Polarized cross-correlation spectroscopy on a quantum dot charged with a
single hole shows the sequential emission of photons with common circular
polarization. This effect is visible without magnetic field, but becomes more
pronounced as the field along the quantization axis is increased. We interpret
the data in terms of electron dephasing in the X+ state caused by the
Overhauser field of nuclei in the dot. We predict the correlation timescale can
be increased by accelerating the emission rate with cavity-QED

Bennett, A. J.

Cao, Y.

Farrer, I.

Ritchie, D. A.

Shields, A. J.

English

arXiv

Polarized cross-correlation spectroscopy on a quantum dot charged with a single hole shows the sequential emission of photons with common circular polarization. This effect is visible without magnetic field, but becomes more pronounced as the field along the quantization axis is increased. We interpret the data in terms of electron dephasing in the X+ state caused by the Overhauser field of nuclei in the dot. We predict the correlation time scale can be increased by accelerating the emission rate with cavity QED

Bennett, A.J.

Ritchie, D.A.

Shields, A.J.

White Rose Research Online

RAPID COMMUNICATIONSPHYSICAL REVIEW B 92, 081302(R) (2015)Polarization-correlated photons from a positively charged quantum dotY. Cao,1,2 A. J. Bennett,1 I. Farrer,3 D. A. Ritchie,3 and A. J. Shields11Toshiba Research Europe Limited, Cambridge Research Laboratory, 208 Science Park, Milton Road, Cambridge CB4 0GZ, United Kingdom2Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, United Kingdom3Cavendish Laboratory, Cambridge University, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom(Received 13 May 2015; revised manuscript received 14 July 2015; published 7 August 2015)Polarized cross-correlation spectroscopy on a quantum dot charged with a single hole shows the sequentialemission of photons with common circular polarization. This effect is visible without magnetic field, but becomesmore pronounced as the field along the quantization axis is increased. We interpret the data in terms of electrondephasing in the X+ state caused by the Overhauser field of nuclei in the dot. We predict the correlation timescale can be increased by accelerating the emission rate with cavity QED.DOI: 10.1103/PhysRevB.92.081302 PACS number(s): 78.66.Fd, 78.67.HcSpins in quantum dots (QDs) provide a promising platformfor manipulating and storing quantum information in thesolid state. Optical measurements have demonstrated spinpreparation [1,2] coherent spin control [3], and electron-spin–photon entanglement [4,5]. There are also proposalsfor achieving photon entanglement [6] and nondestructivemeasurement of photons [7] using charged QDs. However,the time evolution of the carrier spin is unavoidably affectedby the 104–105 nuclei in the dot, all with nonzero spin.One example of the utility of the electron-nuclear interactionis its use in spin pumping the hole into the spin downstate in zero external field, by pumping on the spin upstate of X+ [2]. From a fundamental point of view then,the hyperfine interaction provides an interesting system formanipulating a mesoscopic nuclear ensemble and observing itsdynamics.It has now been established that the electron-nuclearhyperfine interaction is dominated by the contact interactionand is isotropic in a QD [8]. The dynamics of this interac-tion manifests itself in studies of polarized photolumines-cence [9,10]. Contrastingly, the hole’s p-like wave function hasa node at each nucleus, leaving the dipole-dipole interactionbetween the hole and nuclear spins to dominate [11]. Thisinteraction has a strength one order of magnitude below thatof the electron [12,13]. Thus, there has been interest in usingthe hole spin as a quantum bit with reduced decoherence.Direct measurements of the hole-spin relaxation time in avertical magnetic field T h1 have shown it is hundreds ofmicroseconds [2,14]. Without applied magnetic field, someexperiments suggest the hole spin T h1 time is 13 ns [8]. Severalstudies have now estimated the hole dephasing time T h2 inmagnetic field is approximately 1 μs [3,15].We study here the emission from the X+ state in a dotdeterministically charged with a hole using a diode [Fig. 1(a)].We show photons from this transition display polarizationcorrelation over a time scale one order of magnitude greaterthan the radiative lifetime when excited by a linearly