Measurement and control of electron wave packets from a single-electron source
J. Waldie, P. See, V. Kashcheyevs, J.P. Griffiths, I. Farrer, G.A.C. Jones, D.A. Ritchie, T.J.B.M. Janssen, M. Kataoka
MMeasurement and control of electron wave packets from a single-electron source
J. Waldie,
1, 2
P. See, V. Kashcheyevs, J.P. Griffiths, I. Farrer, G.A.C. Jones, D.A. Ritchie, T.J.B.M. Janssen, and M. Kataoka National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, UK Cavendish Laboratory, University of Cambridge,J.J. Thomson Avenue, Cambridge CB3 0HE, UK Faculty of Physics and Mathematics, University of Latvia, Zellu Street 8, LV-1002, Riga, Latvia (Dated: November 16, 2018)We report an experimental technique to measure and manipulate the arrival-time and energydistributions of electrons emitted from a semiconductor electron pump, operated as both a single-electron source and a two-electron source. Using an energy-selective detector whose transmissionwe control on picosecond timescales, we can measure directly the electron arrival-time distributionand we determine the upper-bound to the distribution width to be 30 ps. We study the effectsof modifying the shape of the voltage waveform that drives the electron pump, and show that ourresults can be explained by a tunneling model of the emission mechanism. This information was inturn used to control the emission-time difference and energy gap between a pair of electrons.
I. INTRODUCTION
The ability to emit, coherently control and detect sin-gle electrons is highly desirable for quantum informationprocessing applications and experiments exploring thefermionic quantum behavior of electrons. Semiconduc-tor two-dimensional electron systems (2DES) in perpen-dicular magnetic fields offer the possibility of ballistic,coherent electron transport over tens of microns in chi-ral one-dimensional (1D) quantum Hall edge channels, the electronic equivalent of the fiber-optic for pho-tons. Several experiments have used electrons in 1Dedge channels, with quantum point contacts as elec-tron beam splitters, to perform electron-quantum-optics-type experiments. The realization of triggered single-electron emitters in semiconductors allows these ex-periments to be performed using single-particle states.Using a mesoscopic capacitor as a source of singleelectron-hole pairs, Bocquillon et al performed noise-correlation measurements using the Hanbury Brown andTwiss and Hong-Ou-Mandel geometries. In these ex-periments, the single particles emitted by the mesoscopiccapacitor lie close to the Fermi energy. In contrast,the tunable-barrier quantum dot electron pump caninject single electrons into edge states more than 100meV above the Fermi level.
The high electron energylimits the mixing of the emitted electrons with the low-energy Fermi sea, and this enabled Fletcher et al to mea-sure the emitted electron wavepackets, distinct from theFermi sea, with temporal resolution of ∼
80 ps. Usinga similar device geometry, Ubbelohde et al measured thepartitioning noise of electron pairs from an electron pumpincident on an electronic beam splitter, revealing regimesof independent, distinguishable or correlated partition-ing, with the origin of the latter not yet understood. These results call for more detailed studies of the electronemission process with higher temporal resolution, with aview to controlling the emitted wavepackets.Here we study the arrival-time and energy distribu- tions of electrons emitted by a single-electron pump, us-ing an energy-selective detector which we controlon picosecond timescales using an arbitrary waveformgenerator (AWG). This time-resolution allows us to mea-sure directly the time distribution of electrons arrivingat the detector, which we find to have a full-width-at-half-maximum (FWHM) of 30 ps or less. We also usethe AWG to engineer the ac voltage waveform drivingthe pump, so as to study the link between the electronemission process and the shape of the driving waveform.We observe distinct features in the electron energy dis-tribution linked to the digital nature of the waveform,which we explain using a tunneling model of the emissionprocess. Using these insights, we demonstrate manip-ulation of the electron emission by operating the pumpas a two-electron source and modifying the emission-timedifference and energy gap between the two electrons.
