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Coherent optical control of the spin of a single hole in a quantum dot T. M. Godden,1 J. H. Quilter,1 A. J. Ramsay,1,∗ Yanwen Wu,2 P. Brereton,2 S. J. Boyle,1 I. J. Luxmoore,1 J. Puebla-Nunez,1 A. M. Fox,1 and M. S. Skolnick1 1Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom 2Cavendish Laboratory, University of Cambridge, Cambridge, CB3 OHE, United Kingdom (Dated: January 9, 2012) WedemonstratecoherentopticalcontrolofasingleholespinconfinedtoanInAs/GaAsquantum dot. Asuperpositionofholespinstatesiscreatedbyfast(10-100ps)dissociationofaspin-polarized 2 electron-hole pair. Full control of the hole-spin is achieved by combining coherent rotations about 1 twoaxes: Larmorprecession of thehole-spin about an externalVoigt geometry magnetic field,and 0 2 rotationabouttheoptical-axisduetothegeometricphaseshiftinducedbyapicosecondlaserpulse resonant with the hole-trion transition. n a PACSnumbers: 78.67.Hc,42.50.Hz,03.67.Lx J 6 The principal source of dephasing of an electron spin a magnetic field is applied in-plane. A reverse bias is ] trapped on a quantum dot is the nuclear spins of the appliedsuchthatthe electrontunneling rateisfastcom- l l crystal-lattice [1]. Since the heavy-hole has a p-type, pared with the splitting between the energy-eigenstates a h rather than s-type wavefunction, the hyperfine interac- of the neutral exciton spin states. Due to a larger effec- - tionexperiencedbytheholeisaboutonetenthofthatof tivemass,theholetunnelingrateismuchslowerthanfor s e the electron due to the suppression of the contact term the electron. The sample is excited at normal incidence m [2–4]. This has stimulated interestin using the hole spin by two or three circularly polarizedpicosecondGaussian . as a qubit, encouragedby measurementsofms-scalelife- laserpulsesof0.2-meVFWHMderivedfromasingle100- t a times [5]andhighvisibilitydips incoherencepopulation fs Ti:sapphire laser. A photocurrent detection technique m trapping (CPT) experiments suggesting coherence times is used [13]. A background photocurrent is subtracted - in the microsecond regime [6]. Key requirements for the from all data. For more details on the sample and the d qubit are the ability to prepare, detect [7–9] and rotate preparation of the laser pulses, see ref. [14]. n a single hole spin. However, whilst the coherent opti- o Theprecessionofasingleholespininanappliedmag- c calcontrolofasingleelectronspinisrelativelyadvanced netic field of 4.7 T, is observed by exciting the dot with [ [11], there are no reports of the control of a hole spin. two laser pulses termed preparation and detection, sep- 2 Here we report the full coherent optical control of arated by a time-delay τ . In step (i) of fig. 1, the σ d + v a single heavy-hole spin, mJ = ±3/2, confined to an circularly polarized preparation pulse is tuned on reso- 2 InAs/GaAs quantum dot in an in-plane magnetic field. nancewiththebrightneutralexcitontransition,andhas 8 Acoherentsuperpositionoftheenergy-eigenstatesofthe a pulse-area of π. This creates a spin-polarizedelectron- 2 6 hole spin is created through the ionization of a spin- hole pair |↓⇑i. This is a superposition of the linearly . polarized electron-hole pair, where the electron tunnels polarized eigenstates of fine-structure splitting 17 µeV, 6 from the dot to leave a spin-polarized hole [8], which causing the exciton spin to precess. (ii) If the frequency 0 1 then precesses about the applied magnetic field along mismatch between the exciton and hole spin precessions 1 the x-axis. From the decay of the hole spin precession, issmallcomparedwith the electrontunneling rate[8,9], v: a dephasing time T2∗ = 15.4+−53..52 ns is deduced. This when the electron tunnels from the dot it leaves a hole i value