Two-photon- photoluminescence excitation spectroscopy of single quantum-dots

Y. Benny,∗ Y. Kodriano, E. Poem, S. Khatsevitch, and D. GershoniThe Physics Department and the Solid State Institute,

Technion – Israel institute of technology, Haifa, 32000, Israel.

P. M. PetroffMaterials Department, University of California Santa Barbara, CA, 93106, USA

(Dated: July 25, 2011)

We present experimental and theoretical study of single semiconductor quantum dots excited bytwo non-degenerate, resonantly tuned variably polarized lasers. The first laser is tuned to excitonicresonances. Depending on its polarization it photogenerates a coherent single exciton state. Thesecond laser is tuned to biexciton resonances. By scanning the energy of the second laser for variouspolarizations of the two lasers, while monitoring the emission from the biexciton and exciton spectrallines, we map the biexciton photoluminescence excitation spectra. The resonances rich spectra of thesecond photon absorption are analyzed and fully understood in terms of a many carrier theoreticalmodel which takes into account the direct and exchange Coulomb interactions between the quantumconfined carriers.

I. INTRODUCTION

Semiconductor quantum dots (QDs) confine chargecarriers in three spatial directions. This confinement re-sults in discrete spectrum of energy levels and energet-ically sharp optical transitions between these levels1,2.These “atomic-like” features, together with their com-patibility with modern semiconductor based microelec-tronics and optoelectronics, make QDs promising build-ing block devices for future technologies involving single-photon emitters3 and quantum information processing(QIP)4–6. In particular, QDs are considered to be anexcellent interface between photons, whose polarizationstate may carry quantum information from one site to an-other, and confined carriers’ spins, whose states can becoherently manipulated locally7,8. For these reasons, it isvery important to study and to understand in detail light-matter interactions in such nanostructures. Deep under-standing of these interactions is required in order to im-plement protocols and schemes relevant to QIP9, in theseman-made, technology compatible, ‘artificial atoms’.

In this work we present a comprehensive study of sin-gle, neutral semiconductor QDs subject to excitation bytwo different variably polarized resonant lasers. Thefirst laser is tuned to an excitonic resonance and gen-erates a coherent single exciton state, while the secondlaser is scanned through biexcitonic resonances. Depend-ing on the particular resonance and the direction of thelight polarization relative to the direction of the excitonspin, it photogenerates a biexciton8. The absorption isthen monitored through the emission intensities of vari-ous biexcitonic and excitonic spectral lines.

The manuscript is organized as follows: Section II isdevoted to set the theoretical background which is re-quired to analyze the experimental data. In section IIIwe describe the experimental methods and the measure-ments that we performed. In section IV we present theexperimental results and analyze the data, using the the-ory outlined in section II. The last part of this section

provides a short summary of the results.

II. THEORY

We use a simple one-band model to describe the single-particle wavefunctions of electrons in the conductionband and heavy-holes in the valence band of a singleQD. Since in these InAs/GaAs lattice mismatch straininduced self-assembled QDs, the light-holes band is en-ergetically separated from the heavy-hole band by thestrain and the quantum size effect, light holes are notconsidered in our model. The lateral extent of these QDsis typically about an order of magnitude larger than theirextent along the growth direction. Therefore, for simplic-ity, our model considers only the two lateral directions.The exact composition and strain distribution in theseQDs are not accurately known, therefore we use a verysimple, two dimensional parabolic potential model to de-scribe the QD influence on the carriers that it confines.This simple model is general enough to describe the C2υ

symmetry of these QDs10, and it contains four parame-ters (see below) which permit its adjustment to the ex-perimental observations2. Two separated infinite ellipticparabolic potentials are thus used, one for the electronsand one for the heavy-holes11. The resulting envelopewavefunctions or orbitals of the carriers, are thereforedescribed analytically by the 2D harmonic solutions:

ψpnx,ny(x, y) =

Hnx( xlxp)Hny ( y

lyp)√

2(nx+ny)nx!ny!πlxp lyp

e− 1

2 [( xlxp

)2+( y

lyp

)2]

(1)where p=e(h) stands for electron (heavy-hole) and Hnx(y)

are the Hermite polynomials of order nx(y).

lx(y)p =

√~

M∗⊥,pωx(y)p

(2)

arX

iv:1

105.

4861

v2 [

quan

t-ph

] 2

2 Ju

l 201

1

2

is a characteristic length, which describes the extent ofthe parabolic potential along the x(y) direction and Eq. 2relates this length to the charge carrier’s in-plane effectivemass M∗⊥,p and the harmonic potential inter-level energy

separation ~ωx(y)p . We fit the four characteristic lengthes

lx(y)e(h) to best describe the observed spectral lines.

Equipped with the single carrier’s eigenenergies andenvelope wavefunctions we proceed by calculating themany-carrier energies and states using configuration in-teraction (CI) model2,12,13. A detailed description of themodel, which takes into account the direct and exchangeCoulomb interactions between any pair of carriers in theQD, is presented elsewhere2.

Previous studies dealt mainly with optical transitionsto ground excitonic states as a tool to describe polar-ization sensitive photoluminescence (PL) experiments1,2.Here, motivated by our progress in performing resonantPL excitation (PLE) spectroscopy, using one and twolaser sources, we use the same model to consider transi-tions from various other levels. First, we consider tran-sitions which result from the resonant absorption of onephoton. Then, we add a second photon, resonantly tunedto the resonances of the optically excited QD. Since, as weexplain below, the situation in this case is much richerthan for PL only, we have to modify the notation fordescribing the QD many-carrier states. We use the fol-lowing notation; A single carrier state is described byits envelope wavefunction or orbital mode (O=1,2,...,6),where the number represents the energy order of the levelso that O=1 represents the ground state. O is followedby the type of carrier, electron (e) or heavy-hole (h) anda superscript which describes the occupation of the sin-gle carrier state. The superscript can be either 1 (openshell) or 2 (closed shell), subject to the Pauli exclusionprinciple (non occupied states are not described). All theoccupied states of carriers of same type are then markedby subscripts which describe the mutual spin configura-tion (σ) of these states.

A. Characterization of excitonic resonances

The ground exciton state [X01,1 ≡ (1e1)(1h1)] is a two-

carrier state, formed mainly by one electron and oneheavy-hole in their respective ground states. The ex-change interaction between the electron and the heavy-hole 2,14,15 is described, using the method of invariants16,by the following spin Hamiltonian14:

HX01,1

=∑

i=x,y,z

(a1,1i SiJi + b1,1i SiJ

3i ) (3)

where Si (Ji) denotes the ith cartesian component of the

electron (hole) spin and a1,1i and b1,1i are spin-spin cou-

pling constants. The total spin projection on the ith di-rection is thereby given by Fi = Si + Ji. Here as well,the interaction with light-holes is neglected so that in Ji

only heavy-hole spins are considered14. In matrix form,for the basis |Sz〉 ⊗ |Jz〉:

| − 1/2, 3/2〉 = ↓1⇑1 , Fz = 1|1/2,−3/2〉 = ↑1⇓1 , Fz = −1|1/2, 3/2〉 = ↑1⇑1 , Fz = 2| − 1/2,−3/2〉 = ↓1⇓1 , Fz = −2

(4)

where ↑j (⇓j) indicates spin-up (-down) electron (heavy-hole) in the orbital j, the Hamiltonian is given by