II. EXPERIMENTAL METHODS
Our device consists of a tunable-barrier quantum dotelectron pump and a tunable potential barrierdetector, defined in a 2DES in a GaAs/AlGaAs het-erostructure [see Fig. 1(a)]. Negative bias voltages V dcG1 and V G2 applied to gates G1 and G2 define a quantumdot region between entrance and exit barriers. The en-trance gate voltage is modulated with an ac waveform V rfG1 of frequency f = 120 MHz and peak-to-peak am-plitude 1 V, from a two-channel AWG. This modu-lation periodically lowers the entrance barrier, so thatelectrons tunnel from the left reservoir into the dot, andthen raises the dot potential so that the electrons tun-nel out to the right reservoir. An integer number, n , ofelectrons is pumped per cycle, resulting in dc pumpedcurrent I P = nef , where e is the electron charge. Wenote that the waveform produced by the AWG is a dig-ital reconstruction of a sinusoidal wave, with samplingrate 12 GS/s and analog bandwidth 5 GHz, and is trans-mitted to the sample via 50 Ω impedance co-axial cables a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l FIG. 1. (a) Scanning electron micrograph of the device, in-dicating the measured currents and applied voltages. (b) I T as a function of the static detector voltage V dcG3 . (c) Deriva-tive dI T /dV dcG3 , which is proportional to the electron energydistribution ρ E ( E ). (d) Evolution of ρ E ( E ) with V dcG1 . Datataken at V G2 = − .
535 V and B = 14 T. In (b) and (c), V dcG1 = − .
566 V, corresponding to the upper dashed line in(d). Colored dashed lines and symbols (square, circle, dia-mond) in (d) indicate values of V dcG1 that we study in Fig. 2(a)-(c). (Color online) and a bias-tee of bandwidth 12 GHz. Measurements arecarried out at 300 mK in a perpendicular magnetic fieldof 10 −
14 T (corresponding to Landau level filling fac-tors ν < µ m. At the detectorthey are either transmitted or reflected, depending on theelectron energy E relative to the detector barrier height E D = E o − βV G3 ( E o and β are constants), giving dctransmitted and reflected currents I T and I R . For a suf-ficiently low detector barrier height, we find I T ≈ I P and I R ≈ as expected for chiral edge state transport.From measurements of I T we determine the time and en-ergy distributions for electrons arriving at the detector.With constant detector voltage V G3 = V dcG3 , assumingthe detector transmission T ( E − E D ) is 1(0) for E > E D ( E < E D ), we can estimate the energy distribution from ρ E ( E ) ∝ dI T /dV dcG3 (corrections to this approximationwill be discussed later). To investigate the time distribu-tion, we add a 120 MHz square wave V rfG3 ( t ), with peak-to-peak amplitude 32 mV, to the detector, with a con-trollable time delay τ d relative to the pump drive wave-form, giving V G3 = V dcG3 + V rfG3 ( t + τ d ). The square wave is generated by the second channel of the AWG and istransmitted to the sample via 50 Ω impedance co-axialcables and a bias-tee of bandwidth 6 GHz. For suit-able values of τ d the electron wave packet will arrive atthe detector just as the square wave V rfG3 changes frompositive to negative and (for suitable V dcG3 ) the detectortransmission probability switches rapidly from 1 to 0,so only the fraction of the electron wave packet arrivingbefore the switch will be transmitted. For a perfectlysharp switch, the arrival-time distribution is given by ρ t ( t ) ∝ dI T /dτ d . A similar technique has recently beenused to measure the time-of-flight of edge magnetoplas-mons in a 2DES. This method of estimating ρ t ( t )has advantages compared to the method previously ap-plied to pumped electrons in Ref. 16, where the detectormodulation was sinusoidal and the arrival-time distribu-tion was deduced from changes in the apparent energybroadening with τ d . First, our method gives a more di-rect measurement of ρ t ( t ) and, second, it allows us to usea much smaller detector modulation amplitude, reducingback-action of the detector on the electron pump. III. ENERGY DISTRIBUTION
In Fig. 1(b)-(d) we present measurements of the elec-tron energy distribution for single-electron pumping,where I P = ef ≈
19 pA. Figure 1(b) is a typical plotof the transmitted current I T as a function of V dcG3 fora static detector ( V rfG3 = 0). As the detector barrier israised, I T decreases from ≈ I P to 0, in two main steps(there is an additional small step barely visible around V dcG3 = − .
