Fast and selective phonon-assisted state preparation of a quantum dot by adiabatic undressing

0.34 MB·

by  A. M. Barth

Fast and selective phonon-assisted state preparation of a quantum dot
by adiabatic undressing
A.M. Barth1, S. Lu¨ker2, A. Vagov1, D.E. Reiter2,3, T. Kuhn2 and V.M. Axt1
1Institut fu¨r Theoretische Physik III, Universita¨t Bayreuth, 95440 Bayreuth, Germany,∗
2Institut fu¨r Festko¨rpertheorie, Universita¨t Mu¨nster, 48149 Mu¨nster, Germany and
3Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
(Dated: January 29, 2016)
6
We investigate theoretically the temporal behavior of a quantum dot under off-resonant optical
1
excitation targeted at fast acoustic phonon-assisted state preparation. We demonstrate that in a
0
preparation process driven by short laser pulses three processes can be identified: a dressing of
2
the states during the switch on of the laser pulse, a subsequent phonon-induced relaxation and an

n undressingattheendofthepulse. Byanalyzingexcitationscenarioswithdifferentpulseshapeswe
a highlight the decisive impact of an adiabatic undressing on the final state in short pulse protocols.
J Furthermore, we show that in exciton-biexciton systems the laser characteristics such as the pulse

8 detuning and the pulse length as well as the biexciton binding energy can be used to select the
2 targeted quantumdot state.

]
l
l I. INTRODUCTION previous studies mostly concentrated on the phonon in-
a
duced relaxationandthe resulting exciton andbiexciton
h
- Many of today’s proposals for quantum information occupationobtainedwhenapplyinganoff-resonantdriv-
s applications1 rely on the controlled and fast manipu- ing pulse29, here we will explain the impact of all three
e
m lation of the discrete states of the corresponding de- different processes and examine in detail the role of the
vices’ underlying structures. Semiconductor quantum switch-onandswitch-offphaseofthe excitation. We will
.
t dots (QDs) are frequently discussed as building blocks show that within the relaxationprocess there is a trade-
a
m for such materials because they hold out the prospect off situation between a sufficiently fast preparation and
of tailor-made energy spectra and a high integrability anoptimalpreparationfidelityandthatforahighfidelity
-
d in a solid-state environment. The excitonic QD states preparation of the bare QD states an adiabatic undress-
n are promising candidates to be used as qubits for quan- ing,thatcanberealizedbyalongenoughswitch-offtime,
o tum computing2–7, while the radiative decay from the is indispensable in short pulse protocols. Our analysis
c biexciton cascade offers the possibility of an on-demand also shows that even though Gaussian pulses, as used in
[ creation of indistinguishable entangled photon pairs8–10. veryrecentexperiments30–32, arenotthe idealchoice for

1 It has been shown that both exciton and biexciton thephonon-assistedprotocoltheyfulfilltherequirements
v states of a QD can be prepared by using ultra-fast laser for fast state preparation surprisingly well for a wide in-
6 pulses under a variety of excitation conditions11,12. The tensity range provided that the regime of non-adiabatic
8 most commonly known schemes for this purpose are res- dynamics is not yet reached. The latter condition sets a
8
7 onant excitation leading to Rabi rotations13–16, differ- lower bound to the pulse duration.
0 ent protocols using chirped laser pulses exploiting the The pulse characteristics not only allow to control
. adiabatic rapid passage effect17–25, and phonon-assisted the fidelity of the achieved inversion, but in addition
1
off-resonant driving26–29. Recently, the latter method can be used to select the QD state that is targeted by
0
6 hasalsobeenexperimentallydemonstrated30–32. Indeed, the phonon-assisted process. For the two-level case the
1 there is an increased interest in this approach, because signofthe pulse detuning determines whether the QDis
: it is not only stable against fluctuations of the applied driventowardstheground-stateortheexcitonstate. For
v
fieldintensity,butalsoleavesthequantumdottransition the exciton-biexciton system also the biexciton binding
i
X laser-free,whichcanbeimportantwhentheemittedpho- energy and the pulse length play a critical role in deter-
r tonsneedtobespectrallyseparatedfromthelaserpulse. mining the targeted QD state. Understanding that the
a Furthermore, in contrast to the other two protocols the preparation is a three-step process gives us an intuitive
phonon-assisted scheme makes active use of the phonon answer to the important question which state is selected
couplingandevenworksthebetterthestrongerthiscou- by the phonon-assisted preparation scheme.
pling is.
In this paper, we examine the influence of the pulse
shapeonthephonon-assistedstatepreparation. Weiden- II. MODEL
tify three distinct processes that take place during the
laser-driven evolution of the QD states. When the pulse WeconsiderastronglyconfinedGaAsQDdrivenbyan
startsthecouplingtothelaser-fieldleadstoadressing of externallaserfield andcoupledto a continuumof acous-
thebareQDstates. Thisenablesaphonon-inducedrelax- tic bulk phonons. Our model is defined by the Hamilto-
ation between the dressedstates26,33,34 and finally when nianH =H +H ,i.e.,the electronicsystem
dot,light dot−ph
thepulseisswitchedoffanundressing takesplace. While coupledto the laserfield andanadditionalphononpart.
2
Let us first concentrate on singleexcitonstate |Xiinthe three-levelsystemhasdif-
ferentpolarizationthaninthetwo-levelsystemdescribed
Hdot,light= ~ων|νihν|+ ~Mνν′|νihν′|, (1) furtherabove. AsisillustratedinFig.1(b),thebiexciton
Xν Xνν′ statehastheenergy~ωB =2~ωX−∆B,where∆B isthe
biexciton binding energy. In the exciton-biexciton sys-
where|νiaretheelectronicbasisstateswithcorrespond-
tem, the non-vanishing dipole matrix elements are given
ing energies ~ων. The matrix element Mνν′ describes by
the couplingbetweenthe QDandthe classicallaserfield
using the common dipole and rotating wave approxima- 1
M =M = f(t)eiωLt,M =M =M∗ . (6)
tions. In the first part of the paper we will restrict our- 0X XB 2 X0 BX 0X
selvestoatwo-levelsystemconsistingoftheground-state
|0i, for which we set ~ω =0, and a single exciton state Let us now focus on the coupling of the QD to the
0
|Xi with energy ~ω as illustrated in Fig. 1(a). The phonon environment. We model the electron-phonon in-
X
teraction by a pure-dephasing Hamiltonian which, to-
two-level approximation is valid when considering cir-
gether with the free phonon Hamiltonian, has the form
cularly polarized light with a single polarization orien-
tation and the exchange interaction is negligibly small.