HX01,1

=1

2

1 −1 2 −2

1 ∆1,10 ∆1,1

1 0 0

−1 ∆1,11 ∆1,1

0 0 0

2 0 0 −∆1,10 ∆1,1

2

−2 0 0 ∆1,12 −∆1,1

0

. (5)

where ∆1,10 = 3(a1,1

z + 2.25b1,1z ), ∆1,11 = 1.5(b1,1x −

b1,1y ) and ∆1,12 = 1.5(b1,1x + b1,1y )14. ∆1,1

0,1,2 are threeconstants, which fully characterize the exchange in-teraction between the carriers in the ground states16.It is clearly seen that in a C2υ symmetry the ex-change interaction completely removes the degeneracybetween the four possible various combinations of theelectron-hole pair spin states10. The eigen-energies andthe eigenstates are schematically described in Fig. 1.The lowest energy state is the symmetric dark exci-ton state which in our notation is described as follows:|X0

1,1,D+〉 ≡ 1√

2[(1e1)1/2(1h1)3/2 + (1e1)−1/2(1h1)−3/2].

∆1,12 above it lies the anti-symmetric dark exciton state:|X0

1,1,D−〉 ≡ 1√

2[(1e1)1/2(1h1)3/2 − (1e1)−1/2(1h1)−3/2].

∆1,12 , is known to be quite small and believed to be or-

bit independent16. It was recently measured from thetemporal period of the coherent precession of the darkexciton spin to be 1.4 µeV17. The bright exciton eigen-states in which the electron and heavy-hole spins areanti-aligned lie ∆1,1

0 above the dark exciton states. The

isotropic e-h exchange, ∆1,10 , was previously found to be

about 300 µeV, by magneto-optical measurements18.The symmetric and antisymmetric bright exci-

ton states |X01,1,B±〉 ≡ 1√

2[(1e1)−1/2(1h1)3/2 ±

(1e1)1/2(1h1)−3/2], are split by the anisotropic e-h

exchange, ∆1,11 . The magnitude and sign of ∆1,1

1 = −34µeV, is directly measured by polarization sensitive PLspectroscopy. Since ∆1,1

1 is negative, the antisymmetricstate (|X0

1,1,B−〉) is higher in energy than the symmetric

one (|X01,1,B+〉)2,15.

Conservation of angular momentum dictates that whena ↓⇑ (↑⇓) e-h pair radiatively recombines, a right- (left-)hand circularly polarized photon is emitted. It followsthat radiative recombination from the symmetric (anti-symmetric) bright exciton state is linearly polarized H(V) along the major (minor) in-plane axis of the QD16,19.We note that the symmetric and antisymmetric dark andbright exciton states are by no means unique to the firstsingle carrier spatial levels (Oe=Oh=1). In fact, similar

3

1 11e 1h

H V

1 1B

1 1B

2 2D

2 2D

1,1

2

1,1

0

1,1

1

0

FIG. 1: Schematic description of the energy levels of the(1e1)(1h1) exciton and the allowed optical transitions from itsstates to the vacuum. The major parts of the spin wavefunc-tions are described to the right of each level. ↑(⇓) representsan electron (hole) with spin up (down) and a blue arrow rep-resents a carrier in its first level. A blue (red) vertical arrowrepresents linearly polarized optical transition along the ma-jor (H) [minor (V)] axis of the QD. The bracketed numbersstand for the total spin projection of the carriers along theQD growth direction.

bright and dark excitonic states are formed for any com-bination of Oe and Oh single carrier states. In general,

∆Oe,Oh0,1,2 depend on the orbital mode of the carriers2,15.When the laser is resonantly tuned into one of the

excited bright exciton states and its light is polarizedcorrectly, the light is absorbed and a single electron-hole pair is photogenerated. For example, let us con-sider the states: |X0

1,2,B±〉 ≡ 1/√

2[(1e1)−1/2(2h1)3/2 ±(1e1)1/2(2h1)−3/2]. These states are similar to the ground

bright states |X01,1,B±〉, albeit, here the hole is in its sec-

ond orbital mode (Oh = 2). Electron and hole pair willbe photogenerated in these levels, and then the hole willrapidly relax non-radiatively (within ∼20 psec20,21), byemitting phonons, to the ground level (Oe = Oh = 1).This relaxation is faster than the radiative recombinationrate (∼1 nsec).

Experimental identification of single photon or singleexciton transitions is conventionally done by polarizationsensitive PL and PLE spectroscopies. In PL, a QD isoptically excited. The excitation gives rise to light emis-sion from various long-lived states which do not relaxto lower energy states within their radiative lifetime. Po-larization and intensity sensitive PL spectroscopies are inparticular useful for these identifications. For example,the bright exciton typically gives rise to PL doublet com-posed of two cross-linearly polarized components. Thesecomponents are due to recombination from each of itsnon-degenerate eigenstates.

The second orbital wavefunction (pH) has one nodealong the major symmetry axis of the QD. As a result

∆1,21 is positive and therefore the symmetric eigenstate|X0

1,2,B+〉 is higher in energy than the anti-symmetric

eigenstate15. It thus follows that the V linearly polar-ized transition to this exciton is lower in energy thanthe H polarized one. When the |X0

1,2,B−〉 (|X01,2,B+〉)

state is excited, the hole rapidly relaxes non-radiativelyto its ground state, releasing its energy into acousticalphonons. Since phonons do not interact with the car-riers’ spin8,20,21, the spin wavefunction’s symmetry re-mains the same, and the recombination occurs from the|X0

1,1,B−〉 (|X01,1,B+〉) state. Thus, polarization sensitive

PLE spectroscopy can be efficiently used to identify andsort various excitonic resonances.

B. Characterization of biexcitonic resonances

The ground biexciton state is formed mainly by twospin paired electrons and two spin paired heavy-holes intheir respective ground states. In our notation this stateis described as follows: |XX0

1,1,1,1〉 ≡ (1e2)(1h2). We notethat spin paired carriers can only form an antisymmetricspin singlet state and therefore the σ subscripts in thiscase is redundant and it is omitted from the pair’s statedescription. For unpaired carriers, however, the situationis different. Two unpaired carriers can form either a oneantisymmetric singlet state, or three symmetric tripletstates. Therefore, in this case we do assign σ subscriptsfor describing the unpaired carriers’ wavefunctions. Fortwo unpaired carriers the σ can either be S, to indicate asinglet state, or Tm, to indicate a triplet state. Here m isthe total spin projection, 0 or ±1 (0 or ±3) of the pair ofelectrons (heavy-holes), along the QD growth direction.A full description of a state with two unpaired electronsand two unpaired holes has therefore the form

|XX0Oe1,Oe2,Oh1,Oh2,σe,σh〉 ≡ (Oe1e1Oe2e1)σe(Oh1h1Oh2h1)σh .

(6)

We note that for a given set of 4 unpaired spatial co-ordinates, 16 different states with different spin config-urations are possible. These are naturally divided intothe following 4 subgroups; One state, similar in natureto the ground biexciton state, in which the two electronsform a singlet (e-singlet) and the two heavy-holes alsoform a singlet (h-singlet). Three states in which the elec-trons form an e-singlet and the holes triplet (h-triplet),three in which the holes form a h-singlet and the elec-trons e-triplet and nine in which both the electrons andholes form triplets (e-triplet-h-triplet). These four sub-groups have different energies due to the exchange in-teractions between carriers of same charge. The lowestenergy level includes the 9 e-triplet-h-triplet states, thetwo intermediate groups include the 6 e-triplet-h-singletand e-singlet-h-triplet states and the highest energy oneincludes only a single e-singlet-h-singlet state.

