Measurement of the Transmission Phase of an Electron in a Quantum Two-Path Interferometer
S. Takada, M. Yamamoto, C. Bäuerle, K. Watanabe, A. Ludwig, A. D. Wieck, S. Tarucha
MMeasurement of the Transmission Phase of an Electron in a QuantumTwo-Path Interferometer
S. Takada, a) M. Yamamoto,
1, 2
C. B¨auerle,
3, 4
K. Watanabe, A. Ludwig, A. D. Wieck, and S. Tarucha
1, 6 Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113-8656,Japan PRESTO, JST, Kawaguchi-shi, Saitama 331-0012, Japan Univ. Grenoble Alpes, Institut NEEL, F-38042 Grenoble, France CNRS, Institut NEEL, F-38042 Grenoble, France Lehrstuhl f¨ur Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum, Universit¨atsstraße 150, 44780 Bochum,Germany RIKEN, Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198,Japan (Dated: 12 October 2018)
A quantum two-path interferometer allows for direct measurement of the transmission phase shift of anelectron, providing useful information on coherent scattering problems. In mesoscopic systems, however,the two-path interference is easily smeared by contributions from other paths, and this makes it difficult toobserve the true transmission phase shift. To eliminate this problem, multi-terminal Aharonov-Bohm (AB)interferometers have been used to derive the phase shift by assuming that the relative phase shift of theelectrons between the two paths is simply obtained when a smooth shift of the AB oscillations is observed.Nevertheless the phase shifts using such a criterion have sometimes been inconsistent with theory. On theother hand, we have used an AB ring contacted to tunnel-coupled wires and acquired the phase shift consistentwith theory when the two output currents through the coupled wires oscillate with well-defined anti-phase.Here, we investigate thoroughly these two criteria used to ensure a reliable phase measurement, the anti-phaserelation of the two output currents and the smooth phase shift in the AB oscillation. We confirm that thewell-defined anti-phase relation ensures a correct phase measurement with a quantum two-path interference.In contrast we find that even in a situation where the anti-phase relation is less well-defined, the smooth phaseshift in the AB oscillation can still occur but does not give the correct transmission phase due to contributionsfrom multiple paths. This indicates that the phase relation of the two output currents in our interferometergives a good criterion for the measurement of the true transmission phase while the smooth phase shift inthe AB oscillation itself does not.The transmission phase of an electron plays a crucialrole in various quantum interference phenomena. Fullcharacterization of the coherent transport therefore re-quires a reliable phase measurement, but this is still chal-lenging. One may envisage a quantum two-path interfer-ometer because the interference is measured as a func-tion of the phase difference between the two paths. Forinstance the phase shift across a quantum dot (QD), inwhich one can control the quantum state of single elec-trons, can be measured using a QD embedded in one ofthe two arms of the interferometer. The theory predictsa Breit-Wigner type phase shift across a Coulomb peak(CP) and a π/ and both were experimentally investigated.The Breit-Wigner type phase shift was confirmed by apioneering experiment for a QD embedded in a multi-terminal Aharonov-Bohm (AB) interferometer . Thephase shift was derived from a smooth shift of AB oscil-lation phase. However, unanticipated results have some-times been observed, such as a universal phase lapse be-tween CPs for a large QD and a large phase shift ex-ceeding π across two Coulomb peaks of a spin degen-erated level for a Kondo correlated QD . Although a) Electronic mail: [email protected] several mechanisms have been proposed to account forthe universal phase lapse , origins of the behavior re-main unaccounted. This is also related to the fact thatonly a few experiments have been reported for the phasemeasurement due to difficulty in realizing a reliablephase measurement for QDs. In a two-terminal AB in-terferometer, which is usually considered as a two-pathinterferometer, the phase of the AB oscillation is fixed toeither 0 or π at zero magnetic field due to the bound-ary conditions imposed by the two-terminal geometry,whereas the real transmission phase across the QD isnot. The 0 - π rigidity of the observed phase called phaserigidity therefore implies that the two-terminal ABring is not a true two-path interferometer; because notonly direct two paths but also paths of an electron encir-cling the AB ring multiple times contribute to the inter-ference.A multi-terminal as well as a multi-channel AB interferometer was employed to avoid the phase rigid-ity and to measure the transmission phase shift across agate-defined QD embedded in one of the two arms. Inthese experiments lifting of the phase rigidity was con-firmed by observation of a smooth phase shift with gatevoltage at a fixed magnetic field. On the other handlifting of the phase rigidity does not readily ensure thatthe observed interference is a pure two-path interference. a r X i v : . [ c ond - m a t . m e s - h a ll ] J un FIG. 