Tunable quantum interference between noisy electron sources
aa r X i v : . [ c ond - m a t . o t h e r] J a n Tunable quantum interference between noisy electron sources
Yuanzhen Chen, Samir Garzon, and Richard A. Webb
Department of Physics and USC NanoCenter, University of South Carolina, Columbia, South Carolina 29208, USA (Dated: October 29, 2018)We report shot noise cross correlation measurements in a four terminal beam splitter configuration.By using two tunnel barriers as independent electron sources with tunable statistics and energy, wecan adjust the degree of quantum interference that results when the electrons scatter at a beamsplitter. Even though quantum interference is only weakly affected by noise, it can be stronglysuppressed by detuning the energies of the interfering electrons. Our results illustrate the importanceof indistinguishability for quantum interference, and its resilience to unsynchronized electron sourcesand noise.
PACS numbers: 72.70.+m, 73.23.-b, 73.40.Gk
Shot noise measurements [1] in multi-terminal meso-scopic devices have been used to probe quantum statisticsand interference [2, 3, 4, 5, 6], to understand the dynam-ics of electrons and their interaction [7, 8, 9, 10, 11, 12],and have even been proposed as probes for electron quan-tum entanglement [13, 14, 15, 16, 17, 18, 19], whichhas not yet been observed. To date, most experimentalreports on electron quantum statistics and interferenceuse three terminal configurations with a single electronsource [4, 5, 11, 12], even though configurations which usetwo electron sources are essential for measuring electronquantum entanglement. In the only two-electron sourcemeasurement reported to date, Liu et al. [3] used shotnoise cross correlation measurements to observe destruc-tive quantum interference between two noiseless electronsources. However, it was pointed out that in order to pro-ceed towards the observation of electron entanglement,the effects of noise and lack of synchronization on quan-tum interference had to be understood [20]. In this Let-ter we use shot noise cross correlation measurements toobserve quantum interference between two noisy electronsources. These tunnel barrier electron sources are uniquesince (i) the statistics of the tunneling currents can betuned over a wide range and (ii) the energy of the tun-neling electrons can be precisely controlled. Our data in-dicates that quantum interference between electrons fromthe two uncorrelated and unsynchronized sources can oc-cur even in the presence of noise. Even though quantuminterference is only weakly dependent on the amount ofnoise present in the sources, it can be strongly suppressedby decreasing the energy overlap between electrons fromboth sources. Our experiments thus provide a direct ob-servation of the fundamental relation between indistin-guishability and quantum interference.The samples used in this experiment were definedon a GaAs/AlGaAs heterostructure using electrostaticgates [21]. At low temperatures a two dimensional elec-tron gas (2DEG) with electron mobility µ e = 6 . × cm /Vs and carrier density n = 1 . × m − forms 50nm below the wafer surface. A schematic of the gates isshown in Fig. 1(a). Negative voltages are applied to gates1, 2 and 3 to form two tunnel barriers, and to gates 6, 7,and 8, to form a beam splitter at the thin section of gate -0.32 -0.28 -0.240.000.030.060.09 -0.30 -0.27 -0.240.000.030.06 0.40.60.80.30.60.9 (c)(b)(a) 1AnalyzerSpectrum cryostat C DB8 356 742 A F B F A G ( / h ) V (V) G ( / h ) V (V) FIG. 1: (a) Device schematic. A , B , C , and D are electronreservoirs, while gates 1, 2, and 3 define the two tunnel bar-riers. Large negative voltages are applied to gates 6 and 8so that the partition of electrons occurs only at the narrowsection of gate 7 between the bottom of gate 1 and the top ofgates 6 and 8. (b) & (c) Conductance and Fano factor of the(b) left and (c) right tunnel barriers.