For simplicity, we begin by characterizing optical tran-sitions in which at least one type of carriers forms asinglet. In Fig. 2 we schematically describe the en-ergy levels and the spin wavefunctions of the config-

4

1 11e 1h

H V

1 1 1 1

S S1e 2e 1h 2h 0

1 1 1 1

S T1e 2e 1h 2h 0

3

3

0 1 1 1 1

T S1e 2e 1h 2h

1

1

B

B

D

D

0

FIG. 2: Schematic description of the energy levels and spinwavefunctions of the configuration (1e12e1)σe(1h12h1)σh , for(σe, σh) = (T, S), (S, T ), (S, S). The notations are as in Fig.1. Calculated two-lasers PLE spectra are presented by dash(solid) lines for cross-(co-)linearly polarized exciton and biex-citon transitions. Blue (red) lines represent H (V) polarizedbiexcitonic transitions.

uration (1e12e1)σe(1h12h1)σh

, for the cases (σe, σh) =(T, S), (S, T ), or (S, S). The major parts of the spinwavefunctions are described to the right of each level,where ↑(⇓) represents an electron (hole) with spin up(down) and a blue (red) arrow represents a carrier in itsfirst (second) level. The bracketed numbers stand for thetotal spin of the configuration. H (V) polarized opticaltransitions are represented by blue (red) vertical arrows.

We note that singlet-triplet and singlet-singlet biexci-tonic resonances may, in principle, occur also when thetwo carriers that form the singlet reside in the same singlecarrier orbital mode (the two carriers are paired). Naiveintuitive considerations, which are based on single-bandmodels, predict that these transitions should be weak,due to the small spatial overlap between the electronand hole orbital modes which belong to different O num-bers22. Transitions which involve orbital modes of differ-ent symmetries should be forbidden in particular, sincethen, their dipole moment vanishes. Nevertheless, theseoptically forbidden transitions were previously observedin PLE spectroscopy of quantum wells23 and QDs24,25.

In Fig. 3 we present an example for the case in whichthe electrons are paired in their ground single carrier levelwhile the holes are not. One hole is in the Oh=1 s-likeorbital and the other is in the Oh=4, dHH-like orbital.Since the electrons here are paired, they form a singlet,thus their total spin vanishes. Therefore, the e-h ex-change interaction is not expected to remove the degen-eracy between the holes triplet states. We find, however,that this degeneracy is slightly removed due to many-carrier mixing effects. Previous works attributed this ef-fect to anisotropic h-h exchange interactions1,24, which

our model does not contain.Turning to Fig. 2 again, we note that two absorption

resonances are expected from the bright exciton statesinto an e-singlet-h-singlet state. These two transitionsform a typical cross-linearly polarized doublet, resem-bling the optical transitions from the vacuum to thebright exciton states (see Fig. 1). Four transitions areexcepted from the exciton states into the three e-singlet-h-triplet states and four similar ones into the e-triplet-h-singlet states. Two of these four are cross-linearly po-larized transitions from the bright exciton states into thestate in which the two holes (or electrons) spins are anti-parallel (T0), and two cross-linearly polarized transitionsfrom each one of the dark exciton states into the corre-sponding symmetric and anti-symmetric combinations ofthe parallel hole (or electron) spin states T±3 (T±1) of thebiexciton. By inspecting the wavefunctions of the initialand final state of each optical transition one immediatelysees that the oscillator strength of the optical transitionsfrom the bright exciton states is exactly half that of thetransitions from the dark exciton states. Moreover, sinceboth the dark exciton and corresponding biexciton pairstates are nearly degenerate, these two transitions formone unpolarized spectral line. Therefore, the total inten-sity of this line is four times larger than that of the othertwo transitions. The calculated spectra are presented inFig. 2 and Fig. 3. In obtaining these spectra, the calcu-lated transition energies are convoluted with normalizedGaussians of 50 µeV width, to take into account the finitelifetime of the spin blockaded biexcitons. Transitions inwhich the exciton and biexciton photons are co-(cross-)linearly polarized are presented by solid- (dash-) lines,where blue- (red-) lines represent H- (V-) polarized biex-citon photons.

We now turn to discuss the optical transitions intothe e-triplet-h-triplet states. The electron-hole (e-h)exchange interactions, which in our QDs are typicallyabout an order of magnitude smaller than same-carrierexchange interactions, remove the degeneracy betweenthe states within this subgroup. We actually calculatethe eigenenergies and eigenstates accurately using a CImodel2,13. However, for a more intuitive discussion onecan build an effective biexciton e-h exchange Hamilto-nian for the subspace of (1e12e1)Te

(1h12h1)Th, using the

single exciton effective e-h exchange Hamiltonian of Eq.3, such that an element is defined as follows26:

f 〈J2z , J

1z , S

2z , S

1z |HXX0

1,2,1,2|S1z , S

2z , J

1z , J

2z 〉i

= f 〈J1z , S

1z |HX0

1,1|S1z , J

1z 〉i

+f 〈J2z , S

1z |HX0

1,2|S1z , J

2z 〉i

+f 〈J1z , S

2z |HX0

2,1|S2z , J

1z 〉i

+f 〈J2z , S

2z |HX0

2,2|S2z , J

2z 〉i,

(7)

where HX0i,j

is the single e-h pair spin Hamiltonian for

electron and hole in the orbital modes i and j respec-tively, and the subscript i (f) denotes the initial (final)spin state. We change to a new basis in which the same-carrier exchange states are diagonal. The weak e-h ex-

5

2 1 1

T1e 1h 4h

1 11e 1h

H V

2 1 1

S1e 1h 4h

D

D

B

B

0

0

3

3

0

FIG. 3: Schematic description of the energy levels and spinwavefunctions of the configuration (1e2)(1h14h1). The majorparts of the spin wavefunctions are described to the right ofeach level. The notations are as in Fig. 1, where ↑(⇓) repre-sents an electron (hole) with spin up (down) and a blue (red)arrow represents a carrier in its first (excited) level. Calcu-lated two-lasers PLE spectra are presented by dash (solid)lines for cross- (co-) linearly polarized excitonic and biexci-tonic transitions. Blue (red) lines represent H (V) polarizedbiexcitonic transitions.

change interactions are then treated as perturbations onthese states. A similar approach was previously used fordescribing charged excitons (trions)27,28. Taking only thesubspace of the e-triplet-h-triplet spin states, |σe〉⊗ |σh〉:

| − 1, 3〉 = ↓1↓2⇑1⇑2 Fz = 2

| − 1, 0〉 = ↓1↓2 (⇓1⇑2+⇑1⇓2)√2

Fz = −1

| − 1,−3〉 = ↓1↓2⇓1⇓2 Fz = −4

|0, 3〉 = (↑1↓2+↓1↑2)√2

⇑1⇑2 Fz = 3

|0, 0〉 = (↑1↓2+↓1↑2)(⇓1⇑2+⇑1⇓2)2 Fz = 0

|0,−3〉 = (↑1↓2+↓1↑2)√2

⇓1⇓2 Fz = −3

|1, 3〉 = ↑1↑2⇑1⇑2 Fz = 4

|1, 0〉 = ↑1↑2 (⇓1⇑2+⇑1⇓2)√2

Fz = 1

|1,−3〉 = ↑1↑2⇓1⇓2 Fz = −2

(8)

we obtain the following matrix (neglecting many-bodymixing corrections)