is consistent with dephasing due to fluctuations in with a net spin-up at time zero [10]. (iii) The energy- X a nuclear magnetic field acting on the hole spin, and is eigenstatesofthehole-spinarealignedalongtheexternal r 7-13 times longer than for an electron spin confined to magnetic field B and the spin-up state is a superposi- a x an InAs/GaAs quantum dot [12], as expected from the tion of these states. This causes the hole-spin to precess weaker hyperfine interaction. Rotation of the hole-spin about B at the Larmor frequency of the in-plane hole x about the optical z-axis is achievedusing a 2π circularly Zeeman splitting. (iv) To detect the hole spin, the fre- polarized laser pulse resonant with the hole-trion tran- quency of the circularly polarized detection pulse, also sition to impart a geometric phase-shift on the selected of pulse-area π, is scanned through the hole-trion tran- spin. In this way we demonstrate the ability to perform sition [14] and a change in photocurrent recorded. Due any arbitrary rotation of the hole spin by combining ro- to Pauli blockade, creationof two holes of the same spin tations about two axes. is forbidden. (v) Therefore absorption of the detection The principle of the experiment is sketched in fig. 1. pulse resultsina changeinphotocurrentproportionalto The InAs/GaAs quantum dot which is embedded in the theoccupationoftheholespinup/downstateasselected intrinsic region of an n-i-Schottky diode structure. The by the helicity of the detection pulse. Examples of such sample is held at 4.2 K in a Helium bath cryostat, and two-colorphotocurrent spectra for co- and cross-circular 2 FIG. 1: Preparation, coherent control and detection of a single hole spin. (i) Resonant excitation of the neutral exciton transition by a laser pulse propagating along the z-axis creates a spin-polarized electron-hole pair. (ii) When the electron tunnels it leaves a spin-polarized hole that precesses about the magnetic field applied along the x-axis. (iii) Rotation of hole- spin. The hole (trion) spin-z states are coupled with in-plane Zeeman energies of ~ωh (~ωe) respectively. The σ+-polarized control pulse couples the |⇓i ↔|⇓⇑↓i states only, imparting a phase-shift on |⇓i. (iv) To detect the hole-spin, a circularly polarized laser pulse resonant with the hole-trion transition is absorbed conditional on the spin-z state of the hole. (v) When theadditional carriers created in step (iv) tunnelfrom thedot achange inphotocurrent proportional totheoccupation ofthe hole spin state selected by thehelicity of the detection pulseis measured. FIG.2: Precessionofsingleholespin(Bx =4.7 T,Vg =0.8 V). (a)Changeinphotocurrentvsdetectionpulsedetuningforco (•) and cross (+) circular excitation at various time-delays. The peak corresponds to the hole-trion transition. (b) Precession of hole spin, sz = II−−ppcc+−II++ppcc vs detection-pulse time-delay τd. I±pc is the amplitude of the photocurrent peaks measured for σ± polarized detection pulse as in (a). (solid-line) undamped cosine to guide the eye. (inset) Amplitude of Larmor precession vs τd, the traces are Gaussian decays with TL =12.2, 20.9 ns. The amplitude is determined from a sine fit to the data of (a) in therange τd±T/2, where T is theLarmor period. excitation are presented in fig. 2(a) as a function of the dotisemptyonthearrivalofthenextpreparationpulse, inter-pulsetime-delayτ . Theamplitudeofthepeaksos- but long enough to enable over 40 periods of the pre- d cillateinanti-phaseduetoLarmorprecessionofthe hole cession to be resolved. Due to hole tunneling, the total spin. The energy separation of the peaks also oscillates photocurrent signal of the trion peak becomes weak at with τ . This is probably a result of optical pumping of large time-delays, leading to the increase