The exchange interaction strongly depends on the QD Hdot−ph= ~ωqb†qbq+ ~nν γqbq+γq∗b†q |νihν|.
geometry35 and can be close to zero, as it is favorable q qν
X X (cid:0) (cid:1)
for, e.g., entangled photon creation9. In this case the (7)
non-zeromatrixelementsofthelight-mattercouplingare
The operators b† (b ) create (annihilate) a phonon with
given by q q
wave vector q and linear dispersion ~ω =~c |q|, where
q s
1 c denotesthelongitudinalsoundvelocity. n countsthe
M = f(t)eiωLt, M =M∗ , (2) s ν
0X 2 X0 0X numberofexcitonspresentinthestate|νi. Thecoupling
constants γ are chosen specific for the deformation po-
q
where f(t) is a real pulse envelope function, which in tential coupling to longitudinal acoustic phonons, which
the following is referred to as field strength. The field hasbeenshowntobedominantfortypicalself-assembled
strength f(t) is related to the electric field E(t) with GaAs QDs16,37. As described in more detail in Ref. 37,
f−redq·uEen(ct)y=ωL~fa(ntd)e−thiωeLtQ.D dipole matrix element d by tfuhnecctoiounpslinΨg conastsawntesllγaqsdoenpetnhdeodneftohremealteicotnropnoictewntaivael
2 e(h)
A very useful picture for strongly driven few-levelsys- constants D for electrons (e) and holes (h), respec-
e(h)
tems, which we will employ to analyze our results, is the tively. For simplicity, we assume the wave functions to
dressed state picture. The dressed states are the eigen- be the ground-state solutions of a spherically symmetric
statesofthecoupledlight-matterHamiltonianinaframe harmonic potential, i.e., Ψ (r)∝exp −r2/(2a2 ) ,
rotatingwiththelaserfrequency36. Forthetwo-levelsys- e(h) e(h)
and refer to a as the QD radius. The characteristics
tem driven by a laser-field with a fixed field strength f e (cid:0) (cid:1)
of the exciton-phonon coupling can be expressed by the
and a detuning ∆ they are given by the expressions spectral density J(ω)= |γ |2δ(ω−ω ), which under
q q q
the above assumptions reads37
|ψupi=+cos(Θ)|0i+sin(Θ)|Xi (3a) P
|ψlowi=−sin(Θ)|0i+cos(Θ)|Xi (3b) J(ω)= ω3 D e−ω2a2e/(4c2s)−D e−ω2a2h/(4c2s) 2
where Θ is the mixing angle defined by tan(Θ) = ~f 4π2ρ~c5s e h
∆+~Ω (cid:2) (cid:3)(8)
and Ω is the Rabi frequency given by
with ρ being the mass density. For our present calcula-
~Ω= (~f)2+∆2. (4)
tionsweusematerialparametersspecificforGaAstaken
The corresponding dressped state energies read from the literature38 ρ = 5370 kg/m3, cs = 5110 m/s,
D = 7.0 eV, and D = −3.5 eV. For the QD size we
e h
Eup/low = 12(−∆±~Ω). (5) cphooteonsetiaales=bu3t ntamkinagndinsteotaacec/oauhnt=th1.e15diaffsesruemntinegffeecqtuivael
masses of electrons and holes for GaAs. The spectral
It is worth noting that the contributions of the ground density for these parameters is shown in Fig. 1(c) where
and exciton states to the dressed states vary depending it can be seen that J(ω) exposes a clear maximum at
on the detuning and the field strength. ~ω = Emax ≈ 2 meV. The electron-phonon interaction
J
The final part of this paper will be devoted to excita- leads to a polaron-shift δ of the exciton and biexci-
ph
tions with linearly polarized light. In this case one also tonenergy,suchthatwhendrivingtheexciton-to-ground
has to take into account the biexciton state and the sys- state transition the resonant excitation is on the shifted
temdescribedsofarneedstobeextendedtoathree-level exciton energy ~ω˜ = ~ω −δ . In this paper we are
X X ph
model consisting of the ground state |0i, the single exci- mostly interested in detuned excitations and define the
ton state |Xi and the biexciton state |Bi. Note that the detuning as the difference between the laser energy and
3
a) b) c) (a)1.0 (b)500 1.0
0.25
∆B B> X0.8 400 0.9
| 0.20 C 0.8
n ∞X
∆ 1s)−0.15 upatio0.6 ps)300 C00..67 T=1 K
|X> |X> (pJ(ω)00..1005 exciton occ00..42 ff==01..50 ppss−−11 (τ210000 0.50 1f (2ps3−1)4 5
f=1.5 ps−1
0> 0>
| | 0.00 0.0 0
0 1 2 3 4 5 0 50 100 150 200 250 300 0 1 2 3 4 5
energy ħω (meV) time t (ps) field strength f (ps−1)
FIG. 1. (Color online) Energetic level diagram of (a) the ex- FIG. 2. (Color online) (a) Exciton occupation C as a func-
X
citon system with circular polarized excitation and (b) the tionoftimefordifferentfieldstrengthsf =0.5ps−1,1.0ps−1
exciton-biexcitonsystemwithlinearpolarized excitation. (c) and1.5ps−1. (b)Timeτ afterwhichthetimeaverageofthe
Spectral density of the phonon coupling as a function of en- oscillationsofC hasreached99%ofC∞(seetext)asafunc-
X X
ergy for the parameters used in our simulations (see text). tion ofthefieldstrength f. Insetof (b): C∞ asafunctionof
X
thefield strength f. The detuningis ∆=1.0 meV.