73 V, but we do not yet know the origin ofthis step ). The corresponding energy distribution, es-timated from dI T /dV dcG3 , has two main peaks [Fig. 1(c)].We use the method of Taubert et al to determinethe conversion factor between V dcG3 and electron energy, dE/dV dcG3 ≈ − (0 . ± . e . The separation between thepeaks in the energy distribution is ≈
40 meV, consistentwith the longitudinal optic (LO) phonon energy 36 meVin GaAs within the uncertainty of our energy conver-sion factor. Therefore we attribute the lower energy peakto electrons that have emitted an LO phonon. We findthat the probability of phonon emission decreases as B is increased from 8 T to 14 T, as observed by Fletcher etal . In the following, we focus on the higher-energy peak,due to the electrons that do not emit phonons, which hasFWHM ≈ . Asobserved by Fletcher et al , we find that the energy dis-tribution shifts linearly to higher energy as the exit gatevoltage V G2 is made more negative. This is because witha higher exit barrier the electrons require more energy totunnel out of the pump. However, we see a very different FIG. 2. (a)-(c) dI T /dV dcG3 as a function of τ d at the threevalues of V dcG1 indicated by dashed lines and symbols inFig. 1(d); (a) V dcG1 = − .
566 V, (b) V dcG1 = − .
572 V, (c) V dcG1 = − .
580 V; the approximate shift of ∆ τ d = 82 ps be-tween the pattern in (a) and (c) is indicated; this shift is ap-proximately equal to the AWG sampling interval of 83.3 ps.(d) ρ t ( t ), estimated from dI T /dτ d , as we change V dcG1 from(top) − .
560 V to (bottom) − .
584 V in steps of 2 mV. Eachtrace in (d) is measured at the V dcG3 that ensures the electronenergy distribution is centered between the high and low val-ues of the modulated detector barrier height [dashed lines in(a)-(c) indicate the V dcG3 used]; colored solid traces and sym-bols in (d) correspond to the same V dcG1 as in (a) to (c). Datataken at V G2 = − .
535 V and B = 14 T. (color online) dependence on the entrance gate voltage V dcG1 , as shownin Fig 1(d). The total entrance gate voltage is the sum V G1 = V dcG1 + V rfG1 . We might expect emission to occurwhen the total voltage V G1 reaches a certain thresholdvalue. In this case a shift ∆ V dcG1 would shift the emissiontime by − ∆ V dcG1 / ( dV rfG1 /dt ), but not the emission energy.In contrast to this simple picture, Fig. 1(d) shows thatthe peak in the energy distribution follows a series of di-agonal lines as a function of V dcG1 . As V dcG1 becomes morenegative, the peak in ρ E ( E ) shifts to higher energy andthen diminishes in amplitude, being replaced by a newpeak at lower energy. In the following sections, we showhow these features arise from the details of the pump-ing waveform, giving insight into the electron emissionprocess. IV. TIME DISTRIBUTION
To gain understanding of the behavior in Fig. 1(d),we study the electron arrival-time distribution using thesquare-wave detector modulation. In Fig. 2(a)-(c) weshow how the derivative dI T /dV dcG3 changes as we sweepthe time delay τ d of the detector square wave V rfG3 , for three different values of V dcG1 [corresponding to coloredsymbols in Fig. 1(d)]. This derivative is no longer asimple measure of the energy distribution, because thetransmitted current now depends on whether the elec-trons arrive at the detector when V rfG3 is high or low.For small(large) τ d the peak in dI T /dV dcG3 is shifted tomore negative(positive) V dcG3 because the electrons arriveat the detector in the positive(negative) half of the squarewave. The position (in τ d ) of the crossover between thetwo regimes indicates the electron arrival time at the de-tector (plus a constant offset due to different propagationlengths for the two ac signals). From the horizontal shiftbetween the patterns of Fig. 2(a) and (c), we see that theelectron arrival is shifted earlier in time by approximately82 ps as we change V dcG1 from − .
566 V to − .
580 V.However, the peak in the time distribution does not shiftcontinuously with V dcG1 . At intermediate V dcG1 ( − .
572 V)the arrival-time distribution is split into two [Fig. 2(b)].We note from Fig. 1(d) that the energy distribution isalso bimodal at this V dcG1 .In Fig. 2(d) we present the arrival-time distribution ρ t ( t ), estimated from dI T /dτ d , as we vary V dcG1 from (top) − .