polarizedlaser. This time is short relative to some reports of thehole-spin polarization lifetime [14], and we show that thisis a result of dephasing caused by the electron when thesystem is excited. We investigate the magnitude of the effectas a function of applied external magnetic field and radiativelifetime.The X+ consists of two holes in a singlet S = 0 state andan electron. The zero net-hole spin ensures the electron-holeanisotropic exchange interaction is absent. In zero magneticfield, the X+ eigenstates are degenerate and labeled by the spinof the electron [Fig. 1(b)] [16,17]. Radiative decay from X+to the single hole ground state occurs with a change in totalangular momentum ±1, the polarization of the photon beingcorrelated with the initial and final spin state. In a dot where thehole is “heavy” (mj = ±3/2) only the vertical transitions inFig. 1(b) are allowed: The detection of a left-handed photon (L)photon ensures the decay occurred by the left-hand transitionon Fig. 1(b).The strain, shape anisotropy, and inversion asymmetry inInGaAs/GaAs QDs ensures the optically active transition has amixed heavy-hole (mj = ±3/2) and light-hole (mj = ±1/2)character [18,19]. The state is given by φ± = (| ± 3/2〉 + β| ∓1/2〉)/√(1 + β2), which we denote ⇑ and ⇓. Recombinationof an electron and a mixed heavy/light hole now resultsin elliptically polarized photons from X+, and β may bedetermined from the emission pattern [Fig. 1(d)] [18]. Withinthe sample studied, β values from 0.02 to 0.20 are typical, andfor the data shown here, β = 0.092 [Fig. 1(d)].The diode for controlled charging has a 20 nm GaAs tunnelbarrier between the dot and p contact [Fig. 1(a)]. A 75%AlGaAs barrier on the n side prevents electron charging, so theX+ dominates at 1.2–1.3 V. Emission from X+ at an energy of1349.2 meV is excited quasiresonantly by a linearly polarizedlaser at 1317.2 meV. This excitation scheme equally excitesboth transitions in Fig. 1(b) and the absence of spin pumpingensures there is no buildup of nuclear spin polarization, but itdoes not populate other carrier combinations such as neutraland negatively charged excitons [Fig. 1(c)].The experiment is shown in Fig. 2(a). After filtering, theemission is passed to polarization-maintaining fiber opticswhich enable four simultaneous measurements of correlationin a basis selected by the quarter-wave plate (QWP) and thepolarizing coupler (PBS).Figure 2(b) presents correlations recorded at zero externalfield at an excitation power ×10 below saturation, in thecircular basis. Comparing the sum of the two copolarized andcross-polarized measurements [gco(t) in black and gcross(t) inred, respectively], we see a clear difference. Note that bothgco(t) and gcross(t) show a reduced signal within ∼1 ns of1098-0121/2015/92(8)/081302(4) 081302-1 ©2015 American Physical SocietyRAPID COMMUNICATIONSCAO, BENNETT, FARRER, RITCHIE, AND SHIELDS PHYSICAL REVIEW B 92, 081302(R) (2015)FIG. 1. (Color online) (a) Schematic band structure of the hole-charging diode when the dot contains a single hole. (b) Energylevel diagram for the X+ at zero magnetic field. (c) Spectrum underquasiresonant excitation at 1.2 V. (d) Emission pattern for an X+transition showing β = 0.092.zero-time delay due to the antibunched nature of the light.Outside the central ∼1 ns there is an enhanced probabilityof the source emitting two photons of the same circularpolarization over the case of emitting photons of oppositecircular polarization.The degree of polarization correlation C(t) is definedas C(t) = [gco(t) − gcross(t)]/[gco(t) + gcross(t)] from whicha least squares fit with a function C(t) = C0 exp(−|t |/τd )(a)(b) (c)-20 -10 0 10 20-20 -10 0 10-0.50.00.51.01.52.0 Time, t (ns)g(2)(t) & C(t) (no units)FIG. 2. (Color online) (a) Apparatus to measure the polarizationcorrelation from a dot. (b) Circular copolarized and cross-polarizedemission correlation from X+ at zero external field, gco(t) (black)and gcross(t) (red), respectively. Extracted degree of polarizationcorrelation C(t) (green). (c) The same measurement made in thelinear detection basis.extracts the polarization correlation at zero delay C0 andthe time scale τd . Empirically, this function is a good fit toC(t) [Fig. 2(b)]. C0 = 0.33 ± 0.01 and the decay time ofthe correlation τd = 9.0 ± 0.4 ns. In contrast, measurementsin the linear-polarization basis (H/V ) show an absence ofpolarization correlation [Fig. 2(c)].Nonzero heavy–light-hole