in the scatter d thenuclearspins,butliesoutsidethescopeofthisletter. of the data. Figure 2(b) shows the precession of the hole-spin for Byfactoringouttheholetunneling,thedampingofthe a time-delay up to 8.5 ns. The z-component of the spin Larmorprecessionin fig. 2(b) depends on the relaxation is calculated using sz = II−−ppcc−+II++ppcc, where I±pc is the am- and dephasing of the hole spin only. This assumes that plitude of the hole-trion peak measured for a detection the hole tunneling rate is independent of spin. Since no pulse of σ polarization, and plotted against the time- spin-echotechniquesareemployed,themostlikelysource ± delay τ . The frequency of the oscillation is propor- of hole-spin decoherence is dephasing due to inhomoge- d tional to the magnetic field, confirming that the oscilla- neous broadening. From Gaussian fits to the amplitude tionarisesfromacoherentsuperpositionoftwoZeeman- of the precession, shown in the inset of fig. 2(b), where split hole spin states with an in-plane hole g-factor of s ∝exp(−τ2/T2), a damping time of T =15.4+5.5 ns z d L L −3.2 g = 0.079±0.004. For the 0.8-V gate voltage used, is deduced. This is similar to the hole-spin dephasing hx the hole tunneling time is 4 ns. This is small compared time T∗ measured for an ensemble of InAs/GaAs dots 2 to the 13-ns repetition period of the laser, ensuring the [15]. It is 7-13 times longer than the 1.7 ns measured 3 FIG. 3: Coherent control of hole spin. (B = 1.13 T, V = 0.8 V) (a) (Top,bottom) Orbits of Larmor precession of hole-spin aboutmagneticfieldaxisx,beforeandafterthecontrolpulseareshownasdashedlines. Top-sphereillustratestheexperiments in(b),whereanon-resonancecontrolpulserotatesthehole-spinaboutzbyangleπ,changingphaseofprecession. Thebottom- sphereillustratestheexperimentsin(d). Thecontrolactswhenthehole-spinpointsalongy,rotatingthehole-spinby∆φz(∆c) about z, reducing the amplitude of precession. (a,middle) Pulse sequence. (b) Control of phase of precession. ∆I =Ipc−Ipc − + isplottedvsdetection-timeτd for variouscontrol-times τc. (c)Changeinstart-timeoftheprecession duetothecontrolpulse: τs = 1.99τc−69 ps. (d) Control of rotation angle ∆φz via the detuning ∆c varies the amplitude of the hole-spin precession. The control-time τc =234 ps, where the hole-spin points along y. (e) (•) Ratio R of precession amplitude normalized to total hole population, with and without thecontrol vs ∆c. (line) Calculation of cos∆φz, theideal dependenceof R [20]. by Press et al [12] for an electron spin confined to a sin- measurements of exciton Rabi rotations [19]. gle InGaAs/GaAs quantum dot. This is in-line with the We now present experiments to demonstrate an arbi- ratio of the hyperfine interaction strengths of the elec- trary rotation of the hole spin about a second axis using tronandholemeasuredforInAs/GaAsquantumdots[3]. a third laser pulse termed the control pulse. We use a Therefore we cautiously suggest that the main source of ‘geometric-phase’ approach as proposed theoretically in dephasing is the hole-nuclear spin interaction. To sup- ref. [20] and demonstrated for an ensemble of electron portthisviewpoint,estimatesofT2∗(GaAs)≈13ns,and spins in ref. [21]. If the hole-spin is represented by a T2∗(InAs) ≈ 5.4 ns were calculated [14], in good semi- vector on a Bloch-sphere, as depicted in fig. 3(a), the quantitativeagreementwiththemeasuredTL. TheTL is magnetic field leads to spin precession about the x-axis, smallcomparedto the microsecond-scaledephasingtime andthecontrolpulsetorotationaboutthebeam-pathof measured by Brunner et al [6] in a coherence population the laser, ie. the z-axis. The control pulse has circular trapping (CPT) experiment. In the CPT experiments, polarization and is resonant with the hole-trion transi- the hole-spinis