the polaron-shifted exciton energy, i.e.,
to two electronic levels, as we explained in the previ-
∆=~ω −~ω . ous section. Generally, for a weak coupling between the
L X
electronic states and the phonon environment the stan-
TostudythetimeevolutionoftheelectronicQDoccu- dardexpectationforthe drivenQDdynamics is thatthe
e
pationsunderexcitationwiththelaserfieldweemployan Markovianapproximationiswelljustifiedandthatinthe
implementation of a real-time path-integral approach39. long-timelimitthe relaxationleadstoathermaloccupa-
This method allows a numerically exact treatment of tion of the dot-photon dressed states33,40 [c.f. Eq. (3)].
the above model despite the infinite number of LA In a very good approximation, this has been shown to
phonon modes and yields the dynamics of the reduced hold also true for the two-level model of the QD with
electronic density matrix of the QD including arbitrary standard GaAs-type parameters considered here26. For
multi-phonon processes and taking into account all non- verylowtemperatures,the systemendsupmainlyinthe
Markovian effects. We assume the QD to be initially lowerdressed state |ψlowi, which corresponds to an exci-
in a product state of the electronic ground-state and a ton occupation of
thermalequilibriumofthephononmodesattemperature
1 ∆
T =1 K. C∞ ≈cos2(Θ)= 1+ . (9)
X 2 ~2f2+∆2!
Figure 2(a) shows the simulatedptemporal evolution of
III. RESULTS
the exciton state under constant excitation for a detun-
ing of ∆ = 1.0 meV and three different field strengths
We start by analyzing the phonon-induced relaxation
f. Here, the laser field is switched on instantaneously at
using continuous excitation that is switched on instan-
t = 0 ps. The occupations perform damped Rabi oscil-
taneously. Then we will apply short pulses with differ-
lations around a mean value that approaches a constant
ent pulse shapes to analyze the influence of the adia-
value. For a decreasing field strength the stationary ex-
baticityofthedressingandundressingprocessinviewof
citon occupation rises as can also be seen in the inset of
high-fidelity state preparation. For this analysis we will Fig. 2(b) where C∞ given by Eq.(9) is shownas a func-
also visualize the system trajectoryon the Bloch sphere. X
tionoff for∆=1.0meV.Mostimportantlyforphonon-
Finally, we will demonstrate the selective addressing of
assistedexcitonpreparation,thestationaryexcitonoccu-
allthree states in the exciton-biexcitonsystem using the
pation is very close to one and only limited by the finite
phonon-assisted state preparation protocol.
temperature for sufficiently small field strengths. Larger
values of f, however,strongly reduce C∞.
X
On the other hand, when we look at the time re-
A. Phonon-induced relaxation quired to reach the final state, we find that for small
field strengths f, the time to reach the final state be-
The phonon-induced relaxation can be best analyzed comes longer. This is quantified in Fig. 2(b), where we
by considering a continuous excitation of the QD with a have plotted the time τ it takes for the mean value of
constant field strength. Further, the laser field shall be the oscillations to reach 99% of the exciton occupation
circularlypolarizedsuchthatourmodelcanberestricted expected for a thermal distribution of the dot-photon
4
FdroersesexdamstpaleteastCfX∞=a0s.5apsf−u1ncatiroenlaxofattiohne tfiimelde osftrseenvgetrha.l n CX01..80 (a) rectangular (b) Gaussian
o
hotuhnedrrreedlapxsaitsiorenqpuriorecdes,sweshiocchcumrirginhgteoxncleoendgtehretitmimeescuanlteisl upati 0.6
c
than the phonon-induced relaxation such as the radia- c 0.4
o
tTivheerdeefocraeyswuchhicahloiswndortivcionngssidtreernegdthhedroee,scnoomteyiienltdoapslauyf-. citon 0.2
x
ficiently fast relaxation for phonon-assisted state prepa- e 0.0
ration. Indeed, we find that for f → 0 the time needed Clow 1.0 (c) (d)
for the state preparation diverges. on 0.8
ati
p 0.6
To understand the varying relaxation times for differ- u
c
c
epdnuirtreecdtrditevrpianhngassistitnirogennstgybtpheestwpoehneoennhotahnsecteooluekcptelrienopgniicndosmteasitnendso,ttbhiuanttdutthhceee er d.s. o 00..42 ααα===123000πππ
w
phonon-induced relaxation is only enabled by the laser o 0.0
l
field. Transitions between the QD states only take place 1.0 (e) (f)
due to a non-vanishing overlap of both of the photon eV) 0.5 Eup Eup
m
dressed states with the exciton state, which in turn is s ( 0.0
e
coupled to the phonon-environment. For an efficient re- gi−0.5
r
laxation also the Rabi splitting Ω, i.e. the difference be- ne−1.0
e
tween the two dressed state energies, needs to be close s. −1.5 E E
totypicalphononenergiesandideallymatchesthemaxi- d.−2.0 low low
mumofthespectraldensityofthephononcouplingJ(ω)
5.0
[c.f. Eq. (8)]. Both of these properties of the relax- 1)− (g) (h)
ation rate are also captured by a simple application of ps 4.0
Fermi’s Golden Rule, which yields a relaxation time be- h (f 3.0
tween the upper dressed state without phonons and the gt
en 2.0
lower dressed state with one phonon of r
st
d 1.0
el
fi 0.0
π −1 -30 -20 -10 0 10 20 30-30 -20 -10 0 10 20 30
τ = sin2(2Θ)J(Ω) . (10) time t (ps) time t (ps)
relax
2
(cid:16) (cid:17)
FIG. 3. (Color online) Dynamics for an off-resonant excita-
tion of theQD with rectangular pulses (left) havinga length
Here it can be seen that the mixing angle Θ as well as of 40 psand Gaussian pulses (right) with aFWHM of 20 ps.