7. The transmission coefficient t of the beam splitter canbe adjusted by changing the voltage on gate 7. Electronsinjected from reservoirs A and B tunnel through the bar-riers and are guided by additional gates 4 and 5 towardsthe beam splitter, where they scatter into channels C and D . Mean free paths in our devices are ∼ µ m and thuselectrons travel ballistically from the tunnel barrier tothe beam splitter. The current fluctuations in both chan-nels are measured by two cryogenic preamplifiers, furtheramplified at room temperature, and eventually fed into aspectrum analyzer, which calculates their cross correla-tion S [1]. All measurements are done in a 20 kHz windowaround 220 kHz and at a temperature of 70 mK. Detailsof the measurement setup and of the thermal noise back-ground subtraction procedure used to obtain the currentshot noise are described in detail elsewhere [21]. We stud-ied a total of 4 devices, repeating the measurements oneach device after multiple room temperature thermal cy-cles. The same general behavior was observed every time.Except when explicitly noted, the data reported here isfor a single device on a single cooldown. -0.28 -0.26 -0.24 -0.220.40.50.60.70.80.9 -0.260 -0.255 -0.250 -0.245 -0.240 -10-8-6-4-20 (b)(a) S ( - A / H z ) F A , F B V (V) F A F B V = -0.24 V V (V) t = 0.5 t = 0 FIG. 2: (a) F A and F B as functions of V showing that thestatistics of the two sources can be tuned independently. (b) S as a function of V at t = 0.5 and t = 0 showing that thetwo sources are uncorrelated. Tunnel barriers fabricated in semiconductor het-erostructures usually contain localized electronic states[22, 23, 24] which can be probed with conductivity andshot noise measurements. It has been previously reportedthat by tuning the gate voltages used to form such tunnelbarriers, the energy of the localized states and the cou-pling between them and the electron reservoirs can beadjusted [23]. This modifies not only average transportquantities such as the tunneling current I , but also fluc-tuations dependent on the electron transport statistics,such as the shot noise. Therefore a tunnel barrier is anelectron source with tunable statistics characterized bythe Fano factor F [1], defined as the ratio of the totalshot noise current power and 2 eI .We first characterize each of the tunnel barriers bymeasuring their conductivity and Fano factor as a func-tion of gate voltage [Figs. 1(b) and 1(c)]. Shot noise sup-pression below the ideal tunnel barrier value of F = 1 oc-curs when transport is dominated by conduction throughlocalized states, as also evidenced by the strong conduc-tance modulation with gate voltage. Therefore by ad-justing V ( V ) we can control the statistics of electronscoming from the left (right) tunnel barrier. Furthermore,we observe that the two electron sources can be tunedindependently. Figure 2(a) shows that as V changes,the Fano factor of the left barrier is strongly modulated,while the Fano factor of the right barrier remains con-stant. Similarly (data not shown), the properties of theleft barrier are unmodified when V is changed. In orderto show that the two tunnel barriers inject uncorrelatedelectrons, we measured the cross correlation S of the sig-nals at electron reservoirs C and D for both t =0 (beamsplitter completely closed) and t = 0.5. Figure 2(b) showsthat there is zero cross correlation when the beam splitteris completely closed, but a clear gate voltage dependentcross correlation when t = 0.5, demonstrating that theonly significant source of correlations occurs when theelectrons scatter at the beam splitter. Therefore, we con-clude that the tunnel barriers act as uncorrelated sourcesof electrons with independently adjustable statistics.Our goal is to quantify the quantum interference be-tween these two electron sources by measuring the cross (b)(a) S A S B S S ( e I ) t S N F A + F B FIG. 3: (a) Single source cross correlations ( S A and S B ) anddual source cross correlation ( S ) as functions of t for F A =0.45 and F B = 0.37. Solid curves are predictions using Eq. 1and the measured values of F i , I i , and t . Lower dashed curveis the sum of the two solid curves while upper dashed curveis one half of that sum. (b) S N at t = 0.5 as a function of F A + F B for three different samples (for these measurements, I A = I B = I ). Fluctuating patterns for each sample arereproducible in repetitive measurements (the size of typicalerror bars is ± correlation at reservoirs C and D . However, even sin-gle source injection produces a nonzero cross correla-tion [4, 5, 11, 12, 25]. Therefore, we first perform singlesource cross correlation measurements. Electrons are in-jected from reservoir A ( B ) through a single tunnel barrierwhile the shot noise cross correlation S A ( S B ) betweenreservoirs