mixing is an obvious source ofreduced polarization correlation. Taking the heavy:light-holeoscillator strength of 3:1 [18,19], we see that recombinationof a φ+ hole and an electron in the X+ level leads to anelliptical photon with state ∝√3|L〉 + β|R〉, and a φ− hole.Conversely, decay involving a φ− hole and an electron leadsto a ∝ √3|R〉 + β|L〉 photon. The measurement in Fig. 2(b)is in the circular basis, so detection of a left-handed photonimplies the decay came from φ+ with 3/(3 + β2) probability.In the absence of dephasing in the upper or lower states,this reduces the probability of obtaining sequential left-leftphotodetections to (32 + β4)/(3 + β2)2. For this QD, β =0.092, so the probability of copolarized photon emission isreduced to 0.994. This is higher than we have measured, so weconclude an additional factor must be included.In fact, the data can be explained by the fast dephasingof the electron spin in the upper state, which dominatesany dephasing from the hole spin. A coincidence detectionevent arises as follows: The transition emits a photon that isdetected with circular polarization and the hole spin is leftin the corresponding state. Some time later, the system isreexcited to the upper state, where electron spin dephasingoccurs during the radiative lifetime of the X+ state, followingwhich a second photon is emitted from the spontaneous decay.These two photons form a single coincidence in Fig. 2(b).We stress that our model implicitly assumes the hole-spinlifetime is greater than the measured τd , though we envisagethat in future experiments that reduce the effect of electrondephasing, it will be necessary to include the contribution ofthe hole.Our studies provide four pieces of evidence that electron-spin dephasing is the factor limiting the polarization corre-lation. First, the degree of polarization correlation from theX+ observed in Fig. 2(b) is 1/3. When excited, the unpairedelectron spin evolves through a hyperfine interaction withthe nuclei. Only those nuclear field fluctuations in the twodirections perpendicular to the spin will cause precession. Ifthis precession is faster than the radiative lifetime of the upperstate, its effect is to randomize the spin. The electron spinparallel to the nuclear field is preserved. Thus, the mean spinprojection along z is reduced to 1/3. Second, the time scaleover which polarization correlation is observed is inverselyproportional to the pump rate, as shown in Fig. 3. This cannotbe explained by dephasing occurring in the ground state. Theincreased excitation increases the number of times the systemis excited between photon detection events, and this increasesthe rate at which the polarization correlation is lost. Third, thereis no polarization correlation in the linear basis [Fig. 2(c)]. Thisis consistent with a dephasing of the electron-spin state in atime faster than the X+ radiative lifetime. Finally, we shallshow that the change in C0 with magnetic field is explained bythe dynamics of the electron spin.We next discuss the application of a Faraday magnetic field,which removes the degeneracy of the upper and lower states,081302-2RAPID COMMUNICATIONSPOLARIZATION-CORRELATED PHOTONS FROM A . . . PHYSICAL REVIEW B 92, 081302(R) (2015)0.0 0.5 1.00510Correlation time scale , τd (ns)Exciton intensity (a.u.)FIG. 3. (Color online) A measurement of the time scale of corre-lation at zero external magnetic field τd as a function of the normalizedintensity of the source. The data are fitted with an inverse relationship.shown in Fig. 4(a). The net field experienced by the spins is thesum of the external field Bext and the internal nuclear field BN .This stabilizes the electron spin along z and causes it to precessabout the sum of the two fields, which is predominantlyalong the z when the |Bext| > BN . Thus, the application ofvertical field increases the value of C0, as shown in Fig. 4(b).Figure 4(c) plots the polarization-correlation time scale τdversus magnetic field at constant laser intensity. This valuechanges from 9.0 ± 0.4 ns at zero field to 14.5 ± 0.5 ns at300 mT.A model of the dephasing of electron spin in QDswas presented by Merkolov, Efros, and Rosen [20]. In thisframework it is assumed that on time scales below 1 μsthe hyperfine interaction between the electron spin and thenuclei in the dot can be considered semiclassically as a“frozen” magnetic field, of finite variance, but no directionalpreference. The electron g factor is assumed isotropic. Thetime evolution of the electron spin S(t) (initially along S0) isFIG. 