HXX0TT

=1

2

2 −1 −4 3 0 −3 4 1 −2

2 ∆0 0 0 0 ∆1 0 0 0 0

−1 0 0 0 ∆2 0 ∆1 0 0 0

−4 0 0 −∆0 0 ∆2 0 0 0 0

3 0 ∆2 0 0 0 0 0 ∆1 0

0 ∆1 0 ∆2 0 0 0 ∆2 0 ∆1

−3 0 ∆1 0 0 0 0 0 ∆2 0

4 0 0 0 0 ∆2 0 −∆0 0 0

1 0 0 0 ∆1 0 ∆2 0 0 0

−2 0 0 0 0 ∆1 0 0 0 ∆0

(9)

where ∆0 =∆

Oe1,Oh10 +∆

Oe1,Oh20 +∆

Oe2,Oh10 +∆

Oe2,Oh20

4 and

1 11e 1h

H V

4 4

4 4

0

1 1 3 3

1 1 3 3

1 1 3 3

1 1 3 3

2 2

2 2

B

B

D

D

1

0

2

1 1 1 1

T T1e 2e 1h 2h

0

FIG. 4: Schematic description of the energy levels of the(1e12e1)T(1h12h1)T biexciton and their optical transitions.The major parts of the spin wavefunctions are described tothe right of each level. A blue (red) vertical arrow representslinearly polarized optical transition along the major (H) [mi-nor (V)] axis of the QD.

∆1,2 =∆

Oe1,Oh11,2 +∆

Oe1,Oh21,2 +∆

Oe2,Oh11,2 +∆

Oe2,Oh21,2

8 . In TableI we present the nine eigenenergies and eigenfunctionsof the effective Hamiltonian HXX0

TT. These eigenener-

gies and spin wavefunctions are also presented in Fig. 4.The allowed optical transitions between the ground exci-ton states to these biexciton states, together with theirpolarization selection rules, are presented as well. Wenote that since a photon can carry angular momentumof ±1 only, biexciton resonances of total spin 3 and 1can be reached optically only from the ground dark ex-citon states. Similarly, biexciton resonances of total spin0 and 2 can be reached optically from the bright excitonstates only. Biexciton states with total spin 4 cannot bereached optically.

C. Many-carrier mixing effects

The above discussion assumes that, to first order,the interactions between the carriers are much smallerin comparison with the single-carrier level separations.Therefore, we safely ignore contributions to the biexcitoneigenstates which results from mixing with other config-urations outside the subspace considered. Our model,however, does include these contributions2,13 and as weshow below, in some cases, specifically when otherwisethe transitions are forbidden, mixing with other configu-rations are directly observed in the experimental data.

Our model includes six orbital modes for each car-rier. The many-carrier eigenstates are obtained by thediagonalization of the many-body Hamiltonian, which isconstructed from all the possible configurations of theconfined carriers in a system of six bound levels. Thus,

6

TABLE I: Calculated eigen-energies and their respectivespin configurations for the e-triplet-h-triplet states. The basestates are expressed by their total spin projection |Fz〉.

Energy Configuration

−∆0 |4〉 − | − 4〉−(∆1 + ∆2)/2 (|1〉+ | − 1〉)− (|3〉+ | − 3〉)−(∆1 − ∆2)/2 (|1〉 − | − 1〉)− (|3〉 − | − 3〉)(∆1 − ∆2)/2 (|1〉 − | − 1〉) + (|3〉 − | − 3〉)(∆1 + ∆2)/2 (|1〉+ | − 1〉) + (|3〉+ | − 3〉)∆0 −(|2〉+ | − 2〉)Ri

a, i = 1, 2, 3

R1 → [−∆0] → [|4〉+ | − 4〉]R2 → [∆0 −

√∆2

0 + 2∆21/2] → [|0〉]

R3 → [∆0 +

√∆2

0 + 2∆21/2] → [|2〉+ | − 2〉]

aRi is the ith root of the equation 2R3i − (2∆2

0 + 2∆21 + ∆2

2)Ri +

∆0(∆22−∆2

1) = 0. The expressions in square brackets are obtained

for the case ∆2 � ∆1 � ∆0. Ri is given to order (∆1/∆0)2.

a many-carrier eigenstate always contains contributionsfrom different combinations of single carriers’ orbitalmodes.

An example for transitions in which these contribu-tions become important are the optical transitions fromthe (1e12e1)σe

(1h2) biexciton to the first excited exci-ton state where the leading contribution comes from theconfiguration (1e1)(2h1). Our model calculation resultedin optical transitions between these states, as indeedwe found experimentally (see below). In Fig. 5 we de-scribe the energy level structure of the (1e2)(1h12h1)T

and (1e12e1)T(1h2) biexcitons. Since these excited biex-citon states are spin blockaded for thermal relaxation,they decay radiatively by recombination of a ground statee-h pair. The optical transitions originated from their de-cay are also described in Fig. 5. If one neglects mixing,it follows that the (1e2)(1h12h1)T biexcitons decay intoexcited (1e1)(2h1) excitons and the (1e12e1)T(1h2) biex-citons decay into (2e1)(1h1) excitons. These two excitedexcitons are, however, highly mixed due to the Coulombinteraction between the electron and the hole. Roughlyspeaking, our model shows that each biexciton group de-cays into both excited exciton states, resulting in foursets of three spectral lines. Each group of three spec-tral lines resembles the sets described in Fig. 2 and Fig.3. It contains two lower energy cross-linearly polarizedlines and one, four fold stronger, unpolarized line. Thecalculated PL spectra which result from these transitionsare presented in Fig. 5. The spectral width of the lineswhich results from emission to the lower energy excitedexciton states [mainly (1e1)(2h1)] are obtained by convo-luting the calculated transitions with Gaussians of ∼ 50µeV, accounting for the finite lifetime of the excited holestates. Similarly, the spectral width of the lines whichresult from emission to the higher energy excited excitonstates [mainly (2e1)(1h1)], should have been obtained byconvolution with Gaussians of ∼ 1 meV (not visible, be-

0.7 0.2

0.7 0.2

0.7 0.2

0.7 0.2

0

3

3

0

1

1

+

1 10.7 1e 2h

2 1 11e 1h 2h

1 1 21e 2e 1h

0.7 0.2

0.7 0.2

0.7 0.2

0.7 0.2

+

1 10.7 2e 1h

FIG. 5: Schematic description of the energy levels of(1e2)(1h12h1)T and (1e12e1)T(1h2) biexcitons and the excitedexciton states (1e1)(2h1) and (2e1)(1h1). The notations usedhere are the same as in Fig. 3. The curves describe calcu-lated PL spectra of the transitions to the (1e1)(2h1) excitedexcitons. The calculated transitions to the (2e1)(1h1) excitedexcitons are not visible due to the short lifetime of the finalstate.

cause the convolution results in a nearly uniform, unpo-larized background on the relevant energy scale), due tothe much shorter lifetime of the excited electron states.The difference between the two cases is due to the dif-ference between the relaxation rates of the hole and theelectron. While a hole in the second orbital state re-laxes non-radiatively to the first orbital within ∼ 20 psecby emitting acoustical phonons20,21, the electron does sowithin less than 1 psec, by coupling to optical phonons29.The decay of the electrons is so rapid because the energyof optical phonons in the wetting layer, nearly resonatewith the energy separation between the two electronicorbitals (∼29 meV8).