1. SEM picture of device A and measurement setup.Output currents are measured for a constant voltage biasacross the resistance R = 10 kΩ. Dashed lines indicate elec-tron trajectories for the two-path interference. There is a possibility that contributions from multi-pathinterferences still remain. Previously we have devel-oped a new type of interferometer realized in an AB ringcontacted to tunnel-coupled wires. It can be tuned intoa two-path interferometer in the weak tunnel-couplingregime when the two output currents through the twocoupled wires oscillate with magnetic field but oppositephase . We have used this original device to inves-tigate the transmission phase shift across a Kondo cor-related QD and obtained a very good agreement for thephase shift between experiment and theory by carefullyanalyzing the anti-phase oscillations . In addition wehave noticed that a smooth phase shift as a function ofgate voltage can be observed even when the contributionsfrom other than the direct two-paths exist. Here a ques-tion, on how reliable the phase measurement in such asituation is, is raised. This is indeed a serious problembecause all previous experiments relied on the observa-tion of such a smooth phase shift to derive the phase shiftand the results often showed disagreement with theory.In this letter, we experimentally address this question.We investigate the influence of multi-path interferenceson the phase measurement by analyzing both anti-phaseand non-anti-phase AB oscillations between the two out-put currents through the coupled wires. We show thatthe smooth phase shift at a fixed magnetic field is ob-served even when contributions of interferences from mul-tiple paths are present. In this case, however, we observeno well-defined anti-phase AB oscillations and find thatthe measured phase shift deviates significantly from thetheoretically expected transmission phase shift. In con-trast when we observe the anti-phase AB oscillation, thederived phase shift is in very good agreement with theory.We thus conclude that the anti-phase oscillations of thetwo output currents are a hallmark of a reliable phasemeasurement while the smooth phase shift as observedfor a multi-terminal geometry is not.The device was fabricated on a two-dimensional elec-tron gas formed in a GaAs/AlGaAs heterostructure [elec-tron density n = 3 . × cm − , electron mobility µ = 8 . × cm / Vs at the temperature of T = 4 . -500 ∆ V M , ( m V ) -40 -30 -20 B (mT) -20-1001020 I ( p A ) -20020 I ( p A ) -10010 I ( p A ) I ( p A ) (a)(b)(c)(d) (a)(b)(c) FIG. 2. (a), (b), (c) Quantum oscillations as a function ofmagnetic field B observed in I (black line) and I (red line)in the weak tunnel-coupling regime. Only oscillating partsextracted from raw data by performing a complex fast Fouriertransform (FFT) are plotted. Three figures are measured atthe different gate voltages of ∆ V M1 , , which are indicated in(d). (d) Modulation of geometrical phase as a function of B and ∆ V M1 , . The oscillating components with B extractedfrom a complex FFT of ( I = I − I ) are plotted in the planeof B and ∆ V M1 , . The black solid lines are added to highlightthe change of the slope. see Fig. 1]. The interferometer was defined by applyingnegative voltages on surface Schottky gates and locallydepleting electrons underneath the gates. It consists ofan AB ring at the center and tunnel-coupled wires onboth ends of the ring. The coupling energy of the tunnel-coupled wires can be controlled by the gate voltages V T1 and V T2 . The gate voltages V M1 , V M2 ( V M3 , V M4 ) areused to modulate the wave vector of electrons in the up-per (lower) path. A QD can also be formed by applyingthe gate voltages V L , V p and V R . We measured two sam-ples with a slightly different size of the AB ring and QD(device A and B). The data shown in Fig. 2 and Fig. 3was measured in device A and that in Fig. 4 for deviceB. Electrons are injected from the lower left contact byapplying an AC bias (20 ∼ µ V, 23.3 Hz) and currentsare measured at the two right contacts by voltage mea-surements across the resistance ( I = V /R ) using astandard lock-in technique.We first tuned the tunnel-coupled wires into the weakcoupling regime such that the interferometer works as atwo-path interferometer, where the two output currentsoscillate with anti-phase as shown in Fig. 2(a). For panels(a) - (c) of Fig. 2 we plot the oscillating components ofthe currents as a function of magnetic field, which areobtained by performing a complex fast Fourier transform(FFT) of the raw data, filtering out the noise outside theoscillation frequency and performing a back transform.The two-path interference is sensitive to the difference ofthe transmission phase shift between the two paths acrossthe AB ring θ = (cid:72) k · d l − e ¯ h BS + ϕ dot . The first term isthe geometrical phase depending on the path length l andthe wave vector of an electron k , the second term is theAB phase controlled by the magnetic field B penetratingthe surface area S enclosed by the two paths, and thethird term is the transmission phase shift across the QD,respectively. Fig. 2(a) shows the phase shift induced bythe modulation of the AB phase.We then measured the phase shift induced by modula-tion of the geometrical phase, where the wave vector ofelectrons passing through the upper path is controlled bythe gate voltages V M1 and V M2 . Here V M1 and V M2 areshifted simultaneously by the same amount. The resultis shown in Fig. 2(d). The oscillating part of I = I − I as a function of magnetic field, which mainly consists ofthe anti-phase components, is plotted for the gate volt-age shift V M1 , along the vertical axis around the config-uration used for the measurement of Fig. 2(a). Around∆ V M1 , = 0, where the anti-phase oscillations of the twooutput currents are observed, the phase smoothly shiftsalong the vertical axis with a certain slope. Around thegate voltage shift from − −
