Coherent optical writing and reading of the exciton spin state in single quantum dots

0.49 MB·English

by  Y. Benny

Coherent optical writing and reading of the exciton spin state in single quantum dots
Y.Benny,1 S.Khatsevich,1 Y.Kodriano,1 E.Poem,1 R.Presman,1 D.Galushko,1 P.M.Petroff,2 andD.Gershoni1,∗
1The Physics Department, Technion - Israel institute of technology, Haifa, 32000, Israel.
2Materials Department, University of California Santa Barbara, CA, 93106, USA.
(Dated: January 7, 2011)
Wedemonstrateaonetoonecorrespondencebetweenthepolarizationstateofalightpulsetuned
toneutralexcitonresonancesofsinglesemiconductorquantumdotsandthespinstateoftheexciton
thatitphotogenerates. Thisisaccomplishedusingtwovariablypolarizedandindependentlytuned
picosecond laser pulses. The first “writes” the spin state of the resonantly excited exciton. The
secondistunedtobiexcitonicresonances,anditsabsorptionisusedto“read”theexcitonspinstate.
Theabsorptionofthesecondpulsedependsonitspolarizationrelativetotheexcitonspindirection.
Changesintheexcitonspinresultincorrespondingchangesintheintensityofthephotoluminescence
fromthebiexcitonlineswhichwemonitor,obtainingthusaonetoonemappingbetweenanypoint
1 on the Poincar´e sphere of the light polarization to a point on the Bloch sphere of the exciton spin.
1
0 PACSnumbers: 42.50.Dv,42.50.Md,42.25.Ja
2

n
Coherent manipulation of quantum states is a crit- tum projection is conserved. We associate the spin state
a
ical step towards applications in quantum information of such a pair with the polarization of the light by defin-
J
processing. The atomic-like spectrum of semiconductor ing|R(cid:105)=⇑↓(|L(cid:105)=⇓↑). Hereaspinup(down)electronis
6
quantum dots (QDs) and their compatibility with mod- denoted by ↑ (↓) and a spin up (down) heavy hole is de-

] ernmicroelectronicsmakethempromisingcandidatesfor noted by ⇑ (⇓). With this notation it is straightforward
h
forming the building blocks of these future technologies. to show that the correspondence between the horizontal,
p
In particular, they form an excellent interface between vertical, diagonal and cross-diagonal linear polarizations
-
t flying photonic-qubits and anchored matter-spin-qubits. of the exciting light (H, V, D and D¯, respectively) and
n
a The coherent properties of spins of confined carriers and the spin s√tate of the photogenerate√d pair are given by:
u pairs of carriers (excitons) in QDs have been demon- |H(cid:105) = 1/ 2(⇑↓ + ⇓↑); |V(cid:105) = −i/ 2(⇑↓ − ⇓↑); |D(cid:105) =
√ √
q stratedbyvariousexperimentalwaysovertheyears[1–9]. e−iπ/4/ 2(⇑↓ +i ⇓↑) and |D¯(cid:105) = eiπ/4/ 2(⇑↓ −i ⇓↑).
[
The fundamental optical excitations of QDs, the exciton The direction in space of the H polarization coincides