the spectraldensity evaluatedatthe Rabisplitting J(Ω) The pulse areas are α = 10π (green), 20π (blue) and 30π
enter the relaxation rate. According to the simulations (red). (a),(b) Exciton occupation CX, (c),(d) occupation of
the maximal relaxation for the detuning ∆ = 1.0 meV the energetically lower dressed state Clow, (e),(f) instanta-
used here occurs for a field strength around 2.7 ps−1, neous energy of the upper and lower dressed state Eup/low,
(g),(h) pulse envelopes. The detuningis ∆=1 meV.
which is reflected by the position of the minimum of the
time needed for an almost complete relaxation plotted
in Fig. 2(b). In a good approximation this is expected
fromthe roughestimationinEq.(10). Despite the short B. Dressing and Undressing
relaxation time at the optimal field strength, which is
below 20 ps, such a strong driving field is not applica-
In the previous section the switch-on of the laser field
ble for high-fidelity state preparationeither, because the
has been taken to be instantaneous. This implies that
achievedfinaloccupationofbelow0.8istoo farfromthe
a sudden transformation of the photon dressed states
desired fidelity of one.
occurs at the beginning of the pulse. Similarly, when
Looking at our results, it becomes clear that the max- switching off the laser field the photon dressed states
imum exciton occupation for a given detuning can only are transformed back to the pure ground and exciton
be realized for almost vanishing field strengths where on states. To understand the importance of these dressing
the other hand it takes arbitrarily long times to com- andundressing processesintermsoffastandhighfidelity
pletetherelaxationprocess. Therefore,weconclude,that phonon-assisted state preparation, we now look at exci-
when only the phononrelaxationprocessof the prepara- tations with different pulse shapes. First we compare an
tiontakesplace,thereisatrade-offbetweenreachingthe excitation of the two-level system with rectangular and
targetedstatewithhighfidelity andrealizingfastprepa- Gaussian pulses. Later, we approximate the Gaussian
ration times. pulse using a rectangular pulse with softened edges.
5
Fig. 3 shows the temporal evolution under excitation shown in the right column of Fig. 3. In this case, no
for the rectangular (left) and Gaussian (right) pulses oscillations of the exciton occupation are visible and in-
including (a),(b) the exciton occupation, (c),(d) the stead C smoothly rises to its final value that for pulse
X
lower dressed state occupation, (e),(f) the instantaneous areas α = 20π and 30π is considerably higher than for
dressed state energies and (g),(h) the pulse envelopes of the rectangular pulses and practically reaches 1.0 as it
the electric field. The length of the rectangular pulses can be seen in Fig. 3(b). Like for the rectangularpulses,
is chosen as 40 ps, which is twice the full width of half a phonon relaxation process takes place from the upper
maximum (FWHM) of the Gaussian pulses, which is set to the lower dressed state yielding a steady increase of
to 20 ps. All calculations are performed for a detun- the occupation of the lower dressed state in Fig. 3(d).
ing of ∆ = 1.0 meV and for three different pulse areas However, because the field strength for Gaussian pulses
∞
α = f(t)dt with α = 10π (green curves), 20π (blue is time-dependent, the bare QD state contributions of
−∞
curves), and 30π (red curves). the dressed states change during the pulse. This is also
R
reflectedby the time dependence ofthe dressedstateen-
Let us start with the rectangular pulse (left panels
ergies shown in Fig. 3(f), which can be used to extract
in Fig. 3). The laser pulse sets in instantaneously at
theRabisplittingatagiventime. FortheweakestGaus-
t = −20 ps and besides damped Rabi oscillations there
sianpulse shownhere (α=10π), and similar to the case
is an overall increase of the exciton occupation that de-
ofthe weakestrectangularpulse,the phonon-inducedre-
pends onthe strengthofthe pulse [Fig.3(a)andsee also
laxation is too weak to complete the relaxation process
Fig. 2(a)]. In the dressed state picture this behavior can
within the duration of the pulse. This is because the
be understood as follows: When there is no laser pulse,
Rabi splitting is well below the maximum of the phonon
thedressedstatesareequaltothebarestateswherefora
density at energy Emax ≈ 2 meV [c.f. Fig. 1(c)] even
positivedetuningthegroundstatecorrespondstotheup- J
at the maximum of the pulse where the Rabi splitting
per dressed state E and the exciton state corresponds
up
reaches its maximum value of approx. 1.4 meV [c.f.