aligned along the in-plane magnetic field tion, as shown in fig. 1(iii). On the timescale of the (x), whereas in our experiments, the hole-spin precesses controlpulse,the precessionsofthe holeandtrionstates in the yz-plane. We speculate that the anisotropy [2] of areeffectively stationaryandthe σ polarizedlasercou- + thehole-hyperfinecouplingleadstothedifferencesinthe plesthe|⇓i↔|↓⇑⇓istatesonly. Initially,theholespinis measured T2∗. The overall coherence time is limited by in a superposition state |ψi=h⇑ |⇑i+h⇓ |⇓i. The con- hole-tunneling and the repetition rate of the laser, but trol pulse drives a Rabi rotation between the selected this could be overcome through dynamic control of the hole spin and its corresponding trion state such that tunneling rates as in the experiments of ref. [5]. We | ψi → h |⇑i+h [cosΘ/2|⇓i+isinΘ/2|↓⇑⇓i], where ⇑ ⇓ note that, although the TL measured here is large com- Θ is the pulse-area. In the ideal case of weak trion de- paredto anelectron-spinin anInAs/GaAsquantumdot phasing, and Θ=2π, the state of the dot is returned to [12], it is similar to electron-spin values measured for the hole-spin subspace having acquired a phase-shift of much larger GaAs interface [16, 17] or electrically de- π [20]. This is also true for detuned control pulses with fined [18] quantum dots, where longer dephasing times ahyperbolic secantshape,similarto the Gaussianshape are to be expected, since the variance of the Overhauser used here, except that the z-axis rotation-angle ∆φ de- z field scales with the number of nuclei, N, as ∼ N−1/2. pends on the detuning [20]. If the carrierwavefunctionofourdotis approximatedas We first present experiments demonstrating controlof |ψ |2∼e−r2/a2, then a=3.2−3.5 nm, as deduced from the phase of the hole-spin precession using a 2π control 4 pulse. The magnetic field is reduced to 1.128 T, where which is approximately equal to the bandwidth of the the hole and trion Zeeman splittings of 5.1 and 30 µeV controlpulse,therotationangle∆φ isclosetoπ/2. This z respectively are smallcomparedto the bandwidth of the leaves the hole-spin aligned along the x-axis which sup- controlpulse. Forreference,thehole-spinprecessionwith presses the subsequent precession of the hole-spin about a period of 770 ps is measured without the controlpulse the magnetic field as shown in the blue-trace. Near res- and is shown as the lowest plot in figs. 3(b,d). The de- onance,the amplitude changessignindicating arotation tection pulse is resonant with the hole-trion transition angle of greater than π/2. The amplitude of the hole- and the difference between the photocurrents measured spinprecessionis maximalwhenthe controlisveryclose for σ detection pulses is plotted: ∆I = Ipc−Ipc. The to resonance, as shown in green. ± − + 2π-control pulse is tuned on resonance with the hole- Figure 3(e) is a plot of the ratio of the precession am- trion transition and arrives at a time-delay of τc after plitudes, normalized to the total hole population, with the preparation pulse. The hole-spin precession is mea- and without the control pulse R = s(c)/sno against the z z sured by scanning the detection time τ , and a series d detuning of the control pulse ∆ . This is measured us- c of measurements for different values of τ are presented c ing a series of two-color photocurrent spectra as in fig. in fig. 3(b). The main effect of the control pulse is to 2(b). The red-line in fig. 3(e) is a calculation of R ex- change the phase of the hole-spin precession as seen in pected for the ideal case of no trion dephasing, namely fig. 3(b). For detection times within plus or minus the R = cos(∆φ ), where tan(∆φ /2) = ∆ω /∆ , with a z z c c electron-tunneling time, a fast 138-ps period oscillation bandwidth ∆ω = 0.13 meV [20]. There is