4. (Color online) Energy level diagram for the X+ transitionwith finite z magnetic field. (b) The degree of correlation C0 asa function of external Faraday magnetic field (black points). Thecalculated variation with field is shown as a red line. (c) Variation inthe time scale of correlation τd .FIG. 5. (Color online) (a) Projection of the electron spin alongthe z direction as a function of magnetic field and time, assuming anuclear field fluctuation of width δBN = 100 mT and an electron gfactor of 0.5 [21]. (b) The averaged projection of the z component ofthe spin over the radiative lifetime, scaled in terms of the electron-spindephasing time.given byS(t) = (S · n)n + {S0 − (S0 · n)n} cos(ωt)+ [{S0 − (S0 · n)n} × n] sin(ωt), (1a)W (BN ) ∝ exp[− (BN )2δB2N], (1b)where the distribution of nuclear field strengths W (BN ) isparametrized by the Gaussian width of fluctuations δBN .Figure 5(a) shows how the spin projection along the z directionSz varies with the external magnetic field Bzext [20]. At fieldsof a few times δBN , the spin projection along z is stabilized.This has not eliminated the nuclear spin fluctuations, but ithas merely overwhelmed them at the cost of increased rateof precession about the net field. Any measurement alongan orthogonal polarization direction will reveal the increasedprecession rate.From Eq. 1(a) we extract the expected final electron-spinprojection along z, Sz, which is equal to the polarizationcorrelation C0 [solid line in Fig. 4(b)]. The only fittingparameter is the width of the fluctuations in the nuclear fieldδBN , set to 100 mT. The dephasing time for this electron [8] istherefore T	 = /geμBδBN ∼ 200 ps, which is, as expected,much less than the 1 ns radiative lifetime of the upper state.This provides a good fit to the data, reproducing the value ofC0 and width around zero field. The model fits less well at thehigher fields. Partly, this can be explained by non zero β, butthe discrepancy requires further investigation.The extracted δBN is within the range derived from aspin-noise measurement [22] but is greater than inferred fromdephasing of the X0 state in similar dots [23]. We attributethis to the smaller wave-function extent of the electron whenthe dot additionally contains two holes. As δBN scales with1/√(V ), where V is the volume of spins overlapping with thewave function, there is a variation in the effective δBN betweenstates. This is the same reason the electron g factor changes inthe presence of additional holes [21].To increase C0 one could employ a host semiconduc-tor without nuclear spin, a QD of greater volume, orreduce the fluctuations in the nuclear field. Alternatively,081302-3RAPID COMMUNICATIONSCAO, BENNETT, FARRER, RITCHIE, AND SHIELDS PHYSICAL REVIEW B 92, 081302(R) (2015)Fig. 5(b) shows that reducing τrad to the electron-spin lifetimeleads to a significant increase in polarization correlation.This could be achieved by placing the dot into a cavitythat equally enhances the radiative decay, independent ofpolarization.In conclusion, despite the long hole-spin coherence time inquantum dots, the emission of polarization-correlated photonsfrom the X+ state is limited by electron-spin dephasing. Asignificant increase in the polarization-correlation time shouldbe achieved by reducing the radiative lifetime of the X+ state.Additionally, the degree of polarization correlation can beincreased by applying an external magnetic field greater thanthe nuclear field.We thank the EPSRC for funding the molecular beam epi-taxy (MBE) machine and the Controlled Quantum DynamicsCentre for Doctoral Training (CQD-CDT) which supportedY. Cao.[1] M. 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Polarization-correlated photons from a positively charged quantum dot

Y. Cao

A. J. Bennett

I. Farrer

D. A. Ritchie

A. J. Shields

Crossref

Physical Review B

Cao, Yameng

Bennett, Anthony J.

Farrer, Ian

Ritchie, David A.

Shields, Andrew J.

Lancaster E-Prints

Bennett, Anthony

Online Research @ Cardiff

https://eprints.whiterose.ac.uk/89692/1/PhysRevB.92.081302.pdf

Polarization correlated photons from a positively charged quantum dot

Polarization correlated photons from a positively charged quantum dot

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White Rose Research Online

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