tothe lowerdressedstateE . Notethatinthispicture
low
Fig.3(f)]. Incontrast,forthehigherpulseareasα=20π
the energies of the photons needed for the excitation are
and α = 30π the relaxation is more effective and leads
countedas partof the dressedstate energies. As soonas
to a full occupation of the lower dressed state. At the
thelaserpulsesetsinthedressedstatesbecomeamixture
pulse maximum the Rabi splitting for the 30π pulse al-
of ground and exciton state [cf. Eq. (3)] with shifted en-
ready becomes larger (3.0 meV) than Emax temporarily
ergiesshowninFig.3(e), i.e. thebareQDstatesbecome J
leading to a weakening of the relaxation which is visi-
dressed by the laser field. Due to their overlap with the
ble by a reduced gain of the lower dressed state occupa-
ground-state both dressed states instantly become occu-
tion [red curve in Fig. 3(d)] around t = 0 ps. Most in-
pied, which is reflected by Rabi oscillations in the bare
states28. Phononsinducetransitionsbetweenthedressed terestingly, the phonon-induced relaxation is practically
states11,20 and at low temperatures transitions to the complete around t= 10 ps while the exciton occupation
obtained by the stronger pulses is still far from its final
lowerdressedstate,thatcorrespondto phononemission,
value and thereforethe remaining increaseof C cannot
outweigh. Therefore, during the pulse the lower dressed X
be attributed to the relaxation. The final value of C is
state becomes more and more occupied as it can be seen X
onlyreachedwithinasecondphaseofincreasethattakes
inFig.3(c). Thisalsomeansthattheexcitonoccupation
successively approaches its stationary value C∞, which placewhilethepulseisswitched-off. Thisisonlypossible
X becauseduringtheswitch-offthedressedstatesareadia-
depends onthe excitoncontributiontothe lowerdressed
baticallyundressed,i.e. adiabaticallytransformedbackto
state but is above 0.5 for all field strengths and positive
thebareQDstates. Inthisprocessthegroundstatecom-
detunings[c.f. Eq.(9)]. Whenthelaserpulseisswitched
ponent of the then almost fully occupied lower dressed
off at t = 40 ps the exciton occupation [Fig. 3(a)] keeps
state is reducedto zerowhichyields a drasticincreaseof
the value it has right before the switch-off. At the same
theexcitonoccupation. Importantly,anecessaryprecon-
time thedressedstatesareabruptlytransformedbackto
dition for this increase is that the undressing takes place
the bareQDstatesdue tothe suddenstopofthe rectan-
slowenoughsuchthatthesystemcanfollowthetransfor-
gularpulse,whichmeansthattheundressingtakesplace
mation of the dressed states adiabatically and the occu-
instantaneously. This is reflected by a step-like drop of
pationstaysin the lowerdressedstate. For example this
theoccupationofthelowerdressedstatethatcanbeseen
isnotfulfilledforthe instantaneousundressingincaseof
in Fig. 3(c). For the weakest pulse (α = 10π) the final
the rectangular pulses discussed earlier where the lower
state is not reached within the time window of 40 ps,
dressed state occupation experiences a step-like drop at
while for the higher pulse areas the relaxationprocess is
the end of the pulse [Fig. 3(c)] that does not occur for
mostly completed. Either way, the final exciton occupa-
Gaussian pulses [Fig. 3(d)].
tionstaysbelow0.9forallofthethreerectangularpulses
andinfactnomatterwhichrectangularshapeisassumed As it turns out the adiabatic undressing towards the
for pulse lengths of a few tens of ps one never achieves end of the pulse is in fact essential for a successful fast
a final value close to one due to the trade-off situation exciton preparation. The absence of the adiabatic end-
described in the previous section. ingpreventsthephonon-assistedstatepreparationproto-
The situationis quite differentforthe Gaussianpulses col from working with high fidelity for short rectangular
6
X1.0 (a) (b) Looking at the left panels of Fig. 4, we find that a
n C0.8 longer switch-on time does not alter the resulting exci-
atio0.6 ton occupation after the pulse, which is about 0.75 for
p all three cases. However, we find a significant reduc-
u τ =0.0 ps
c off
citon oc00..42 τττoonn===035...000 pppsss τττooffff===135...000 pppsss rtτieoonsnp=eocf5tttphose.tRhTeahbesiwionitssccehinl-lsaoittniivodnitysyn,aowmfhtiihccsehifiasnlmraelolaosttcecdduipstaaopttipoheneawririfrtoehr-
x on off
e0.0 versiblenatureoftherelaxationprocesswhich,incaseof
w1.0
Clo (c) (d) acompleterelaxation,leadstoacompletelossofmemory
on 0.8 oftheinitialstate. Wethusfindthatthefirstphase,i.e.,
ati the dressing,is necessaryto start the relaxationprocess,
p0.6
cu but for a sufficiently long relaxation phase the details of
c
s. o0.4 the switch-on are irrelevant for the final fidelity.
r d.0.2
we In contrast, as demonstrated in the right panels of
o0.0
l3.0 Fig. 4, a slower switch-off gives rise to a further gain
) (e) (f)
1s−2.5 of exciton occupation when the field strength is reduced
p tozero. Thiseffectisconsiderablylesspronouncedifthe
(f2.0
h switch-off time becomes too short and does not occur at
ngt1.5 all if the pulse stops instantaneously. For example for
e
r1.0 τ = 5 ps (red curve) C increases by more than 0.2
st off X
eld 0.5 duringtheswitch-off,while forτoff =1ps(black)thein-
fi0.0 crease is below 0.05. Thus, we conclude, that to yield a
-30 -20 -10 0 10 20 30-30 -20 -10 0 10 20 30 high fidelity state preparation the transformation of the
time t (ps) time t (ps)
dressed states into the bare QD states needs to happen
slow enoughin orderto allow for an adiabatic evolution,
FIG. 4. (Color online) Time dependent exciton occupation
i.e., the undressing needs to be adiabatic.
C underpulsedexcitation detunedby∆=1.0meVfordif-
X
ferent(a)switch-on(b)andswitch-offtimes(seekey). (c),(d)
corresponding occupations of the energetically lower dressed To sum up, for an efficient and fast phonon-assisted
state Clow and (e),(f) corresponding pulseenvelopes. state preparation there are basically two features of the
pulse envelope that are significant: First, there has to
be a phase of the pulse where the QD thermalizes in the
pulsesandanyotherpulseshapeswithtoofastswitch-off dressed state basis, which must be long enough to com-
times. pletetherelaxation. Thisphasecanbechosentheshorter
Havingseenthatthetemporalevolutionofthedressing the strongerthe phonon-inducedrelaxationis andthere-
andundressingplaysacrucialrole,weinvestigateinmore forecanbeminimizedwhenthefieldstrengthissuchthat
detail the role of the switch-on and switch-off phase. To the strength of the phonon-coupling is maximal. There-
this end we choosea special pulse shape thatis designed fore in this phase a constant field strength correspond-
to highlight the important role of these two processes ing to the maximal relaxation rate is optimal to achieve
on the preparation process. The thermalization of the a complete thermalization with a minimal pulse length.
dressed states is obviously achieved best by a constant Second, the pulse must be switched-off slowly enough to
part of the pulse that is chosen such that the coupling allow the QD system for an adiabatic undressing of the
to the phononenvironmentis maximal(f =2.7 ps−1 for states.