close agree- c isalsoobserved. Thisisduetoprecessionofatrioncom- mentbetweenexperimentandtheory,whichimpliesthat ponent createdby the controlpulse due to the imperfect the control-pulse rotates the hole-spin by a detuning- contrast of the hole-trion Rabi rotation [19]. dependent angle ∆φ , with a maximum value close to z The red-trace in fig. 3(b) presents the case where π, in accordance with model of ref. [20]. the hole-spin points along the z-axis when the control In conclusion, by combining coherent rotations about pulse arrives. Consequently, a rotation about the z- two axes, defined by an external magnetic field and the axis has minimal effect on the hole-spin as seen by opticalaxisofacontrollaser,fullcontrolofthehole-spin comparing the bold and red-traces of fig. 3(b). For ontheBloch-sphereisachieved. Theopticalrotationhas the blue-trace, just before applying the control pulse, a gate-time defined by the 14 ps FWHM of the control the hole-spin points along the y-axis and a rotation pulse,whichismuchshorterthanthemeasuredextrinsic of π about the z-axis phase-shifts the hole-spin preces- dephasing time of the hole spin T =15.4+5.5 ns. sion by π. More generally, the effect of the rotation L −3.3 We thank the EPSRC (UK) EP/G001642, and the is to reflect the hole-spin about the z-axis. The hole- QIPIRC UK for financial support; H. Y. Liu and spin before applying the control pulse can be written M. Hopkinson for sample growth; O. Tsyplyatyev, as s = s(0)(0,sinω τ ,cosω τ ). A reflection about h c h c E. A. Chekhovich and J. J. Finley for discussions. the z-axis maps s → s(0)(0,cosω τ ,sinω τ ), and sub- h c h c Oncompletionoftheexperimentswebecameawareof sequently the measured hole-spin precession evolves as related experiments, see ref. [22, 23]. s =cos(ω (τ −2τ )). In other words, the phase of the z h d c hole-spin is shifted by −2ω τ , as occurs in a spin-echo h c experiment. The expected gradient of 2 for the phase of the hole-spin precession ω τ is confirmed in fig. 3(c), h s where τs, defined with respect to the case of no control ∗ Electronic address: [email protected] pulse, is found by fitting the time-traces of fig. 3(b) to [1] A. Greilich et al, Science313 341 (2006). ∆I(τd)=∆I(c)cos(ωh(τd−τs)), for τd &τc+200 ps. [2] J. Fischer, et al Phys. Rev.B 78 155329 (2008). Inthefinalsetofexperiments,wedemonstratecontrol [3] P. Fallahi, S. T. Yilmaz, and A. Imamogˇlu, Phys. Rev. of the rotation angle ∆φ induced by the control pulse Lett. 105 257402 (2010). z [4] E. A. Chekhovich et al Phys. Rev. Lett. 106 027402 viathe detuning ∆ . The time-delayofthe controlpulse c (2011). is set to τ = 234 ps. On arrival of the control pulse, c [5] D. Heiss et al,Phys. Rev.B 76 241306R (2007). the hole-spin points along the y-axis, where sz is most [6] D. Brunneret al, Science 325 70 (2009). sensitive to rotations about the z-axis. A series of hole- [7] B. D.Gerardot et al, Nature445 451 (2008). spin precessions are measured for different detunings ∆c [8] A.J. Ramsay,et alPhys.Rev.Lett.100197401 (2008). of the control pulse, and the results are presented in fig. [9] T. M. Godden et al Appl.Phys.Lett.97061113 (2010). 3(d). The red-trace shows the case where the control [10] In preparation, see ArXiv soon. [11] D.PressetalNature456218(2008);J.Berezovsky,etal pulse is far detuned from the hole-trion transition. The Science320349(2008);E.D.KimetalPhys.Rev.Lett. precession is relatively unaffected by the control, since 104167401(2010);D.KimetalNat.Phys.7223(2011). thefar-detunedpulseonlyinducesasmallrotationangle. [12] D. Press et al Nat. Photon. 4 369 (2010). As the control is tuned into resonance, the amplitude of [13] A. Zrenner, et al,Nature 418 612 (2002). the precession decreases. 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