2 andpairof excitons(biexciton), havebeenproposed[10] withthedirectionofthenaturalmajoraxisoftheQD[3].
v and demonstrated [3] as coherent physical realizations of ThesespinstatesaredescribedontheBlochsphereofthe
3
qubits and quantum logic gates. exciton spin in Fig. 1(b). An arbitrarily elliptically po-
6
larized pulse is described by a point on the surface of
4 Inthisworkwedemonstrateanewmethodforinitializ-
5 ing(“writing”)thespinstateofaQDconfinedexcitonin the Poincar´e sphere. Such a point can be viewed as hav-
. ing two components. A component on the equator plane
9 any coherent superposition of its eigenstates by a single,
0 resonantly tuned, polarized picosecond light pulse. Like- (containing the L and D directions), deflected by an an-
0 gle φ from L, and a component parallel to the rectilinear
wise,weshowthatthespinstateoftheinitializedexciton
1 H-V axis. Thus, two angles completely define an arbi-
can be determined (“readout”) using a second, delayed,
:
v polarized picosecond light pulse, resonantly tuned into trary polarization, the angle φ and the angle θ between
i the polarization and the H-V axis. In a complete anal-
X specific biexcitonic resonances. Fig. 1(a) is an energy
ogy,anarbitraryexcitonspinstateisdescribedasapoint
leveldiagramwhichschematicallydescribestheprocesses
r
a involved in writing and reading the excitonic spin state. on the Bloch sphere. The north and south poles of the
Blochspheredenotetheexcitonsymmetricandantisym-
The first polarized laser pulse is resonantly tuned into
metric eigenstates, |H(cid:105) and |V(cid:105), respectively. However,
either a ground, or an excited exciton state to photo-
due to the anisotropic e-h exchange interaction, these
generate an exciton. From the excited state, the exciton
eigenstates are not degenerate, even in the absence of
rapidly relaxes non-radiatively to its ground state, while
externally applied magnetic field [3, 11, 14]
preserving its initial spin (as demonstrated below).
In order to relate the polarization of the light to the A resonantly tuned H (V) polarized laser pulse pho-
spin state of the photogenerated exciton we note that togenerates an exciton in its symmetric (antisymmetric)
the angular momentum projection of right (left) hand spineigenstate|H(cid:105)(|V(cid:105)). Whenthepulseresonateswith
circularly polarized light R (L) in the direction of prop- the excited state, the generated excited exciton relaxes
agation is 1 (−1). Upon electron-heavy hole (e-h) pair nonradiativelyintoitsgroundstatebeforerecombination
generation the electron spin (1) is oriented downward occurs. The exciton then remains in its eigenstate until
2
(upward) while the heavy-hole spin (3) is oriented up- it radiatively recombines. However, since the two eigen-
2
ward (downward) such that the total angular momen- statesarenotdegenerate,theyevolveatdifferenttempo-
2
ralpaces. Therefore,anyothercoherentsuperpositionof  XX0*
these eigenstates precesses in time at a frequency given
 XX0
by the difference between the eigenenergies, divided by
the Planck constant. Such excitation requires, however, H
2ndpulse V H
a resonant pulse of spectral width which contains both
eigenstates. For example, the orange circle on the equa- 1 
2 X0* P (θ,)
tsoprinoefxtchiteedspbhyerae rdeessocnraibnetsLthpeoleavroizluedtiopnulosef aenxcietxactiitoonn. 12 D θ 0 L