the parameters chosen here) and that is sufficiently long
tocompletethethermalization. Inourcase,theduration Although a Gaussian pulse shape does not meet these
of this part of constant driving is chosen to be 40 ps. In requirements in an optimal fashion it still works surpris-
ordertosystematicallystudytheinfluencesoftheswitch- inglywell,mostlybecausetheoptimumfortheidealfield
on(switch-off)characteristicsweaddtheleft(right)half strength at which the dressed state relaxation is maxi-
of a Gaussianpulse with FWHM τ (τ ) before (after) mal is not a sharp maximum, but rather a broad peak
on off
the constant part of the pulse envelope as illustrated in as it can be seen from Fig. 1. Therefore, even though a
Fig. 4(e)[(f)]. Here, we compare a strictly rectangular Gaussian pulse does not have a constant plateau, a suf-
pulse with τ = 0 ps (green) with pulses of different ficient relaxation takes place during most of the time of
on/off
switch-on/offtimesτ =1ps(black),3ps(blue)and the pulseprovidedareasonablevalueofthe pulseareais
on/off
5 ps (red). The first row [panels (a) and (b)] shows the chosen. Most importantly, deviations from the Gaussian
corresponding evolution of C , while the lower dressed pulse shape that go towards faster switch-off times, as
X
state occupation is plotted in the second row [panels (c) they might be induced by a pulse-shaping setup, can be
and (d)] and the pulse envelopes are shown in the third harmfulto the efficiency ofthe phonon-assistedprepara-
row [panels (e) and (f)]. tion protocol.
7
trajectory of the Bloch vector of the driven QD system,
|X> |X>
z z which is shown in Fig. 5(a),(b) alongside the time evolu-
(a) (b)
tion ofthe length of the Blochvector presentedin Fig. 5
y y
(c),(d)foranexcitationwithGaussianpulses(leftpanel)
and for rectangular pulses with softened switch-on or
switch-off (right panel). The correspondingpulse shapes
can be seen in Fig. 3 (h) and Fig. 4 (e),(f).
x x
Let us first look at the Gaussian pulses in the left
column. When the pulse is switched on the Bloch vec-
|0> |0> tor leaves the surface of the sphere for all pulse areas
Gaussian rect. with switch-on/off in Fig. 5(a) [α = 10π (green), α = 20π (blue) and
1.0
(c) (d) α = 30π (red)]. The corresponding vector length shown
in Fig. 5(c) decreases as it is expected as the result of
h0.8
ngt the pure-dephasing phonon coupling. Indeed, the loss of
e
r l0.6 coherence turns out to be as high as 95%. The time to
o
ct reach the minimal Bloch vector length depends on the
ve0.4
ch α=10π pulse area and is shorter for the stronger pulses where
Blo0.2 α=20π rect., τon=5.0 ps the relaxationis more efficient. The vector length would
α=30π rect., τoff=5.0 ps vanish at the minimum in the ideal case of a completely
0.0 adiabaticdressingprocessandprovidedthatthephonons
-30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30
time t (ps) time t (ps) realizeafullyincoherentoccupationtransferbetweenthe
dressedstates. Inthatcasethe electronicdensity matrix
FIG. 5. (Color online) (a),(b) Illustration of the system tra- wouldstaydiagonalinthedressedstatebasisallthetime
jectory on the Bloch sphere and (c),(d) length of the Bloch andcoherencesbetweenthe dressedstates wouldneither
vector as a function of time for (a),(c) a Gaussian excitation buildupduetothelaserdrivingnorduetothephonons.
with pulse areas α = 10π (green), 20π (blue) and 30π (red) The continuous occupation transfer from the upper to
and (b),(d) a rectangular excitation with different switch- the lower dressed state would then necessarily lead to a
on/off times τon = 5 ps; τoff = 0 ps (blue) and τon = 0 ps; zeroofthe Blochvectorlengthatsomepointin time. In
τoff =5 ps (red). The detuningis ∆=1 meV. ourcase,however,theminimalBlochvectorlengthhasa
smallbutfinitevaluebecausetheGaussianpulsesusedin
thesimulationsalreadyinducesomesmallcoherencesbe-
C. Interpretation on the Bloch sphere tween the dressed states (not shown). Subsequently, the
coherence between |0i and |Xi is restored to a large de-
Another interesting aspect not highlighted so far is greeandtheBlochvectorlengthapproaches1forthetwo
thattheincoherentphononscatteringcanresultinapure stronger pulses as the phonon-assisted transitions result
state,whichevencanbetransformedtoabareQDstate. in an almost complete occupation of the lower dressed
This is best illustrated using the Bloch vector picture41. state which is again a pure state lying on the surface of
InthispicturetheprojectionoftheBlochvectoronthez- theBlochsphere. Fortheweakerpulse,α=10π (green),
axis represents the inversion, i.e., the difference between the trajectory ends inside the Bloch sphere and the vec-
the occupations of the upper and the lower level of the torlength stayswellbelow0.5,because due to the insuf-
two-level system, while the in-plane component reflects ficient pulse strength the relaxation does not complete.
the polarization. Resonant lossless driving of a two-level Forthestrongerpulses,thedressedstatestransformfrom
systemisreflectedbytheBlochvectormovingatthesur- asuperpositionofgroundandexcitonstateintothepure
faceoftheBlochspheretakingtheshortestpathfromone excitonstateduringtheadiabaticundressing,whichcor-
poleto the othercorrespondingto the wellknowncoher- responds to a motion of the Bloch vector on the surface
entRabioscillations. Preparingtheexcitonstateusinga towards the upper pole. It thus becomes obvious that
πpulsethenmeansgoingfromthelowerpolecorrespond- the laser-driven QD evolution describes a way through
ing to the ground level to the upper pole corresponding the Bloch sphere to the upper pole, which is very dif-
totheexcitedlevel. Adetunedexcitationleadstoatilted ferent to the motion along the surface in the case of an
oscillation, which starting from the lower pole does not inversion yielded by applying a resonant π pulse.