III. THE EXPERIMENTAL SETUP

The sample used in this work was grown by molecular-beam epitaxy on a (001) oriented GaAs substrate. Onelayer of strain-induced InxGa1−xAs QDs was deposited inthe center of a one wavelength microcavity formed by twounequal stacks of alternating quarter wavelength layersof AlAs and GaAs, respectively. The height and compo-sition of the QDs were controlled by partially coveringthe InAs QDs with a 3 nm layer of GaAs and subsequentgrowth interruption. To improve photon collection ef-ficiency, the microcavity was designed to have a cavitymode, which matches the QD emission due to groundstate e-h pair recombinations. During the growth of theQD layer the sample was not rotated, resulting in a gra-dient in the density of the formed QDs. The estimatedQD density in the sample areas that were measured is

7

108 cm−2; however, the density of QDs that emit in res-onance with the microcavity mode is more than two or-ders of magnitude lower30. Thus, single QDs separatedapart by few tens of micrometers were easily located byscanning the sample surface during PL measurements.Strong anti-bunching in intensity auto-correlation mea-surements were then used to verify that the isolated QDsare single ones and that they form single photon sources.

The experimental setup that we used for the opticalmeasurements is described in Fig. 6. The sample wasplaced inside a sealed metal tube immersed in liquidHelium, maintaining temperature of 4.2K. A ×60 mi-croscope objective with numerical aperture of 0.85 wasplaced above the sample and used to focus the lightbeams on the sample surface and to collect the emittedPL. The majority of this work was performed with cw ex-citation. We used one tunable Ti:sapphire laser to scanthe energy. A second Ti:sapphire laser was used for thetwo-photon excitation experiments. We performed alsomeasurements with pulse excitation. In these measure-ments we used two dye lasers, synchronously pumped bythe same frequency-doubled Nd:YVO4 (Spectra Physics-VanguardTM) laser for generating the resonantly tunedoptical pulses, as described in the figure. The repetitionrate of the setup was 76 MHz, corresponding to a pulseseparation of about 13 nsec. The duration of the laserpulses were about 6 psec and their spectral widths about200 µeV. The delay between the pulses was controlledby a retroreflector on a translation stage.

The lasers emission energy could have been contin-uously tuned using coordinated rotations of two platebirefringent filters and a thin etalon. The polarizationsof the pulses were independently adjusted using a po-larized beam splitter (PBS) and two pairs of computercontrolled liquid crystal variable retarders (LCVRs). Thepolarization of the emitted PL was analyzed by the sameLCVRs and PBS. The PL was spectrally analyzed by 1-meter monochromator and detected by either a siliconavalanche photodetector or by a cooled CCD camera.

In polarized PLE spectroscopy, one monitors the po-larized emission from an identified PL line while varyingthe energy and polarization of the exciting light source.From the variations in the intensity of the emitted PL,one can readily identify many carrier resonances in whichthe light is preferentially absorbed. Increased absorption,which results in increased emission intensity of a specificPL line, and its polarization sensitivity are then usedto unambiguously identify the many-carrier state whichforms a specific absorption resonance31,32.

IV. RESULTS

In Fig. 7 we present polarization sensitive PL spectrumof a single QD in resonance with the microcavity mode.The PL was obtained by exciting the QD with a 501 nmcw Ar+ laser. We found that at this excitation energythe QD is on average neutral21. The excitation intensity

PBS

BS

DL

LCVRLCVR

LCVRLCVR

c∆/2

BC

BS

6psec

6psec

PBS

Monochromator

Monochromator

VanguardTM Filter

VBS

12psec

APD

APD

CCD

LCVR

Sample

4K

DL

13nsec

FIG. 6: Schematic description of the experimental setup. Thetwo dye laser pulses can be delayed one with respect to theother by a retroreflector mounted on a computer-controlledtranslation stage. For the cw experiments Ti:sapphire laserswere used. (P)BS stands for (polarizing) beam splitter; VBSfor a variable beam splitter; BC for beam combiner; andLCVR for a liquid crystal variable retarder.

was roughly 1W/cm2 aiming at obtaining equal emissionintensity from the exciton and biexciton lines33. Thespectral neutral excitonic and biexcitonic lines, which arerelevant for this study are identified above the spectralfeatures in the figure. We note that in addition to theground bright exciton (|X0

1,1,B±〉 to vacuum) and ground

biexciton [(1e2)(1h2) to |X01,1,B±〉] lines, three additional

biexcitonic lines are observed. These lines are due torecombination from the metastable biexciton configura-tions (1e2)(1h12h1)T to the excited (1e1)(2h1) excitoneigenstates. Two cross-linearly polarized lines are dueto the transitions from the (1e2)(1h12h1)T0

biexcitonconfiguration to the excited bright exciton eigenstates,|X0

1,2,B±〉 and one, unpolarized, is due to the (almost) de-

generate transitions from the (1e2)(1h12h1)T±3biexciton

configurations to the excited dark exciton configurations,|X0

1,2,D±〉. The observed emission intensity ratios of 1:1:4

is straightforward to understand21, as discussed above.

In Fig. 8 we present PLE spectra of the neutral ex-citonic and biexcitonic PL lines. Each panel in the fig-ure presents PLE spectrum of the PL spectral positionmarked by the vertical arrow on the expanded scale PLspectrum to the left of the panel. This set of mea-surements combined with additional measurements (dis-cussed below) and the intuition that we gained from themodel outlined above, allow us to resolve and identifymost of the observed one- and two- photon resonances.The identified optical transitions are marked above theobserved resonances.

8

1.2768 1.2770 1.2772 1.2799 1.2801 1.2832 1.28340

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Energy (eV)

Inte

nsity

(kc

ts./s

ec/p

ix)

HV

T3

T0

(1e1)(1h1)→0

(1e2)(1h2)→(1e1)(1h1)

(1e2)(1h12h1)T→(1e1)(2h1)

FIG. 7: Linear polarization sensitive PL spectra, showing theneutral exciton and biexciton lines of a single QD excited bya 501 nm cw laser. The spectral transitions are identified inthe figure.

A. Identification of excitonic lines

Fig. 8(a) displays single photon absorption resonances.When a photon is resonantly absorbed by the empty QDit enhances the emission from the exciton lines. Thespectrum is dominated by the (2e1)(2h1) absorption res-onance. This excitonic state in which both the electronand the heavy-hole are in their second pH-like orbitalmode, is particularly strong due to the large overlap be-tween the orbitals of the two carriers. The PLE spectrumcontains additional, almost an order of magnitude weakersharp resonances. These resonances are due to “non-diagonal” excitonic states, in which the electron and theheavy-hole differ in their orbital mode’s symmetry. As aresult, the spatial overlap between their modes is smalland the oscillator strength for the optical transition isreduced. The non-diagonal transitions that we clearlyidentify are the (1e1)(6h1) in which the electron is in itsfirst, s-like orbital mode and the hole is in its dVV-likemodes. The oscillator strength for these transitions doesnot vanish, since there is some amount of overlap be-tween the s-like and dVV-like orbitals which are of evensymmetry23.