560 V to (bottom) − .
584 V in steps of 2 mV. Eachtrace is taken by sweeping τ d at constant V dcG3 , being care-ful to choose V dcG3 such that the entire electron energydistribution is between the high and low values of thedetector barrier. For V dcG1 = − .
560 V (top trace) thearrival-time distribution has a single peak, centered at τ d ≈
709 ps. As V dcG1 becomes more negative the timedistribution remains constant, although we know fromFig. 1(d) that the energy distribution shifts to higherenergy. However, at V dcG1 ∼ − .
570 V this peak in ρ t ( t ) weakens and a new peak emerges at τ d ≈
624 ps(85 ps earlier), which dominates the distribution for V dcG1 < − .
574 V. In the same voltage range, the peak in ρ E ( E ) is replaced by a lower energy peak. This behav-ior is periodic in V dcG1 , with the peaks in ρ t ( t ) and ρ E ( E )being replaced by new peaks at earlier time and lowerenergy, roughly every 15 mV.The narrowest time distribution in Fig. 2(d) hasFWHM ≈
30 ps, significantly narrower than the 80 ps re-sult of Fletcher et al . Thus we have achieved improvedtime resolution in the measurement of the electron wavepacket emitted by an electron pump. The improvementin resolution comes from modulating the detector barrierwith a square wave, giving faster switching of the barrierheight from high to low. From Fig. 2(a), we estimate themaximum rate of change dV G3 /dt ≈ .
26 mV/ps, nearlyfour times faster than in Ref. 16. However, this rate isstill finite and, combined with the width of the electronenergy distribution (3.5 meV), gives the main limitationto our time resolution. Electrons with different energiesare reflected/transmitted for slightly different τ d , broad-ening the measured time distribution. Therefore we be-lieve that 30 ps is likely to be an over-estimate of the truewave packet width. V. EMISSION MECHANISM
The spacing of 85 ps between the peaks in the timedistribution of Fig. 2(d) is close to the AWG samplinginterval, (12 GHz) − = 83 . V rfG1 . The AWG generates a wave-form of frequency 120 MHz by cycling through a list of100 voltage values at 12 GS/s, with the voltage updatedonce every 83.3 ps. The details of the waveform reachingthe device depend on the limited-bandwidth frequencyresponse of the signal line, which includes co-axial con-ductors, connectors, attenuators and a bias-tee. Mea-surements of the AWG waveform with an oscilloscope ofsampling rate 60 GS/s, using the same room temper-ature co-axial cables and bias-tee, show a pronouncedquasi-sinusoidal ripple of frequency 12 GHz and a peak-to-peak amplitude 14 mV superimposed on the intended120-MHz sine wave. The ripple frequency matches theAWG sampling rate 12 GS/s. In this section, we use asimple model of the electron emission process to showthat each of the peaks in the electron arrival-time andenergy distributions is due to electrons being emitted indifferent sampling intervals of the AWG waveform. Wenote that the square wave signal applied to our detectordoes not seem to show a significant 12-GHz ripple, prob-ably because a bias-tee of bandwidth 6 GHz was used forthis signal.Figure 3(a) illustrates the potential profile for the elec-tron bound in the dynamic quantum dot of the pump justbefore the electron is emitted. Following the approach ofRefs. 22 and 23, which describe the “back-tunneling” ofelectrons through the entrance barrier just after electronsare loaded into the pump, we model the electron emissionprocess of “forward-tunneling” through the exit barrier.We approximate the time-dependent entrance gate volt-age close to the emission point as [see Fig. 3(b)] V G1 ( t ) = V dcG1 − | ˙ V G1 | t + V sin(2 πf s t ) , (1)where −| ˙ V G1 | is the rate of change of the ideal 120 MHzsinusoidal waveform close to the emission point and V is the amplitude of the 12-GHz ripple. The rate of elec-tron tunneling through the exit barrier isΓ( t ) = Γ exp (cid:20) − E b ( t ) − E p ( t )∆ b (cid:21) , (2)where E b ( t ) and E p ( t ) are the exit barrier height andthe electron energy level in