reachthe upper pole. If the system is fully coherent,the A similar analysis can be done for the Bloch vec-
Bloch vector stays on the surface, i.e., its length is con- tor trajectory for rectangular pulses where the switch-
stantly one. Ifdecoherence takesplace, the length ofthe on or the switch-off edge is softened. The red curve in
Bloch vector is decreased. Clearly, the phonon-assisted Fig.5(b),(d)hasasharpswitch-onandasmoothswitch-
preparation involves an incoherent phonon-induced re- offwithτ =5ps,whilethebluecurvecorrespondstoa
off
laxation, but as we have seen it can still eventually lead smoothswitch-onwithτ =5psandasharpswitch-off.
on
to an almost perfect exciton state. When the switch-on is instantaneous (red curve), we see
To analyze this in more detail we have calculated the that the Bloch vector trajectory exhibits a spiral move-
8
ns 1.0 (a) ∆B =+2.0 meV (b) ∆B =−2.0 meV rsetaspteonadnidnglonoukminbgeraotftphheortoesnusltnineegdeodrdfeorrorfeatchheinsgtattheast.
o 0.8
ati 0.6 More specifically,zero photons have to be subtracted for
p
cu 0.4 the ground state, one photon for an exciton state and
c
o 0.2 two photons for the biexciton state. The resulting state
el. 0.0 lowest in energy will be occupied predominantly at the
eV) 42 (c) (d) end of the preparation process.
gy (mE −−−0462 ||0X>> lasFeorreneexragmypflreomintthheelatwseor--lferveeelQsDystsetmatesuebnterragcietsinygietlhdes
er −8 |B> dressed state energies that are in the limit of vanish-
n
e−10 ing fieldstrengthseparatedby the detuning betweenthe
−2 −1 0 1 2 3 4 −1 0 1 2 3 4 5
detuning ∆ (meV) detuning ∆ (meV) laser and the polaron-shifted QD transition. For a posi-
tive detuning the energyofthe exciton-likedressedstate
FIG.6. (Coloronline)Occupationsoftheelectroniclevels|0i willbebelowthecorrespondinggroundstate-likedressed
(green), |Xi (blue) and |Bi (red) after optically exciting the state energy in the rotating frame and consequently the
exciton-biexciton system with a Gaussian pulseof pulsearea phonon-assisted preparationprotocol with adiabatic un-
α = 20π and FWHM= 20 ps as a function of the detuning dressingpreparestheexcitonstate. Ontheotherhand,a
∆ for a (a) positive biexciton binding energy ∆ =2.0 meV
B negative detuning reversesthe order of the dressed state
and (b) negative biexciton binding energy ∆ = −2.0 meV.
B energies and at low temperatures where there is mostly
(c) and (d) : corresponding energies of the QD states in the
phonon emission the protocol prepares the QD ground
rotating frame.
state independent of the initial state29.
In the case of an exciton-biexciton system subtracting
mentwithintheBlochspherereflectingthedampedRabi the energy of the corresponding photons gives the ener-
oscillations. In agreement with the essentially complete giesofthe QDstatesinthe framerotatingwiththelaser
relaxation,thespiralends uponapointclosetothe sur- frequencyasE0′ =0,EX′ =−∆,EB′ =−∆B−2∆. Thus,
face of the Bloch sphere. For τ = 5 ps (blue curve), theenergeticorderingdependsalsoonthebiexcitonbind-
on
the oscillations are less pronounced and the motion goes ing energy and we will consider both cases with positive
roughly along the axis of the spiral movement. The spi- and negative biexciton binding energy, which can both
ralingwilleventuallyvanishforevenlargervaluesofτ , be realizeddepending onthe QD geometry42. In the fol-
on
but the end point after the relaxation will be the same. lowing we will refer to the dressedstate that in the limit
If the laserpulse is switchedoff rapidly(blue curve),the of vanishing field strength transforms into the |0i [|Xi,
z-componentofthe Blochvectorstaysconstantafterthe |Bi] state as |0′i [|X′i, |B′i].
end of the pulse and performs a circular motion. In con- Let us first discuss the most commonly encountered
trast, for τ =5 ps an adiabatic undressing takes place situation of a QD with a positive biexciton binding en-
off
during the switch-off and the Bloch vector approaches ergy ∆ = 2 meV. Figure 6 (a) shows the final occu-
B
the upper pole. pation after excitation with a detuned Gaussian pulse
with a FWHM of 20 ps and a pulse area of α = 20π as
a function of the detuning, while Fig. 6 (c) shows the
D. Selective state preparation in the corresponding energies in the rotating frame. It can be
exciton-biexciton system seenthat for detunings below the two-photonresonance,
i.e., ∆ < −∆ /2 = −1 meV, the occupation remains in
B
The clear understanding of the different phases of the the ground state (green curve). This is consistent with
phonon-assistedrelaxationmechanism workedout so far the energetic order of the states, since E′ is the lowest
0
turns out to be highly valuable to predict in an easy energy in this parameter region. At ∆ = −1 meV the
way the QD state that can be obtained by pulsed off- energeticorderofthedressedstateschanges,becausefor
resonantexcitationforarbitraryinitialstatesandalsofor detuningsabovethetwo-photonresonanceE′ isthelow-
B
the exciton-biexciton system illustrated in Fig. 1(b). To est energy. This leads to a significant drop of the final
this end it is required as before that during the switch- ground state occupation in favor of the biexciton occu-
off phase of the pulse the system evolves adiabatically. pation (red curve), which approachesits maximum close
When the pulse duration is sufficiently long such that to one at ∆ ≈ −0.5 meV. At the one-photon resonance
the relaxation all the way to the lowest dressed state at∆=0,the energeticorderchangesonce againandfor
is fully completed the prepared state at the end of the all positive detunings |0′i is the highest energy dressed
pulse is determined exclusively by the energetic order of state,whileE′ remainsbeingthelowestenergy. Forsuf-
B
thedressedstatesinthelimitofvanishingpulsestrength. ficiently long pulses all the occupation would, of course,
Recallingthatthedressedstatesaredefinedwithrespect end up in the lowest branch resulting in the preparation
to the rotating frame, the prepared state can technically of the biexciton. However, for ∆ > 1 meV the energetic
be determined by subtracting from the energies of the splittingE′ −E′ >3meVexceedsthemaximumofthe
X B
QD states without light coupling the energy of the cor- phonon spectral density Emax lying around 2 meV by
J
9
far, which results in a very inefficient relaxation to the tonbindingenergythetargetedstatecanbederivedfrom
|B′i state, which is not completed in the time window theenergeticordershowninFig.6(d),demonstratingthe
set by the pulse length. Therefore,instead, we observea correctness of the description found for the dynamics of
gainoftheexcitonoccupation(bluecurve)assoonasE′ the phonon-induced preparation process also in the case
X
crossesE′. Themaximalexcitonoccupationofabout0.8 of the exciton-biexciton system.