More surprising, is the observation of non-diagonal ex-citonic transitions between orbitals of different symme-tries, like the (1e1)(2h1). This transition, which is thelowest energy resonance in the exciton PLE spectrum,is unambiguously identified by its spectral position andspectral shape. As expected, it is a cross linearly po-larized doublet, with the same splitting and the sameenergy-order of polarizations as the PL line due to theoptical transition from the (1e2)(2h11h1)T0

spin block-aded biexciton to this [(1e1)(2h1)] excited non-diagonalexciton states [Fig. 7]. In both cases the spectral shapeis dictated by the same final exciton states. Similarly, weidentified the next in energy order doublet as the non-

diagonal transitions to the bright levels of the (1e1)(3h1)exciton. In these resonances [(1e1)(2h1) and (1e1)(3h1)],the electron is excited into the first, s-like, symmetricorbital mode, while the hole is excited into the second,pH-like, and third, pV-like, antisymmetric mode, respec-tively. These optical transitions are therefore expectedto be forbidden since the orbital modes’ overlap van-ishes. Their appearance indicates some symmetry break-ing, possibly resulting in mixing with other bands24,25.

Another important mechanism which permits thesesymmetry forbidden transitions is provided by phononinduced mixing. This mixing is particularly strong whenthe phonon energy resonates with the single carrier’s en-ergy levels separation29. Clear evidence for such type ofmixing induced excitation is seen in the spectrally broadresonance 29 meV above the exciton line. This energyseparation characterizes the energy of LO phonons incompounds of GaAs and InAs34–37. The InxGa1−xAsoptical phonon closely resonates with the 1e-2e energylevels separation, resulting in an enhanced absorption inthis spectral domain. This observation is also supportedby the fact that the (2e1)(2h1) resonance is higher in en-ergy by about 29 meV from the (1e1)(2h1) resonance, asexpected.

B. Identification of biexcitonic lines

In Fig. 8(b-d) PLE spectra of the biexcitonic lines arepresented. During these measurements one laser wastuned into the broad excitonic resonance at 29 meV,thereby populating the QD with a bright exciton. Thesecond laser energy was then continuously varied whilethe emission from one of the biexciton lines was moni-tored.

The PLE spectrum of the ground biexciton dou-blet, (1e2)(1h2) −→ (1e1)(1h1) is presented in Fig.8(b). The allowed transitions from the bright excitonstates (total spin ±1) into the e-triplet-h-triplet biexci-ton states: (1e12e1)T0

(1h12h1)T0(total spin zero) and

(1e12e1)T±1(1h12h1)T∓3

(total spin ±2) are clearly ob-served, dominating, as expected, this spectrum.

In Fig. 8(c) the PLE is monitored by the PL linewhich corresponds to the decay of the spin blockadedmetastable biexciton, (1e2)(1h12h1)T±3 , by recombina-tion of a ground e-h pair, to the excited dark exci-ton states, |X0

1,2,D±〉. This PLE spectrum is dominatedby e-singlet-h-triplet resonances, just like the resonancefrom which the light is monitored. The absorption reso-nance transitions from the ground dark exciton |X0

1,1,D±〉directly to the monitored resonace [(1e2)(1h12h1)T±3

],by photogeneration of an Oe=1 Oh=2 e-h pair, isclearly identified as the lowest energy resonance in thisPLE spectrum. Likewise, the resonances in whichthe hole is excited into the Oh=3 and Oh=4 orbitals,[(1e2)(1h13h1)T±3

and (1e2)(1h14h1)T±3, respectively]

are clearly identified as well. Photogenerated holes inthese resonances nonradiatively relax to the Oh=2 level,

9

10 15 20 25 30 35 40 450

500

1000

1500

0

100

200

5

0

200

400

2

0

200

400

600

800

-6.5 -6.3

50

100

150

-6.5 -6.3

50

100

150

-3.4 -3.2

100

200

300

400

-0.1 0.10

100

200

300

9103

0XE-E (meV)

Inte

nsi

ty (

cts.

/sec

)

2

1 1

T0

1e

1h 2h

2

1 1

T 3

1e

1h 2h

2

1 1

T 3

1e

1h 4h

2

1 1

T0

1e

1h 4h

1 1

T0

1 1

T0

1e 2e

1h 2h

1 1

T 1

1 1

T 3

1e 2e

1h 2h

1

1

1e

2h

1

1

1e

6h

1

1

2e

2h

1 1

T

2

1e 2e

1h

1

1

1e

3h

2

1 1

T 3

1e

1h 3h

(a)

(b)

(c)

(d)

1

1

1e

1h

2

2

1e

1h

2

1 1

T

1e

1h 2h

2

1 1

T

1e

1h 2h

1

1

2e

1h

1 1

S

1 1

T 3

1e 2e

1h 2h

1 1

S

1 1

T0

1e 2e

1h 2h

FIG. 8: Linearly polarized PL spectra [left panels and the lower energy region in (a)] and PLE spectra (right panels) of asingle QD. The PLE in (a) is measured by continuous scan of one laser’s emission energy. The PLE spectra in (b)-(d) aremeasured with one laser’s energy tuned to the excitonic resonance at 29 meV as shown in (a), while the energy of the secondlaser is continuously scanned. The PL line monitored in each case is marked on the corresponding left panel by a vertical blackarrow. The assignment of the measured resonances is given by the state, which the QD is excited to, above each resonance.

where recombination occurs, since further non-radiativerelaxation is spin blockaded20,21.

In addition, a broad resonance is observed ∼29meV above the (1e2)(1h12h1)T±3 biexciton reso-nance. This resonance is due to absorption into the(1e12e1)S(1h12h1)T±3 . This state is strongly coupled tothe (1e2)(1h12h1)T±3

, by a one LO phonon, in a simi-lar way to the coupling between the (2e1)(1h1) and the(1e1)(1h1) bright exciton states [Fig. 8(a)].

Similar spectral features are observed in Fig. 8(d)where the PLE is monitored through the decay ofthe metastable biexciton (1e2)(1h12h1)T0

to the excitedbright exciton state |X0

1,2,B+〉. In this spectrum the ab-sorption resonances from the bright exciton states tothe (1e2)(1h12h1)T0

and the (1e2)(1h14h1)T0states are

identified. The weaker resonant absorption into the(1e2)(1h13h1)T0

state, is missing from this spectrum dueto poor signal to noise ratio.

We note that the energy difference between the op-tical transitions (1e1)(1h1) −→ (1e2)(1h12h1)T0

and(1e2)(1h12h1)T0

−→ (1e1)(2h1), is 15.7 meV. As ex-pected, this difference exactly matches the energy of theoptical transition from the vacuum into the first excitedexciton state (1e1)(2h1).