C and D is measured. The subindexes A and B indicate which tunnel barrier was used as an electronsource. It was previously found [25] that S A ( S B ) is re-lated to F A ( F B ), the Fano factor of the source barrier,by S i = 2 eI i ( F i − t (1 − t ) , (1)where i = A, B . The solid symbols in Fig. 3(a) showtypical single source cross correlation measurements as afunction of the beam splitter transmission coefficient. Forthe data shown here F A = 0.45 and F B = 0.37, but goodagreement of the single source cross correlation with Eq. 1(solid curves) was found for all other measured values ofthe Fano factors.For the dual source experiments, electrons are injectedfrom both reservoirs while the shot noise cross correlation S between reservoirs C and D is measured. The dualsource cross correlation data [open circles in Fig. 3(a)]show a similar dependence on t , but with larger negativevalues. For uncorrelated sources which inject distinguish-able particles, electrons from different sources scatter atthe beam splitter independently, so the total cross corre-lation is simply S indep. = S A + S B [lower dashed line inFig. 3(a)]. On the other hand, for uncorrelated sourcesof identical particles, quantum interference due to elec-tron wavefunction overlap at the beam splitter needs tobe taken into account. Theory predicts that for two un-correlated, noiseless electron sources ( F A = F B = 0), S ideal = − eIt (1 − t ) [26], where I is the average currentin each channel, and thus S ideal is only one half of S indep. [upper dashed line in Fig. 3(a)]. Since for every t themeasured dual source cross correlation is less than thatfor the case of uncorrelated and distinguishable particles( | S | < | S indep. | ), we can conclude that there is quantuminterference between electrons arriving from the two dif-ferent sources as they scatter at the beam splitter. It waspointed out that quantum interference can only occurwhen there is simultaneous arrival of electrons from bothsources at the beam splitter, or equivalently, that thereshould be enough wavefunction overlap between pairs ofelectrons at the beam splitter to make the particles in-distinguishable [20]. However, this is not guaranteed ifnoisy sources are used, since in that case the time in-terval between successive electron arrivals at the beamsplitter is a random variable, and thus interfering elec-trons are unsynchronized. As such, electrons from twouncorrelated and noisy sources could have little or nospatial overlap and quantum interference might not oc-cur. Nevertheless, the data of Fig. 3(a) clearly show thatsuch quantum interference does exist for noisy sources.To better characterize the effect of quantum inter-ference, we now define a dimensionless quantity S N = S/ ( S A + S B ). Explicitly, S N = S/ ((2 eI A ( F A −
1) + 2 eI B ( F B − t (1 − t )) . (2)With this definition, uncorrelated and distinguishableparticles, which present no quantum interference, have S N =1. On the other hand, for uncorrelated and noise-less sources emitting indistinguishable particles, quan-tum interference is maximum and S N =0.5. The resultsof measurements of S N as a function of F A + F B for t =0.5for three different samples are shown in Fig. 3(b). Foreach of the samples and each of the F A , F B combinationswe first obtained data similar to that shown in Fig. 3(a).Good agreement of S A and S B with Eq. (1) was alwaysobserved. Figure 3(b) shows that for all F A , F B combi-nations, there is always some degree of quantum interfer-ence ( S N < S N > . S N and F A , F B . This is in con-trast with the results of single source cross correlationmeasurements where S i is determined by F i as shown byEq. (1). As we will now show, the degree of quantuminterference measured by the dimensionless cross corre-lation S N is determined mainly by the energy overlap ofthe electrons coming from the two sources, and not bythe tunnel barrier Fano factors.Figure 4(a) shows S N as a function of gate voltage V at seven different values of gate voltage V . By varying V and V the energy of the localized states through whichelectrons tunnel in the two source barriers are changed.In this set of measurements, S is measured at each V , V combination and S N is calculated using Eq. (2). Again,in all measurements S N is always between 0.5 and 1, sug-gesting partial quantum interference of electrons. Fur-thermore, there is a strong dependence of S N on V and -0.28 -0.27 -0.26 -0.25 -0.24 -0.230.500.751.001.251.501.752.002.252.50 (c)(b)(a) V = -0.25 V V = -0.256 V V = -0.262 V V = -0.268 V V = -0.274 V V = -0.28 V V = -0.286 V S N V (V ) V bias (mV ) S N F A + F B V = -0.262 VV = -0.25 V FIG. 4: a) S N at t = 0.5 as a function of gate voltage V atdifferent values of gate voltage V . Curves are offset verticallyby 0.25 units for