0
isreachedaround∆=2meV,wheretheenergysplitting As we have seen, to selectively address a state in an
E′ −E′ =2 meVagreeswith Emax. Therefore,it turns exciton-biexcitonsystembyphonon-assistedstateprepa-
0 X J
out that for a system with more than two states like the ration the biexciton binding energy needs to be consid-
oneconsideredhereanincomplete relaxationcanalsobe eredcarefully,whilethedetuning∆andthepulselength
advantageous for preparation purposes if a preparation can act as control parameters to choose the targeted
of a QD state that in the rotating frame is not the low- state.
est lying state, like in our case the exciton, is intended.
Anevenhigherexcitonoccupationispossibleforalarger
biexcitonbindingenergywhichfavorstransitionsto|X′i, IV. CONCLUSIONS
because the coupling to the lowest dressed state |B′i in
this case gets evenmore out ofresonance. It alsofollows We have analyzed the dynamics of an optically
thatwhetherthepreparedstateistheexcitonorthebiex- drivensemiconductor QD coupledto longitudinal acous-
citon can in principle be selected by suitably adjusting tic phonons for different off-resonant excitation condi-
the pulse length for all positive detunings, because the tionsfocusingontwomainquestions: a)whatarethere-
final state depends on whether the relaxation completes quirementsforafaststatepreparationusingoff-resonant
during the pulse orwhether the intermediate levelcorre- driving and b) how is the prepared state selected. We
sponding to the exciton is still predominantly occupied demonstratedthatafasthighfidelitypreparationprocess
when the pulse is switched off. Detunings higher than not only relies on an efficient phonon-induced relaxation
2.0 meV lead to a decrease of both the final exciton and betweenthedot-photondressedstates,butalsoonasuc-
the biexciton occupations, because the phonon-induced cessfuladiabaticundressingofthestates. Tounderstand
relaxationbecomesweakerandweakerasthephononen- the influence of the adiabatic undressing we compared
vironmentbecomes moreandmoreoutofresonance. Fi- the exciton occupations produced by Gaussian and rect-
nally,forverylargedetuningsabove∆=4.0meVtheen- angularpulseshapesandsystematicallystudiedtheinflu-
ergetic splittings between the dressed states are so large ence of the switch-onand switch-offtimes. This analysis
that for the given pulse length practically no relaxation revealed that while the switch-on time plays a subordi-
takes place and the QD remains in the ground-state. nateroleitiscrucialthatthe pulse isswitchedoffslowly
enough. Furthermore, we analyzed the coherence prop-
The case of a negative biexciton binding energy of erties during the state preparation process and revealed
∆ = −2 meV is shown in the right column. Fig. 6(b) thattheincoherentphononscatteringcanalsorestoreco-
B
shows the final occupations and the corresponding ener- herence and the system trajectory takes a way through
gies are plotted in Fig. 6(d). Similar to the case of posi- theBlochspheretotheoppositepole. Finally,weshowed
tivebiexcitonbindingenergiesthesystemremainsinthe thatthe conceptofphonon-assistedstatepreparationby
groundstateuptoadetuningof0,sinceE′ isthelowest adiabaticundressingalsoappliestotheexciton-biexciton
0
energy. Between ∆ = 0 and 2 meV |X′i is the dressed system, where an easy prediction of the prepared state
state with the lowest energy. This energetic order ap- is possible. When the pulse is long enough to support a
pearsexclusively fornegativebiexcitonbinding energies, fullrelaxationthepreparedstateistheonethatislowest
resultinginabroadregionwhereacompletepreparation in energy in the rotating frame in the limit of vanishing
of the exciton occurs. This finding is somewhat surpris- field strength. We demonstrated that even pulses too
ing, because the two-photon resonance at ∆ = 1 meV short for a full relaxation can be used for preparing in-
also lies within this interval, where an efficient prepara- termediatestatesthatarenotthelowestinenergyinthe
tion of the exciton state might be unexpected, but can rotatingframe. Altogether,itisdemonstratedthatinan
easily be understood in the context described here. At exciton-biexciton system the decisive parameters for se-
∆=2 meV the energetically lowest state changes a sec- lectingthepreparedstatearethedetuning,thebiexciton
ond time and a sharp transition of the prepared state binding energy and the pulse length.
from the exciton to the biexciton occurs. However, for
∆ > 2.5 meV the energy splittings between the dressed
states that are connected to the excitonic states and |0′i V. ACKNOWLEDGMENTS
already become too large for a complete thermalization
to take place during the length of the pulse. This leads A.M.B. and V.M.A. gratefully acknowledge the finan-
to a significant reduction of the biexciton occupation for cial support from Deutsche Forschungsgemeinschaft via
higherdetuningsandabove∆=5meVtheQDdoesnot the Project No. AX 17/7-1. D.E.R. is thankful for fi-
getaffected by the pulse anymorestayingin the ground- nancial support from the German Academic Exchange
state. Importantly, also in the case of a negative biexci- Service (DAAD) within the P.R.I.M.E. programme.
10
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