The transitions to the states (1e2)(1h12h1)Tmwhich

are clearly observed in Fig. 8(c) and (d), are absentfrom the PLE spectrum of the ground biexciton [Fig.8(b)]. This is due to the fact that in these cases theemitting state is directly excited and no intermediatenon-radiative relaxation process is required. This isnot the case when the (1e2)(1h14h1)T0

state is excited.Here, since non-radiative relaxation of the hole must oc-cur prior to the recombination, the resonance is weaklyobserved in the PLE spectrum of the ground biexci-ton state, as well. This means that in the relaxationprocess of the hole from the Oh=4 to the Oh=2 or-bital state, its spin may slightly scatter20. Last, wenote that the resonances (1e12e1)T±1

(1h12h1)T∓3and

(1e12e1)T0(1h12h1)T0

which are due to optical transi-tions from the bright exciton states are only observedin the PLE spectrum of the ground biexciton state [Fig.8(b)]. Similarly, the resonances (1e12e1)T±1

(1h12h1)T0

and (1e12e1)T0(1h12h1)T±3

, which are due to opticaltransitions from the dark exciton states are only observedin PLE spectra of the spin blockaded biexcitons [Fig. 8(c)and (d)]. We note however, that the bright exciton res-onances (1e12e1)T±1

(1h12h1)T∓3spectrally overlap with

the dark exciton resonances (1e12e1)T±1(1h12h1)T0

and

10

(1e12e1)T0(1h12h1)T±3

, and therefore their final identi-fication is also based on polarization sensitive and timeresolved spectroscopy as explained below.

In Fig. 9, we present examples for the use of polar-ization sensitive spectroscopy as a tool for verifying theidentity of the observed spectral resonances. The PLEresonances as monitored by the four biexcitonic PL tran-sitions and by the ground exciton state, are displayed inthe figure for various combinations of rectilinear polariza-tions of the exciting two lasers and the detected PL. Sincethe figure describes e-singlet-h-triplet resonances, the ex-perimentally measured optical transitions and their po-larization selection rules can be directly compared withthe theoretical expectations outlined in Fig. 3. The char-acteristic three lines structure of the optical transitioninto the e-singlet-h-triplet state is clearly resolved in Fig.9. The lowest energy biexcitonic doublet is crossed lin-early polarized, since each component is due to excitationof a different bright exciton eigenstate. The high energyline is unpolarized, and its intensity is twice stronger,(even in rectilinear polarization) since it gets contribu-tions from the two optically allowed transitions of thedark exciton eigenstates17. We note in particular, thatthe energy separation between the cross-linearly polar-ized components of the lower energy doublet, exactlymatches, as expected, that of the bright exciton (-34µeV). The two lasers PLE spectrum of the ground stateexciton [Fig. 9(c)] reveals a striking difference betweentransitions from the bright exciton states and transitionsfrom the dark ones. In The first type of transitions pop-ulation from the bright exciton is transferred into thebiexciton state, in which polarization memory is totallylost, and thus the polarized PL emission is reduced. Inthe latter type population is transferred from the darkexciton into the bright exciton state due to the rxcita-tion to the biexciton state. Therefore, in this case thePL emission from the bright exciton states is enhanced.

In Fig. 10, we use similar methods for studyingthe richer spectrum of the e-triplet - h-triplet reso-nances. In this figure the optical transitions into the(1e12e1)T(1h12h1)T biexciton states are studied and theexperimentally measured transitions should be comparedwith the theoretical considerations outlined in Fig. 4. Ascan be seen in Fig. 4, there are six optical transitionsfrom the bright exciton states into the e-triplet - h-tripletstates, and four optical transitions from the dark exci-ton states. The lowest energy transitions are the cross-linearly polarized doublet due to optical transitions fromthe bright exciton states into the (1e12e1)T0(1h12h1)T0

biexciton. As mentioned above, this biexciton resonanceis only observed in the PL from the ground biexcitonstates. For these optical transitions to occur, both lasersshould be co-linearly polarized, as indeed the data show[Fig. 10(a)]. In addition there is a higher energy doubletdue to the four optical transitions from the bright excitonstates into the symmetric and anti-symmetric biexcitonstates of total spin projection 2. Since these biexcitonstates are almost degenerate (in a similar way to the dark

100

200

300

400

Inte

nsity

(ct

s./s

ec)

(1e2)(1h12h1)T

±3

→(1e1)(2h1)

1500

2500

3500

(1e1)(1h1)→0

23.6 23.8 24 24.2 24.4 24.6 24.8 25 25.2

50

100

150

200

E−EX

0 (meV)

(1e2)(1h12h1)T

0

→(1e1)(2h1)

HH(H)

HV(V)

VH(H)

(b)

(c)

(a)

FIG. 9: Expanded energy scale presentation of the twolasers polarization sensitive PLE spectra revealing the opti-cal transitions into the biexcitonic resonances (1e2)(1h14h1)T.The spectra are monitored by the PL lines of the transi-tions from the metastable biexcitons (1e2)(1h12h1)T0 (a) and(1e2)(1h12h1)T±3 (b) to the excited exciton (1e1)(2h1) andthe PL doublet of the transitions from the ground state exci-ton (1e1)(1h1) to the vacuum (c). Various rectilinear polariza-tions of the exciting lasers and the detected, PL light are used.The first letter denotes the polarization of the laser tuned tothe exciton resonance, the second, that to the biexciton, andthe third, in parentheses, that of the detected emission. Thetwo cross-polarized curves in the ground state exciton (c) aremultiplied by 3.

exciton states), the four transitions form an unpolarizeddoublet which is twice as strong as the lower energy cross-polarized one. Again, this is exactly what one sees in thepolarization sensitive PLE spectrum of the ground biex-citon [Fig. 10(a)].

In Fig. 10(b) a strong resonance in the PLE spectrumof the PL line (1e2)(1h12h1)T±3 to (1e1)±1/2(2h1)±3/2

is observed. The resonances in this spectrum are ex-pected to result mainly from excitations of the dark ex-citon states. As seen in see Fig. 4, the optical transitionsfrom the dark exciton states are expected to form a crosslinearly polarized doublet. Unfortunately, this doubletspectrally overlaps the unpolarized doublet due to tran-sitions from the bright exciton states. We use time re-solved pulsed PLE spectroscopy in order to resolve thesetransitions.

In Fig. 10(c) we show two-pulse polarization sensitivePLE spectra of the bright exciton lines when the tempo-ral separation between the two pulses is relatively short(30 psec), while in Fig. 10(d) these spectra are shownfor the case in which the temporal separation is 13 nsec.While in the first case, immediately after the photogen-eration the exciton population is bright, in the secondcase only dark exciton population lasts. The PLE spec-troscopy reveals this fact in the following way: When thesecond pulse is tuned into a bright exciton to biexcitontransition, the PL signal from the exciton lines is reduced,

11

0

50

100 (1e2)(1h12h1)T

±3

→(1e1)(2h1)

Inte

nsity

(ct

s./s

ec)

33 33.5 34 34.5 350

300

600 (1e2)(1h2)→(1e1)(1h1)

E−EX

0 (meV)

4000

8000(1e1)(1h1)→0, 13 nsec

4000

8000 (1e1)(1h1)→0, 30 psec

HH(H)

VH(H)

VV(V)(c)

(b)

(a)

(d)

FIG. 10: Expanded energy scale presentation of thetwo-color, polarization sensitive PLE spectra revealingthe optical transitions into the biexcitonic resonances(1e12e1)T(1h12h1)T. The spectra are monitored by the PLfrom the ground state biexciton lines (a), the metastablebiexciton (1e2)(1h12h1)T±3 (b), and the ground state exci-

ton (1e1)(1h1) with two pulsed lasers at 30 psec delay (c) and13 nsec delay (d). Various rectilinear polarizations of the ex-citing lasers and the detected, PL emission are used. Thefirst letter denotes the polarization of the laser tuned to theexciton resonance, the second, that to the biexciton, and thethird, in parentheses, that of the detected emission.

since from the biexciton state part of the population doesnot return to the monitored exciton state. This is partic-ularly true for co-polarized pulses, since the polarizationmemory is lost in the biexciton states. Therefore, brightexciton transitions are seen as dips in the PLE spectrumof the exciton for co-polarized pulses and as peaks forcross-polarized pulses. Dark exciton transitions are al-ways obtained as peaks in the PLE spectrum of the exci-ton, since they transfer dark population into bright onethrough the biexciton states. Thus, the polarized natureof the optical transitions from the dark exciton statesinto the J = ±1;±3 biexciton states are clearly revealedin the polarization sensitive PLE spectra of the excitonin Fig. 10(d).