25 mV and the mag-netic field range from −
15 mT to −
30 mT, the phasesmoothly shifts as well but with a slightly different slopeas indicated with the black solid lines, where the two out-put currents do not oscillate with anti-phase as shown inFig. 2(b). For the more negative voltage shift and themagnetic field range from −
30 mT to −
45 mT, abruptphase jumps of π along the vertical axis are observed sim-ilarly to a two-terminal device that suffers from the phaserigidity. In this region the two output currents oscillatein phase as shown in Fig. 2(c).The anti-phase oscillations of the two output currentsindicate that the total current ( I + I ) is independenton θ . This is a clear indication that interferences com-ing from encircling paths around the AB ring are absentand hence the realization of the pure two-path interfer-ence as depicted by the dashed lines in Fig. 1. On theother hand, when the two output currents do not oscil-late with anti-phase, paths encircling the AB ring alsocontribute to the interference even though the magnetooscillations still show a smooth phase shift as a functionof gate voltages at a fixed magnetic field. In such a case,however, the observed phase shift is modified from the true transmission phase shift as we will demonstrate inthe following.First we show that the phase relation between the twooutput currents is a good criteria to exclude the contri-butions of multi-path interferences and allows for a re-liable measurement of the transmission phase shift. Forthis we carefully tuned the interferometer to observe theanti-phase oscillations as shown in Fig. 3(b). For this con-dition, we observed a smooth phase shift induced by the P ha s e ( π ) -0.60 -0.58 -0.56 V p (V) I ( n A ) -100-500 ∆ V M , ( m V ) -20-1001020 I ( p A ) I ( p A ) -35 -30 -25 -20 B (mT) -30 -25 -20 -15 -10 B (mT) -101 I ( p A ) (a)(b) (c)(d) FIG. 3. (a) Transmission phase shift by modulation of geo-metrical phase in anti-phase configuration. Quantum oscilla-tions as a function of magnetic field extracted from the FFTanalysis of ( I = I − I ) are plotted for different gate volt-age shifts of ∆ V M3 , . (b) The oscillating part of I (black)and I (red) of the data shown in (a) at ∆ V M3 , (blue dashedline). (c) Transmission phase shift across a Coulomb peak inthe anti-phase configuration. The phase obtained from exper-iment is shown by the red circles for left axis with the phasebehavior expected theoretically (red solid line). The I aver-aged over one oscillation period of magnetic field is plottedon the right axis with the Lorentzian fit of I (black solidline). (d) Oscillating part of I (black) and I (red) of thedata shown in (c) at V p indicated by the blue dashed line. modulation of the geometrical phase through V M3 , witha single constant slope [Fig. 3(a)]. At the same time wealso measure the transmission phase shift across a QD,where the experimental results can be compared withtheory [Fig. 3(c) and (d)]. The QD is formed bytuning the gate voltages V L , V p and V R and the phaseshift across a CP is observed by recording quantum os-cillations as a function of magnetic field at each value ofthe plunger gate voltage V p . This result is presented inFig. 3(c). The current I averaged over one oscillationperiod of the magnetic field mimics the shape of the CPwith a finite background current coming from the currentthrough the upper path of the AB ring. The black solidline is a Lorentzian fit of the CP, which is used to calcu-late the transmission phase shift expected from Friedel’ssum rule and depicted by the red solid line . Thenumerical values of the observed phase shift are obtainedfrom a complex FFT of ( I − I ). The observed phaseshift is in good agreement with the theoretically expected π -phase shift. This result confirms that the phase evolu-tion obtained under the condition of anti-phase oscilla-tions of the two output currents is the true transmissionphase shift observed for the pure two-path interference.We now turn to the phase shift measurements whenthe two output currents are not kept anti-phase over theentire gate voltage ( V p ) scan across a CP. The measuredphase shift is shown in Fig. 4(a). The phase smoothlyshifts across the CP by 1 . π , which is inconsistent withthe π -phase shift expected from Friedel’s sum rule (redsolid line). In this data the two output currents oscil-late with anti-phase for V p only around the center ofthe CP (red circles) as shown in Fig. 4(b). For the en- P ha s e ( π ) -0.38 -0.36 -0.34 V p (V) I ( n A ) -101 I ( p A ) -0.40.00.4 I ( p A ) -80 -75 -70 -65 -60 B (mT) (a)(b)(c) (b)(c) FIG. 4. (a) Influence of multi-path interference on the trans-mission phase shift across a Coulomb peak. The phase shiftindicated by the blue (red) circles are extracted from ( I − I )for the oscillating two output currents with poorly (well) de-fined anti-phase. The red solid line shows the calculation ofthe phase using Friedel’s sum rule. The black triangles indi-cate the measured current I at each V p and the black solidline is the Lorentzian fit. (b), (c) Quantum oscillations of thetwo output currents I in black and I in red measured at the V p indicated by (b) and (c) in (a). Oscillating componentsextracted by a complex FFT are plotted here. tire other range (blue circles) they do not oscillate withanti-phase as shown in Fig. 4(c) and hence the measuredphase shift must contain contributions from multi-pathinterferences. The larger phase shift observed here musttherefore come from the additional multi-path contribu-tions. Such contributions from multi-path interferencesmight explain the unexpected large phase shift acrossKondo correlated Coulomb peaks observed in the previ-ous experiments . Note that we consider the oscillationsas non-anti-phase when the phase difference between thetwo outputs is deviating more than 10% ( ∼ . π ) fromthe anti-phase. The phase measurements with anti-phaseoscillations within this error are in good agreement withtheoretical expectations as shown in Fig. 3(c).In the weak tunnel-coupling regime the device has fourterminals and hence each output current is not boundto the phase rigidity. This allows for observation of asmooth phase shift induced by the gate voltage at a fixedmagnetic field. However in case we fail to keep an anti-phase relation between the two output currents, the ob-tained phase shift can be modified by multi-path contri-butions and the phase shift is inconsistent with theory.Finally we discuss the key to realize a pure two-path interference in an AB ring contacted to tunnel-coupled wires. As we already pointed out in our ear- lier experimental and theoretical works, the mostimportant factor is to make the tunnel-coupling weakenough to suppress the encircling paths. In addition asmooth potential connection between the AB ring andthe tunnel-coupled wires is important. As seen fromFig. 2(d) the gate voltages V M1 and V M2 play a crucialrole to realize the anti-phase oscillations or two-path in-terference. The gate voltage V M1 and V M2 are not ef-fective for the tunnel-coupling strength but effective forthe potential profile at the transition regions between thering and the coupled wires. This suggests that the key isnot only the weak tunnel-coupling but also a smooth po-tential connection between the AB ring and the tunnel-coupled wires. In other words, one needs to suppressbackscattering of an electron into the other path at thistransition region. Indeed the importance of the smoothpotential connection is also mentioned in ref. 21. How-ever, note that “smooth here is not with respect to theFermi wave-length: since the 2DEG is 100 nm away fromthe gate electrodes, the potential profile is smooth withrespect to the Fermi wavelength for all gate voltagesin Fig. 2(d). The required smoothness depends on thetunnel-coupling energy and the potential profile of thetwo wires at the transition regions, although it is difficultto explore experimentally the detail of the connection ofthe wave function due to the existence of many channelsin each path.In summary, we employed an AB ring with tunnel-coupled wires to demonstrate how to measure the true transmission phase of an electron. We find that liftingthe phase rigidity, i.e., the observation of a smooth phaseshift at a fixed magnetic field in a multi-terminal AB in-terferometer does not ensure a correct measurement ofthe true transmission phase. Our original AB interfer-ometer, on the contrary, allows for the measurement ofthe true transmission phase shift by simply tuning it intoa regime where the two output currents oscillate anti-phase. This interferometer is hence extremely suitableto investigate unsolved problems related to the transmis-sion phase such as the universal phase behavior for largequantum dots .S. Takada acknowledges financial support from JSPSResearch Fellowships for Young Scientists, French Gov-ernment Scholarship for Scientific Disciplines and theEuropean Unions Horizon 2020 research and innovationprogram under the Marie Sklodowska-Curie grant agree-ment No 654603. M.Y. acknowledges financial supportby Grant-in-Aid for Young Scientists A (No. 23684019)and Grant-in-Aid for Challenging Exploratory Research(No. 25610070). C. B. acknowledges financial supportfrom the French National Agency (ANR) in the frameof its program BLANC FLYELEC Project No. anr-12BS10-001, as well as from DRECI-CNRS/JSPS (PRC0677) international collaboration. A.L. and A.D.W. ac-knowledge gratefully support of Mercur Pr-2013-0001,DFG-TRR160, BMBF - Q.com-H 16KIS0109, and theDFH/UFA CDFA-05-06. S. 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