clarity. (b) Minimum values of the sevencurves in (a) plotted as a function of F A + F B . (c) S N ( t =0.5) as a function of the bias voltage applied across sourcebarriers at fixed gate voltages V and V . V . For every value of V , S N always has a minimum ata certain value V = V ,min and approaches 1 as V istuned away from V ,min . As V changes from -0.25V to-0.286V, the position of the minima shifts linearly from-0.242V to -0.266V, giving a ratio of ∆ V /∆ V = 1.5.The sensitivity of the localized state energy to changesin gate voltages can be independently measured by calcu-lating the full width at half maximum of the conductancecurves in Figs. 1(b), (c), which are 52 mV and 34 mVrespectively. The ratio ∆ V /∆ V obtained in this wayis 1.53, in agreement with the ratio obtained from thecross correlation measurements, suggesting that the gatevoltage dependence of S N in Fig. 4(a) is a measure ofthe degree of alignment of the energies of the localizedstates in the two tunnel barriers.The importance of the energy overlap between elec-trons from the two sources for quantum interference isnow explained. In most theoretical studies of quantuminterference in a beam splitter configuration, electronsare assumed to be in plane wave states with well definedwave vectors and energy. However, a more realistic pic-ture is to view electrons as energy wave packets with afinite energy broadening defined by the coupling betweenthe localized states and the reservoirs [20]. If electronsfrom the two sources have similar energies, that is, thetwo wave packets have a significant overlap on the en-ergy scale, then these electrons become indistinguishablewhen their wavefunctions overlap at the beam splitter.In such a case, quantum interference occurs and the shotnoise cross correlation is suppressed [minima in Fig. 4(a)].On the other hand if electrons from the two sources havevery different energies, then we could in principle iden-tify each electron by measuring its energy (since the dis-tance from the source to the beam splitter is smaller thanthe mean free path, electron motion is ballistic and thuselectron scattering can be ignored). In such a case, theseelectrons become effectively distinguishable even whentheir spatial wavefunctions overlap. As a result, no quan-tum interference occurs, and thus there is no shot noisecross correlation suppression [saturation towards S N =1in Fig. 4(a)].Figure 4(b) shows the minimum values (maximumquantum interference) of the seven curves shown inFig. 4(a) as a function of F A + F B . We want to pointout that the seemingly random point to point fluctu-ations are actually reproducible in repetitive measure-ments. The size of the error bars ( ∼ S N,min valueswill be obtained. Our data indicates that the changes in S N,min with F A + F B are much smaller than the changesin S N obtained when the energies of electrons from bothsources change from aligned to not aligned [Fig. 4(a)].This shows that even as the Fano factor, and thereforethe noise, of the sources varies by over a factor of two,there is very small change in the degree of quantum in-terference. This evidences that quantum interference isextremely resilient to noise and lack of synchronization.We also observed that varying the bias voltage by al-most a factor of two (for the sample studied here between0.15 mV and 0.25 mV), has no effect on quantum inter-ference. Figure 4(c) shows that as a function of the bias voltage (equal for both barriers), and for fixed gate volt-ages V and V , S N varies by less than 1%, a few timessmaller than the variation of S N with F shown in Fig.4(b). As long as the bias voltage is varied over this range,tunneling should occur through the same set of localizedstates in each barrier. Since V and V are fixed, the en-ergy of these states remains unchanged, so varying thebias voltage should have negligible effect on tunneling,and thus no effect on the quantum interference occurringat the beam splitter. Therefore quantum interference isvery insensitive to the bias voltage, weakly dependenton the source noise, but strongly affected by the energyoverlap of the electrons from each source.In summary, we performed shot noise cross correla-tion measurements in a four terminal beam splitter con-figuration using two uncorrelated tunnel barriers to in-ject electrons. The observed shot noise suppression ofelectrons leaving the beam splitter is a direct manifesta-tion of quantum interference. We observe that quantuminterference occurs even for noisy sources and that thedegree of quantum interference is only weakly sensitiveto the amount of noise present in the electron sources.Therefore, our observations show that the synchroniza-tion of electrons is not critical for observing quantum in-terference. 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