C. The non-diagonal optical transitions

In Fig. 11 we focus our attention on the “non-diagonal”optical transitions that we identified in the PL andthe one and two color PLE spectra. Only transitionswhich include pH -like orbitals are considered. Fig. 11(a)schematically describes the relevant excitonic and biex-citonic energy levels and the optical transitions betweenthem, once the non-diagonal transitions become allowed.The (un-normalized) spin wavefunctions are described tothe left of each level. Downward (upward) vertical arrowdescribes emission (absorption) and blue (red) stands for

TABLE II: Spectroscopically extracted e-h exchange interac-tion energies.

Energy separation Measured value [µeV]

∆1,10 123

∆1,11 -34

∆1,20 200

∆1,21 151

∆2,11 60

∆2,21

a 60

aextracted from the (2e1)(2h1) doublet and from the e-triplet-h-triplet resonances. (see Fig. 4).

H (V) polarization. Gray arrows describe unpolarizedtransitions. We note that the transitions between the(1e2)(1h12h1)T biexciton states and the (1e1)(1h1) exci-ton states are observed both in PL and in PLE spectra.

The spectra are characterized by two repeating pat-terns: The first one is a cross-linearly polarized doublet.This doublet is due to the anisotropic e-h exchange in-duced splitting of the bright exciton states. In thesedoublets the symmetric state is lower (higher) in energythan the antisymmetric state for diagonal (non-diagonal)optical transitions. The second pattern has three spec-tral lines: a higher energy unpolarized line and a lowerenergy cross-linearly polarized doublet. This pattern isdue to transitions from exciton to biexciton singlet-tripletstates. The doublet is due to transitions from the brightexciton states and the unpolarized line is due to transi-tions from the dark exciton states. The optical transi-tions which are schematically described in Fig. 11(a) arelinked to the experimentally measured transitions in thePL and in the PLE spectra [Fig. 11(b)].

Our ability to unambiguously identify all these non-diagonal optical transitions allows us to fully characterizethe QD in terms of single carriers’ orbital mode energiesand various interaction terms between carrier pairs. Theenergies extracted from our spectroscopy are summarizedin Table II.

In Fig. 12 we compare between the measured exci-tonic and biexcitonic optical transitions and the calcu-lated ones. Good agreement is achieved using the QDparameters listed in Table III. One notes that transitionswhich include the second single electron state 2e, deviatethe most from the calculated ones. We believe that it isdue to the optical phonon induced coupling between thisstate and the first electronic state. Our model does notconsider this coupling.

Finally, we note that the three optical transitions fromthe (1e12e1)T(1h2) to the (1e1)(2h1) states are observedin the measured PL spectrum. This confirms our many-body description, as discussed in section II C.

12

-6.55 -6.30 -3.33 0.00 9.45 9.70 10.10 10.35 15.75 16.00 28.00 28.25 28.50 28.75 29.00 29.25 29.50 29.75 30.000

100

200

300

400

500

600

91032

5

,

,

1 1 21e 2e 1h

2 1 11e 1h 2h

2 21e 1h

1 12e 1h

1 11e 2h

1 11e 1h

(a)

(b)

-6.41 -3.35 0.00 9.57 10.22 15.82 20.80 38.630

100

200

300

400

500

600

E-EX

0 (meV)

Inte

nsi

ty(c

ts./

sec)

10

-6.41 -3.35 0.00 9.57 10.22 15.82 20.80 38.630

100

200

300

400

500

600

E-EX

0 (meV)

Inte

nsi

ty(c

ts./

sec)

10

5

2

FIG. 11: (a) Schematic description of the excitonic and biexcitonic energy levels and carriers’ spin wavefunctions associatedwith the first and the second orbital modes. The optical transitions between these levels and in particular the ‘non-diagonal’ onesare represented by vertical arrows. Blue (red) downward (upward) arrow describes horizontally (vertically) linearly polarizedemission (absorption) transition. Gray arrows represent unpolarized transitions. The optical transitions in (a) are linked withthe experimentally measured polarization sensitive PL and PLE spectra in (b) and in (c), respectively. Solid blue (red) linerepresent horizontal (vertical) polarization.

−10 0 10 20 30 40 50 −10

0

10

20

30

40

50

Calculated E−EX0 (meV)

Mea

sure

d E−E

X0 (

meV

)

(1e1)(1h1)→(1e2)(1h12h1)T

(1e2)(1h2)→(1e1)(1h1)(1e2)(1h12h1)T→(1e1)(2h1)

0→(1e1)(3h1)

0→(2e1)(2h1)(1e1)(1h1)→(1e12e1)S(1h12h1)T

(1e1)(1h1)→(1e12e1)T(1h12h1)T

(1e1)(1h1)→(1e2)(1h14h1)T

0→(2e1)(1h1)

(1e1)(1h1)→(1e2)(1h13h1)T

0→(1e1)(6h1)

0→(1e1)(2h1)

FIG. 12: Comparison between the measured and calculatedexcitonic and biexcitonic transitions.

TABLE III: The parameters used for the model.

Parameter ValueM∗⊥,h

a 0.25m0

M∗ea 0.065m0

lxh 53Alxe 74A

ξ =lyelxe

=lyhlxh

0.87

aReference2.

V. SUMMARY

In summary, we presented a comprehensive study ofsingle, neutral semiconductor quantum dots subject toexcitation by two variably polarized resonant excitations,one to exciton resonances, and the other to biexcitonresonances. By monitoring the emission intensity fromvarious exciton and biexciton lines we completely char-acterize the rich one- and two-photon absorption spectra

13

of single semiconductor quantum dots. The measureddata is compared with a many carrier theoretical model,based on simple, one band parabolic potentials for elec-trons and heavy-holes. While the model provides fullunderstanding of the observed resonances, in terms ofline shapes, energies and polarization selection rules, itis short of quantitatively describing intensities of various“non-diagonal” optical transitions and spectral featureswhich involve strong coupling with optical phonons. Webelieve that the understanding that our study provides,should be very useful in applying semiconductor quan-tum dots as devices for quantum logical gates.

Acknowledgments

The support of the US-Israel binational science foun-dation (BSF), the Israeli science foundation (ISF), theministry of science and technology (MOST), Eranet NanoScience Consortium and that of the Technion’s RBNI aregratefully acknowledged. We also acknowledge helpfuldiscussions with Dr. Garnett Bryant.

14

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