the pump, and ∆ b dependson the shape of the exit barrier. Equation (2) is validprovided that Γ( t ) (cid:28) Γ , i.e. electron emission is bytunneling, rather than ballistic. We assume that thelever-arm factors α b ( p ) = − dE b ( p ) /dV G1 are frequency-independent. Then we can re-write Eqn. (2) asΓ( t ) = 1 τ exp (cid:20) t − t e τ + A πf s τ sin(2 πf s t ) (cid:21) . (3) FIG. 3. (a) Electron potential profile in the electron pump atthe point of electron emission. (b) Model of V G1 ( t ) close tothe emission point; the ideal 120 MHz waveform rises approx-imately linearly (dashed line) but the AWG adds a 12-GHzripple (solid line). (c) Modeled ρ t ( t ) at the same values of V dcG1 shown in Fig. 2(d); colored solid traces and symbols indicatethe correspondence. (d)-(f) ρ E ( E ) as a function of V dcG1 (d)measured ρ E ( E ), (e) modeled ρ E ( E ) without accounting forexperimental broadening, (f) model including broadening dueto the energy-dependence of the detector barrier transmission.(color online) Here, τ − = ( α p − α b ) | ˙ V G1 | / ∆ b , A = 2 πf s V / | ˙ V G1 | and t e = V dcG1 / | ˙ V G1 | +const. Similarly, the electron energylevel E p ( t ) can be written as E p ( t ) = E + ∆ ptb (cid:20) t − t e τ + A πf s τ sin(2 πf s t ) (cid:21) , (4)where ∆ ptb is the “plunger-to-barrier ratio”, α p ∆ b / ( α p − α b ). From Eqn. (3) and the rate equation dp ( t ) /dt = − Γ( t ) p ( t ) we calculate the probability p ( t ) that the elec-tron remains in the pump at time t , and the correspond-ing emission-time distribution ρ t ( t ) = − dp/dt . Here, forthe sake of simplicity, we assume the measured arrival-time distribution to be the equal to the emission-timedistribution; we neglect the electron dispersion and thetime of flight between the pump and the detector, whichwill be the subject of a future publication. Also, we as-sume that the energy of an electron arriving at the de-tector is the same as the energy level E p ( t ) at the time ofemission, so we find the energy distribution ρ E ( E ) from ρ t ( t ) and Eqn. (4). We estimate the amplitude of the 12-GHz ripple to be V ≈ | ˙ V G1 | = 16 mV per sampling interval, and we have sep-arately estimated the lever-arm factor α p ≈ . e . Thesevalues give A ≈ .
75 and ∆ ptb ≈ τ isthe only adjustable parameter in our model, apart fromadditive constants. We note that τ is the characteris-tic timescale over which the integrated tunneling rate[Eqn. (2)] becomes large, so we expect τ to be comparableto the wave packet width in the time domain.Figure 3(c) presents the modeled emission-time distri-bution ρ t ( t ) for the same values of V dcG1 as in Fig. 2(d),using τ = (4 f s ) − ≈
20 ps. The modeled ρ t ( t ) shows aseries of peaks with separation ∼ ( f s ) − ∼
83 ps, withgradual shift in weight to earlier-time peaks as V dcG1 be-comes more negative. The peaks in ρ t ( t ) come at, or justbefore, the local maxima in − V G1 ( t ), where the emissionrate Γ( t ) is also maximized. Thus the approximately con-stant peak positions in the time distribution of Fig. 2(d)arise because the tunneling rate does not rise monotoni-cally with time but has a series of local maxima, approx-imately 83 ps apart. The experimentally measured sepa-ration between the peaks in ρ t ( t ) (85 ps) differs slightlyfrom this, probably because the ripple in the AWG wave-form is only quasi-periodic.The measured and the modeled energy distributions ρ E ( E ) for a range of V dcG1 are shown in Fig. 3(d) and(e). As in the experimental results, each peak in themodeled ρ E ( E ) shifts linearly towards higher energy as V dcG1 is made more negative, then gradually fades andis replaced by another peak at lower energy. The lin-ear shift occurs because emission is concentrated aroundthe local maxima in − V G1 ( t ) and the energy E p at thesetimes increases as we make V dcG1 more negative. Emis-sion only shifts to an earlier, lower-energy local maxi-mum for a sufficient change in V dcG1 . However, althoughthe model reproduces the positions of the peaks in ρ E ( E ),it predicts a rather different peak shape from the approx-imately Gaussian peaks in the measured energy distribu-tion. The modeled peak shape is narrower, and shows asharp peak on the higher energy side. This sharp peakcorresponds to electron emission at the local maximumin − V G1 ( t ), where dE p ( t ) /dt = 0 (a smaller sharp peakoccurs due to the small amount of emission at the localminimum). We suggest these sharp peaks in ρ E ( E ) arenot observed experimentally due to several factors thatbroaden the measured energy distribution, including theenergy-dependence of the detector barrier transmission,gate voltage noise and inelastic scattering. These broad-ening mechanisms are not easy to distinguish from one another experimentally. We include such broadening bymodeling the barrier transmission T as a non-ideal stepfunction T ( E ) = 11 + exp [ − ( E − E D ) / ∆ D ] , (5)where E D = − . eV dcG3 + const. is the height of the de-tector barrier and ∆ D quantifies the broadening. UsingEqn. (5) and the model energy distribution of Fig. 3(e),we find the energy distribution that would be measuredfrom dI T /dV dcG3 , with results shown in Fig. 3(f). The in-clusion of broadening gives much better agreement withthe experimental results and for ∆ D = 0 . τ ≈
20 ps. For muchlarger τ , the peaks in the modeled ρ t ( t ) become too broadto be consistent with the measured peak width ( ∼
30 ps).On the other hand, for τ (cid:28)
20 ps the electron emissionshifts from the local maxima in − V G1 ( t ) to the risersbefore the maxima, and this destroys the linear depen-dence of the peaks in ρ E ( E ) on V dcG1 . Therefore a valueof τ ≈
20 ps is most consistent with our experimentalobservations.We note that our observation of a 30 ps wave packetwidth may be specific to the details of the AWG pump-ing waveform that we used [Fig. 3(b)] and that a differentresult might be obtained using, for example, a pure si-nusoidal waveform as in Ref. 16. Using AWG waveformsto drive the pump gives the possibility of manipulatingthe electron wave packet. In the following sections, wedemonstrate such a technique with the pump operatedas a two-electron source.
VI. TWO-ELECTRON PUMPING
The electron pump can be operated as a source of pairsof electrons, by making V G2 less negative so that twoelectrons are trapped in the dot and pumped per cycle.Figure 4(a) shows the pumped current as a function of V dcG1 and V G2 at B = 10 T, showing clear regions where I P = nef for n = 1 , V rfG1 and for each subsequent line theemission moves to earlier time by one sampling interval( ∼
83 ps). By repeating the map of Fig. 1(d) at differ-ent V G2 , we can plot contours of constant emission time,shown as dashed lines in Fig. 4(a). To our knowledge FIG. 4. (a) I P , as a function of V dcG1 and V G2 . Dashed lines arecontours of constant electron emission time. (b) ρ E ( E ) as afunction of V dcG1 for two-electron pumping, at V G2 = − .
505 V[solid vertical line in (a)]. (c) dI T /dV dcG3 as a function of τ d for V dcG1 = − .
552 V [solid horizontal line in (b)]. (d) In-dicates the relative emission points of electrons A and B inthe pumping waveform V rfG1 . Data taken at B = 10 T. (coloronline) this is the first measurement of the specific time in thepumping cycle when electron emission occurs.We note some jitter in the edges of the constant-currentregions in the map of I P in Fig. 4(a). We believe this jit-ter is also linked to high-frequency ripples in the AWGpumping waveform. However, the period (in V dcG1 ) of thisjitter is different to the period of the dashed lines indicat-ing the emission point. The number of electrons pumpedper cycle (and hence I P ) depends on both the electroncapture and electron emission regions of the pumpingwaveform. We suggest that the jitter in the edges ofthe constant- I P regions may be more linked to the high-frequency ripple in the electron capture region, ratherthan the emission region. This point requires further in-vestigation.To study two-electron pumping, we set V G2 = − .