Suchapulseinitiatesthetwospineigenstateswithequal
1stpulse R
probabilities. Theinitiatedspinthenprecessesclockwise
with time, such that the angle φ equals π2, π, 32π and 2π 1 D
after 1, 1, 3 and 1 period, respectively, while the spin 2 X0
state b4ec2om4es |D¯(cid:105), |R(cid:105), |D(cid:105) and |L(cid:105) again, respectively. 1
2
The purple circle on the sphere describes precession of 1stpulse V H V
anexcitonspin,photogeneratedwithP (θ,φ)spinbyan (a) (b)
0 0
arbitrarily polarized pulse.
Delayedby∆τ fromthefirstpulse,asecond,polarized
FIG.1: (a)Schematicdescriptionofthewritingandreading
pulse, is then applied. This pulse is tuned to an excited
processes. Horizontal lines describe the states’ energies. The
resonance of the biexciton, to avoid scattered light from spin wavefunctions of these states are depicted to the left.
the detector. The probability to photogenerate a biexci- ↑ (⇓) represents spin up (down) electron (hole). Blue (red)
tondependsontheorientationoftheexcitonspinrelative arrow represents carrier in the ground (excited) energy level.
tothepolarizationofthesecondpulse,sincethebiexciton Green arrows denote resonantly tuned light pulses, 1st short
(long) to a ground (excited) exciton and 2nd to an excited
resonance used here contains two electrons in a singlet
biexciton state. Curled lines denote nonradiative relaxation
state (Fig. 1). Therefore, by monitoring the photolumi-
andablue(red)lineradiativeH(V)polarizedrecombination.
nescence (PL) from the biexciton doublet as a function
The linewidth of the laser pulses are given by the curves to
of∆τ,oneobtainsdirectinformationontheevolvingex- the right. (b) A Bloch sphere representation of the exciton
citonspinstate. Asschematicallydescribedbythepoint spin state initiated by the polarized laser pulse. The point
P0(θ,φ) in Fig. 1(b), the spin state of the initiated exci- P0(θ,φ) represents an arbitrarily polarized spin state. A and
ton can be determined by measuring the phase (thus φ) I0, relate to the biexciton PL intensity (see Fig. 3(a)).
and the amplitude (thus θ) of the biexciton signal.
Thesampleusedinthisworkwasgrownbymolecular-
beam epitaxy on a (001) oriented GaAs substrate. One are each indicated with its cross-linearly polarized dou-
layer of strain-induced InGaAs QDs was deposited in blet. Fig. 2(b) shows polarization sensitive PL excita-
the center of a one wavelength microcavity [14, 15]. For tion(PLE)spectraoftheexcitondoubletusingonelaser
the measurements the sample was placed inside a metal source. Fig. 2(c)shows PLE spectrum of the biexciton
tube immersed in liquid Helium, maintaining tempera- doublet using two laser sources. The spectral position of
ture of 4.2K. A ×60 objective of 0.85 numerical aper- the first resonant laser, which “writes” the exciton spin
ture was used to focus the light on the sample surface state, was either tuned to the exciton PL doublet (Fig.
and to collect the emitted PL. Two dye lasers, syn- 2(a))ortoitsbroadresonance(Fig.2(b))asmarkedbya
chronously pumped at a repetition rate of 76 MHz by greenupwardarrow. Thisstrong,linearlypolarizedreso-
thesamefrequency-doubledNd:YVO (SpectraPhysics- nance, about 29 meV above X0, originates from the first
4
VanguardTM) laser were used for generating the reso- single hole and second single electron state (Fig. 1(a)).
nantly tuned light pulses. The duration of the pulses Itswidthisduetoresonantcouplingtothefirstelectronic
were 10 ps and their spectral widths about 100 µeV. level via one opticalphonon [16]. The lifetimeof this ex-
They were continuously tuned using coordinated rota- cited state as judged by its linewidth (∼1meV) is much
tions of two plate birefringent filters and an etalon. The shorter than its precession period as judged by the en-
polarizations of the pulses were independently adjusted ergydifferencebetweenitsHandVco-linearlypolarized
by a polarized beam splitter and two pairs of computer components (60µeV - Fig. 2(b)). The relaxation here,
controlledliquidcrystalvariableretarders(LCVRs). The resonantly mediated by optical phonons, is much faster
polarizationoftheemittedPLwasanalyzedbythesame than that between the corresponding heavy hole levels,
setup. The delay between the pulses was controlled by where acoustic phonons mediate it [12, 13]. The initial
a moving retroreflector. The PL was filtered by a 1- spin is thus, predominantly preserved in the relaxation.
meter monochromator, and detected by either a silicon The PLE spectrum of the biexciton doublet (Fig.
avalanche photodetector or a CCD camera. 2(c))was obtained by scanning the frequency of the sec-
Fig. 2(a) shows polarization-sensitive PL spectra of a ond laser, while the first laser was tuned into excitonic
singleQD.Theneutralexciton(X0)andbiexciton(XX0) resonance. The spectral position of the second resonant
3

