

c) + sz

esz

c, we obtain

T(2)

LM=

?J(ν)

=1

4

ρ

?σ|? se·? sc

?2?

?2?

1

ν+− H∗

LM

? se·? sc|σ?

(S18a)

?J(ν)

?J(ν)

ρ

?Gi|sz

c

1

ν+− H∗

f(εk)(1 − f(εq))

ν+− εq+ εk

LM

sz

c|Gi? + ?Gi|s¯ σ

c

1

ν+− H∗

LM

sσ

c|Gi?

?

(S18b)

=3

8ρ

kq

.(S18c)

Inserting Eq. (S18c) into Eq. (S13), with J(ν) = ln−1[ν/TK], we recover Eq. (3b):

ALM

σ

=3π

4ν

1

ln2(ν/TK).(S19)

Nonzero temperature

For T ?= 0, the calculations are analogous, with only minor changes: The FPPT expansion

in powers of H?

is f(ε) = 1/[e?/T+ 1], and for the local moment regime the exchange coupling now takes

the form J(ν) = ln−1[max(|ν|,T)/TK]. We consider only the local moment regime, with

|ν| ? min[|εf

position lies so far above the Fermi surface (εi

eσ? Γ) that the initial density matrix contains

no correlations between e-level and Fermi reservoir, we again arrive at Eq. (S18c), which now

yields

ris performed on Eq. (S7) (instead of Eq. (S8)), the Fermi occupation function

eσ|,εf

eσ+ U] and max[|ν|,T] ? TK. Assuming (as before) that the initial level

ALM

σ

=3π

4ν

1

1 − e−ν/T

1

ln2[max(|ν|,T)/TK].(S20)

For large positive detuning, ν ? T, we recover Eq. (S19), while the line-shape at large

negative detuning, ν ? −T, is suppressed by an extra factor e−|ν|/T.

In the limit of small detuning, |ν| ? T, Eq. (S20) reduces to

3π

4ν2

ALM

σ (ν)=

T

ln2[T/Tk].(S21)

The apparent ν−2divergence indicates that in this limit, the expansion (S10) of TLMcan not

be truncated at second order, as done above, but must be summed to all orders. Instead of

doing this explicitly, one may use methods which were applied to treat the dynamic magnetic

susceptibility at finite temperature in Ref. [2]. These yield

ALM

σ (ν) =3π

4

ν/T

1 − e−ν/T

γKor(ν,T)/π

ν2+ γ2

Kor(ν,T),(S22)

(Eq. (7) of the main text), which contains a Lorentzian factor involving a frequency-dependent

Korringa relaxation time,

γKor(ν,T) =

πT

ln2[max(|ν|,T)/TK]. (S23)

4

Page 9

For |ν| ? γKor(ν,T), Eq. (S22) reproduces Eq. (S20). For |ν| ? T (but T ? TK), Eq. (S22)

reduces to a pure Lorentzian

ALM

σ (ν) =3π

4

γKor/π

ν2+ γ2

Kor

,(|ν| ? T)(S24)

of width γKor ? πT/ln2[T/TK] (which is ? T). This represents the properly regularized

version of Eq. (S21), to which it reduces for γKor? |ν| ? T.

Generalized Hopfield rule

The generalized Hopfield rule that holds when ν ? TKand arbitrary B is stated in Eq. (8)

of the main text can be found as follows, using arguments similar to those in Refs. [1, 3]:

First, write the T = 0 spectral function of Eq. (4) as

?∞

where |ψ0? = e†

e−i¯ Hft|ψ0? its time-evolved version.

namics is governed by the final ground state, |Gf?, characterized by a screened spin singlet.

Once the latter begins to dominate (for t ? 1/TK), the FR experiences the QD as a site of

pure potential scattering (no spin-flips), just as at t = 0+, but with changed strength. The

adjustment of the FR to this changed potential (via changes in the scattering phase shifts of

its single-particle wave-functions) causes an increasing Anderson orthogonality [4] between

|ψt? and |ψ0?: their overlap decays as ?ψ0|ψt? ∼ t−?

rule [7, 6], ∆n?

from the scattering site to infinity as e†

local occupation difference between |Gf? and |Gi?. Fourier-transforming ?ψ0|ψt? according to

Eq. (S25) yields the powerlaw-decay of Eq. (3c), with exponent

?

This is the generalized Hopfield rule, Eq. (8).

Aσ(ν) = 2Re

0

dteitν+?ψ0|ψt? ,(S25)

σ|Gi? is the state just after photon absorption (at t = 0+) and |ψt? =

In the t → ∞ limit (relevant for ν → 0), the dy-

σ?(∆n?

eσ?)2[5], where, by Friedel’s sum

eσ? = δσσ? −∆neσ? is the displaced charge (of spin σ?), in units of e, that flows

σ|Gi? evolves to |Gf?, with ∆neσ? = ¯ nf

eσ? − ¯ ni

eσ? the

ησ= 1 −

σ?

(∆n?

eσ?)2.(S26)

References

[1] R. W. Helmes et al., Phys. Rev. B 72, 125301, (2005).

[2] M. Garst, P. W¨ olfle, L. Borda, J. von Delft, L. Glazman, Phys. Rev. B 72, 205125

(2005).

[3] J. J. Hopfield, Comments Solid State Phys. 2, 40 (1969).

[4] P. W. Anderson, Phys. Rev. Lett. 18, 1049 (1967).

[5] P. Nozi` eres and C. T. De Dominicis, Phys. Rev. 178, 1097-1107 (1969).

[6] D.C. Langreth, Phys. Rev. 150, 516 (1966).

[7] J. Friedel, Can. J. Phys., 1190, 34 (1956).

5