505 V [vertical line in Fig. 4(a)]. First we considerthe two-electron energy distribution as a function of V dcG1 ,using measurements with a static detector, as shownin Fig. 4(b). Compared with the single-electron case [Fig. 1(d)], there are now two sets of diagonal-line fea-tures, corresponding to two pumped electrons arriving atthe detector barrier with different energies. The higher-energy set of diagonal lines evolves continuously from thesingle-electron features of Fig. 1(d) as we make V G2 lessnegative. These features are due to the electron that re-mains in the pump for longest, which we label as electron“A”. The lower-energy features are due to the electronthat leaves the pump first (labelled “B”). It is notice-able that each diagonal-line feature has a slightly differ-ent length and slope, which we link to irregularities in the12-GHz ripple of the pumping waveform. Looking closelyat Fig. 4(b), we see that the features due to electron B aretranslated vertically with respect to the features due toelectron A by approximately four diagonal-lines, towardsless negative V dcG1 . Recalling that, in the single-electroncase, diagonal-lines at more negative V dcG1 are due to elec-tron emission from earlier sampling intervals, this sug-gests that electron B is emitted four sampling intervals( ≈
330 ps) earlier than electron A. We attribute this tothe increase in electrochemical potential from adding asecond electron to the pump, which has a similar effectto making V dcG1 more negative and causes earlier electronemission. We note that Fletcher et al observed a simi-lar emission-time gap between two electrons for an elec-tron pump similar to our device. We also find that theemission-time gap is increased to five sampling intervals( ≈
415 ps) on increasing the magnetic field from 10 T to14 T, which may be linked to the effect of the magneticfield on the shape of the bound-state wavefunctions inthe pump and hence the tunneling rates. .For a more accurate measurement of the two-electrontime gap, we use the square-wave detector modulation.In Fig. 4(c) we plot the derivative dI T /dV dcG3 at V dcG1 = − .
552 V [horizontal line in Fig. 4(b)] as a function ofthe square wave delay τ d . This data is the two-electronequivalent of Fig. 2(a). We see that electron B arrivesat the detector 340 ps before electron A, consistent withour estimate of four sampling intervals, and with energy ≈
12 meV below the energy of electron A. Fig. 4(d)shows schematically the relative emission points of thetwo electrons in the pumping waveform, based on our ear-lier modeling. The observed energy gap is in contrast tothe results of Ubbelohde et al , who found that for two-electron pumping using a sinusoidal waveform the elec-trons had equal emission energy, which they attributedto the out-tunneling rate Γ( t ) depending only on the dif-ference between the energy of the top-most electron andthe detector barrier height. However, our result agreeswith that of Fletcher et al , who found that the first-emitted electron (B) had lower energy and argued thatadditional factors may enhance the emission rate whentwo electrons are bound in the pump, so electron B canbe emitted with lower energy than in the single-electroncase. This may depend on the device geometry, leadingto the differing results of Refs. 16 and 18. VII. MANIPULATION OF TWO-ELECTRONEMISSION
Finally, we show how engineering of the pumping wave-form can be used to manipulate the two-electron time andenergy gap. We modify the pumping waveform V rfG1 sothat the voltage step ∆ V rfG1 during one particular sam-pling interval within the emission region is much largerthan for the rest of the steps (large step 128 mV, com-pared to 16 mV for the other steps). Figure 5(e) illus-trates this waveform schematically, including the 12-GHzripple, for the emission part of the pumping cycle. In-troducing the large step causes profound changes in theelectron emission, as shown in Fig. 5(a), where we plotthe two-electron energy distribution resulting from themodified pumping waveform as a function of V dcG1 . Fea-tures due to electron A(B) are marked with green solid(red dashed) lines as a guide to the eyes. Compared topumping with the digital sine wave [Fig. 4(b)], the mainchange is that emission during the large-step samplinginterval occurs over a much wider interval of V dcG1 , with avery large increase in energy for the most negative V dcG1 of this interval. We do not attempt to understand thissituation quantitatively, because it is not known how thefinite-bandwidth signal line will transmit the modifiedwaveform to the sample and because the large step maycause excitation of electrons to higher orbital states ofthe dot. However, we believe