XX0 (a) X0 SeTh XX0 (c) (a) (b) 1.75 (c)
T 1.5
HV ∆τ/1.25 HV
).xip/c pix.) I0 LD¯ 1LR LD¯DRLL LD¯ D¯R
es/.stc( y 2ndpulse X0 X0*(b) (cts./sec/ LL RR RR
tisnetnI 100 CCCooro--ppssoo-llpaaorriilzzaeerddiz,,e VHdVH 60eV ntensity LD 400 DR DR
I
LR LR LR
100 A
2
-3.4-3.3-0 .1 0 0.1 24 25 26 27 28 29 0 200 400 600 0 200 400 600 0 200
1stpulse E-E 0 (meV) 1stpulse ∆τ (ps)
X
FIG. 2: (a) V (red) and H (blue) polarized PL spectra from FIG. 3: (a) Emission intensity of the PL from the biexciton
the resonantly excited QD. (b) [(c)] Polarization sensitive spectral lines as a function of the delay time between the
PLEspectraoftheexciton[biexciton]. (c)wasobtainedwhile pulse into the excited exciton resonance and the pulse into
resonantlyexcitingtheexcitonasindicatedbythegreenver- the biexciton resonance, for various pulse polarizations. The
ticalarrowin(b). Forthereadoutthesecondlaserwastuned points are measured data and the solid curves are guides to
tothebiexcitonresonancemarkedbytheverticalgreenarrow the eye. The first (second) letter describes the polarization
in (c), while the PL was monitored from the XX0 doublet as of the first (second) pulse. The phase and amplitude of the
marked by the downward purple arrow in (a). oscillations are related to the polarization of the first pulse
[see Fig. 1(b)]. The phase differences between the various
curves are summarized in the inset to (b). (b) Similar to (a)
but for various polarizations of the first laser pulse and fixed
laser (the “readout”) is indicated by the green vertical
R second pulse. (c) Similar to (b) but here the first pulse is
arrow on the biexciton PLE spectrum. Full character-
tuned directly to the ground exciton states.
ization of the two-photon biexciton PLE spectrum will
be published elsewhere. The particular resonance used
here, is due to a ground state two-electron singlet and a
ground and excited state two-heavy holes triplet [SeTh - eigenstates with equal probabilities. Since these eigen-
Fig. 1(a)]. This resonance differs from the conventional states are energetically separated by 34 µeV, the exciton
biexciton transition in which both carrier pairs form sin- spinstateprecessesintimealongtheequatorofitsBloch
glets in their respective ground levels. Here, the heavy sphere with a period T = h/(34 µeV) = 122 ps. The
hole pair form a triplet [12, 13]. Therefore, while in the differences between the various curves are only in their
first case the biexciton is excited through the exciton relative phases. This dependence is summarized in the
eigenstates by co-linearly polarized photon pair, in the inset to Fig. 3(b). The inset to Fig. 3 presents the delay
latter, the excitation requires a cross-linearly polarized times ∆τ on which the second maxima are observed in
pair[13]. Weusethelatterresonanceandnottheground each one of the four curves in Fig. 3(a), in units of T.
biexciton state, to avoid blinding the PL detector. One clearly sees that there is a constant phase shift of a
In Fig. 3 we plot the PL emission intensity from the quarter of a period between the various polarizations of
biexcitondoubletasafunctionofthedelaytimebetween the second “readout” pulse. The probability to photo-
the two laser pulses, for various combinations of the two generate a biexciton by a second pulse depends on the
pulses polarizations. The first (second) capital letter de- pulsepolarizationwithrespecttotheexcitonspinpolar-
notes the polarization of the first (second) laser pulse. ization. Therefore,thesecondpulseeffectively“projects”
TheuppermostblackcurvesinFig.3(a)and(c)present theexcitonspinstateontothecomplementarydirection.
photogeneration of a biexciton by a cross linearly polar- ThismeansthataR(L)pulseprojectsthespinontothe
izedsecondpulse. ThebiexcitonPLsignalhasmaximum |L(cid:105) (|R(cid:105)) state and a D¯ (D) pulse projects the spin onto
immediately after the first pulse and it decays exponen- the |D(cid:105) (|D¯(cid:105)) state. Similarly, a H (V) pulse projects
tially as the exciton radiatively recombines. The rest of the spin onto the |V(cid:105) (|H(cid:105)) state. Thus, the state of any
thecurvesinFig.3(a)showtheexcitation(“writing”)of polarized exciton spin can be determined by the polar-
the exciton by L pulse and various polarizations of the ization of the initial pulse and a second projective light
secondpulsewhichexcitesthebiexciton(“reads”theex- pulse. Similar behavior is observed when one varies the
citon spin). In these cases, the first photon polarization polarization of the writing pulse while keeping fixed the
lies on the equator of the Poincar´e sphere, and it gen- polarizationofthereadoutpulse,asweshowinFig.3(b)
erates a coherent superposition of the two exciton spin and(c). Fig.3(c)clearlyshowsthatthereisnodifference
4
oscillations are observed in this case. The projection on
0  2
 the D direction, however, undergoes maximal variations

10 H with the angle φ. Indeed, large periodic oscillations in
the signal are observed. Clear maxima (minima) in the
8
intensity are obtained when the Bloch vector of the ex-
6 D θ L citon is antiparallel (parallel) to that of the probe. As
4 (a) expected for this particular biexcitonic resonance, maxi-