electron emission still oc-curs by sequential tunneling, rather than ballistically, sowe can gain some insight using the tunneling model pre-sented earlier.The tunneling model predicts that electron emission isgenerally pinned to the local maximum of − V rfG1 in oneparticular AWG sampling interval, so that the emissionenergy rises linearly as V dcG1 is made more negative, un-til emission from the local maximum in the precedingsampling interval becomes possible. For the large-stepwaveform, a very large negative shift in V dcG1 is requiredto shift emission from the local maximum at the top ofthe large step to the local maximum of the previous sam-pling interval. Therefore emission is pinned to the top ofthe large step for a wide range of V dcG1 . The total voltage V dcG1 + V rfG1 at the top of the large step increases linearlywith increasingly negative V dcG1 , pushing up the emissionenergy. However, this effect seems to saturate, shown bythe longest diagonal line features in Fig. 5(a) becomingcurved for the most negative V dcG1 , perhaps because theemission rate becomes fast enough for emission on theriser of the large step, before the local maximum.Figure 5(a) shows that with the modified pumpingwaveform it is possible for the energy gap E A − E B tobe reduced and even reversed. Once again, we use thesquare-wave detector modulation to reveal the details ofthe two-electron arrival-time difference and energy gap,with the results shown in Fig. 5(b)-(d) for three valuesof V dcG1 . In Fig. 5(b), both electrons are emitted after thelarge step in the pumping waveform so the situation isthe same as in Fig. 4(c), with the electrons emitted four sampling intervals apart, and E A − E B ≈
12 meV. Aswe make V dcG1 more negative, emission of electron A ispushed to earlier sampling intervals but emission of elec-tron B stays fixed on the large step, with a consequentincrease in E B . This makes it possible for the two elec-trons to be emitted with equal energies only ≈
200 psapart [Fig. 5(c)], or for the two electrons to be emit-ted in the same sampling interval (time gap ≈
60 ps)with reversed energy gap E B − E A ≈
13 meV [Fig. 5(d)].Figures 5(e)-(g) show the approximate emission pointsfor the two electrons, based on the emission times fromFig. 5(b)-(d). These results demonstrate the potential ofthis technique to control the two-electron wave packet.
VIII. CONCLUSION
In conclusion, we have measured the arrival-time andenergy distributions of single electrons and pairs of elec-trons emitted from a semiconductor electron pump, withsufficient time-resolution to determine an upper boundof 30 ps FWHM for the width of the single-electronarrival-time distribution. Our measurement techniquehas the potential for even further improvement in time-resolution, by increasing the rate of the detector mod-ulation and by reducing cross-talk between the pumpand the detector. We have shown how the details ofthe waveform used to drive the electron pump affect theelectron emission, in agreement with a tunneling modelof the emission process. This enables manipulation ofthe electron time and energy distributions, which wasdemonstrated using the example of controlling the timedifference and energy gap between a pair of electrons.Measurement and control of the two-electron time andenergy gap could be particularly useful when combinedwith measurements of the electron partitioning noise, for studying the factors that determine the degree ofcorrelation within the emitted electron pairs. Our re-sults highlight the potential of the semiconductor elec-tron pump as an on-demand emitter of single electronsand pairs of electrons, with fine control of the emissiontime, energy and wave packet shape. We expect this tofind application in studies of fermionic quantum behaviorand preparation of electron states for quantum informa-tion processing. ACKNOWLEDGMENTS
We thank J.D. Fletcher, S.P. Giblin andD.A. Humphreys for useful discussions and C.A. Nicolland R.D. Hall for technical assistance. This research wassupported by the UK Department for Business, Innova-tion and Skills, NPL’s Strategic Research Programmeand the UK EPSRC. V.K. has been supported by theLatvian Council of Science within research projectno. 146/2012.
FIG. 5. (a) ρ E ( E ) as a function of V dcG1 using the modified pumping waveform; features due to electron A(B) are coded withgreen solid (red dashed) lines. (b)-(d) dI T /dV dcG3 as a function of τ d for the three values of V dcG1 shown by dashed lines in(a). (e)-(g) Schematic sketches of the modified pumping waveform, showing the approximate electron emission points for cases(b)-(d). Data taken at B = 10 T and V G2 = − .
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