mumabsorptionisobtainedforcross-linearpolarizations
R

10 [13]. In a complementary set of measurement [Fig. 4(b)]
D the opposite behavior is observed. Here the angle θ is
8
continuously varied, while φ = 0. Now, maximal oscil-
6
lations occur for the V readout pulse and diminishing
(c) V
4 oscillations for the D readout pulse. The small oscilla-
(b)
2  tions observed in the latter are probably due to small
0  2
inaccuracies in the calibration of the LCVRs and in the
alignment of the setup axes relative to those of the QD.
FIG.4: (a)[(b)]biexcitondoubletPLintensity(locked-into Fig. 4 demonstrates that the spin state of the exciton
thesecondlaser)asafunctionofthepolarizationangleφ[θ] can be prepared at any point on the Bloch Sphere [Fig.
asdefinedin(c),forD(triangles)andforV(circles)polarized 4(c)],incorrespondencetotheellipticpolarizationofthe
readout pulses. ∆τ between the two pulses was T=122 ps. writing light pulse on its Poincar´e sphere.
In summary, we establish clear correspondence be-
in the exciton spin evolution when it is excited into the
tween the polarization of a light pulse tuned to excited
ground or into the excited state. This proves that the
or ground excitonic resonances, and the initial spin state
non-radiative relaxation is mostly spin conserving. Our
of the photogenerated exciton. We directly map the po-
experimentalmeasurementsdemonstrateunambiguously
larization of the light pulse, as represented by a point on
that a resonantly tuned H, V, D, D¯, R or L picosecond
the Poincar´e sphere, to exciton spin state as represented
pulsephotogenerates(writes)anexcitonwithinitialspin
byapointontheBlochsphere. Forthisweuseasecond,
state |H(cid:105), |V(cid:105), |D(cid:105), |D¯(cid:105), |R(cid:105), or |L(cid:105), respectively.
delayed polarized pulse tuned to particular - electronic-
In Fig. 4 we present a set of measurements in which
singlet, biexcitonic resonance. The second pulse projects
an arbitrary initial positioning of the exciton spin on its
the excitonic spin state onto a predetermined direction,
Blochsphereisdemonstrated. Thefiguredescribeswrit-
providing thus a way for reading the exciton spin.
ing of the exciton spin state by continuous variation of
the polarization of the first pulse along a given circle on The support of the US-Israel binational science foun-
thePoincar´espherewhileleavingthetemporaldelaybe- dation (BSF), the Israeli science foundation (ISF), the
tween the pulses fixed at ∆τ = T. The polarization of ministry of science and technology (MOST) and that of
the readout pulse is left fixed, either in a normal direc- the Technion’s RBNI are gratefully acknowledged.
tion to the plane of variation of the first pulse or in that
plane. Inthefirstcase,oneexpectsthebiexcitonPLsig-
nal to remain constant since the spin projection on the
probe direction is constant, independent of the in-plane
angle. In the second case, however, the spin projection ∗ Electronic address: [email protected]
[1] H. Kosaka et al., Phys. Rev. Lett. 100, 096602 (2008).
on the probe direction is expected to undergo maximal
[2] D. Press et al., Nature 456, 218 (2008).
periodic oscillations resulting in the largest amplitude of
[3] X. Li et al., Science 301, 809 (2003).
oscillations of the signal of the biexciton. In one set of
[4] S. J. Boyle et al., Phys. Rev. B 78, 075301 (2008).
measurements [Fig. 4(a)] the polarization angle is varied [5] T. Flissikowski et al., Phys. Rev. Lett. 86, 3172 (2001).
abouttheV-Haxis(φintheL-Dplane)andintheother [6] H. Kamada et al., Phys. Rev. Lett. 87, 246401 (2001).
set[Fig.4(b)]theangleisvariedabouttheD-D¯ axis(θin [7] G. Chen et al., Phys. Rev. Lett. 88, 117901 (2002).
the H-L plane). The readout in both cases is performed [8] H. Htoon et al., Phys. Rev. Lett. 88, 087401 (2002).
[9] S.J.Boyleet al.,Phys.StatusSolidiB246,824(2009).
with D pulse (triangles) and with V pulse (circles). The
[10] E. Biolatti et al., Phys. Rev. Lett. 85, 5647 (2000).
observed oscillations can be described as change in the
[11] M. Bayer et al., Phys. Rev. B 65, 195315 (2002).
magnitudeoftheprojectionoftheBlochvectoralongthe
[12] E. Poem et al., Phys. Rev. B 81, 085306 (2010).
direction of the polarization of the readout pulse. The [13] Y. Kodriano et al., Phys. Rev. B 82, 155329 (2010).
points on the equator, the circle that is defined by vary- [14] N. Akopian et al., Phys. Rev. Lett. 96, 130501 (2006).
ing φ, while leaving θ = π/2 have the same projection [15] E. Poem et al., Phys. Rev. B 76, 235304 (2007).
(equal to 0) on the V direction. Therefore, almost no [16] S. Hameau et al., Phys. Rev. Lett. 83, 4152 (1999).