Theoretical, numerical, and experimental study of a flying qubit electronic interferometer
Tobias Bautze, Christoph Süssmeier, Shintaro Takada, Christoph Groth, Tristan Meunier, Michihisa Yamamoto, Seigo Tarucha, Xavier Waintal, Christopher Bäuerle
TTheoretical, numerical, and experimental study of a flying qubit electronicinterferometer
Tobias Bautze,
1, 2
Christoph Süssmeier,
1, 2
Shintaro Takada, Christoph Groth, Tristan Meunier,
1, 2
Michihisa Yamamoto,
3, 5
Seigo Tarucha,
3, 6
Xavier Waintal, and Christopher Bäuerle
1, 2 Univ. Grenoble Alpes, Inst. NEEL, F-38042 Grenoble, France CNRS, Inst. NEEL, F-38042 Grenoble, France Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113-8656, Japan CEA-INAC/UJF Grenoble 1, SPSMS UMR-E 9001, F-38054 Grenoble, France PRESTO, JST, Kawaguchi-shi, Saitama, 332-0012, Japan Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan (Dated: April 23, 2014)We discuss an electronic interferometer recently measured by Yamamoto et al. This "flyingquantum bit” experiment showed quantum oscillations between electronic trajectories of two tunnel-coupled wires connected via an Aharanov-Bohm ring. We present a simple scattering model as well asa numerical microscopic model to describe this experiment. In addition, we present new experimentaldata to which we confront our numerical results. While our analytical model provides basic conceptsfor designing the flying qubit device, we find that our numerical simulations allow to reproducedetailed features of the transport measurements such as in-phase and anti-phase oscillations of thetwo output currents as well as a smooth phase shift when sweeping a side gate. Furthermore, we findremarkable resemblance for the magneto conductance oscillations in both conductance and visibilitybetween simulations and experiments within a specific parameter range.
PACS numbers: 73.23.-b, 73.21.Hb, 73.63.Nm ,72.10.-d, 73.23.Ad, 03.67.-a
I. INTRODUCTION
In quantum information science, solid state approachesare attractive as they are easily scalable. The coherenttransport of information in such systems is usually codedinto the degrees of freedom of the electron, either spin orcharge . The advantage of the spin degree of freedomlies in the fact that it is well protected from the electro-static environment whereas the charge degree of freedomis easily measurable . Over the last decade, a varietyof interesting electronic devices have been proposed andtested, such as Fabry-Perot or Mach-Zehnder interfer-ometers in the quantum Hall regime . In particular,these quantum Hall systems are particularly appealing asthey could be operated as flying qubits, where the quan-tum information can be manipulated during flight, dueto the absence of backscattering. The fact that now anindividual electron charge can be controlled and manip-ulated in such systems, opens the possibility to performelectron quantum optics experiments at the single elec-tron level . In a recent experiment by Yamamoto etal. , the charge degree of freedom has been exploited togain full electrical control of a flying qubit. The systemis composed of an Aharonov-Bohm (AB) ring connectedto two tunnel-coupled wires at each side, which can actas beam splitters for the ballistic electrons. In this case,the system behaves like a Mach-Zehnder interferometerfor ballistic electrons . Compared to quantum Hallsystems, this system is more easily scalable and no mag-netic field is in principle necessary to operate the de-vice. It is made from a semiconductor heterostructureand electrostatic surface gates define the borders of theinterferometer by locally depleting the two-dimensional electron gas while the two middle gates allow to tunethe tunnel-coupling of the two wires at the entrance andexit of the AB ring (see Fig. 1) . In the original ex-periment, several interesting experimental features havebeen observed. By applying a relatively small negativegate voltage on the right tunnel gate V t and connectingthe left tunnel gate to ground, only the middle islandwill be depleted, but essentially no tunnel barrier will beformed in the tunnel-coupled wire region. In this case thesystem behaves as a two-terminal device and the currents I u and I d are identical. When sweeping the gate voltagemore negative, the tunnel barrier between the upper andlower channel can be tuned. For a sufficiently negativevoltage the in-phase oscillations of I u and I d turn sur-prisingly into anti-phase oscillations. Other interestingobservations have been made such as the possibility tocontrol the partition between the two output currents byusing the tunnel-coupling gates. It has also been shownthat this interferometer does not suffer from backscat-tered electrons which encircle the AB loop and hence al-lows to perform reliable phase shift measurements . Inthis article, we address all the experimental findings ofref. by means of a simplified theoretical model that canaccount for several features observed in the experiments.In order to capture the more subtle features we performnumerical simulations and confront them with the ex-periment. Our simulations show that the majority of theexperimentally observed features can be well explainedwithin the Landauer-Büttiker scattering formalism . a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r II. SUMMARY OF THE MAINEXPERIMENTAL FEATURES
The flying qubit sample is tailored within a two-dimensional electron gas (2DEG) of density n s = 3 . · cm − and a mobility µ = 0 . · cm / Vs made froma GaAs/AlGaAs heterostructure by metallic (Ti/Au)surface gates (see Fig. 1). The 2DEG is situated 90 nmbelow the surface, hence the electrostatic potential ap-plied to the surface gates leads to a smooth potentialchange in the 2DEG over roughly the same distance. Forthe measurements, a bias voltage is applied to the lowerleft contact (the upper left contact is floating) and thecurrent is measured simultaneously in the right upperand lower contact. To observe Aharonov-Bohm oscilla-tions, the magnetic field is swept over a field range of ap-proximately ±
80 mT. Above this field one suffers fromShubnikov-de-Haas oscillations. As the right tunnel gate
FIG. 1: Scanning electron microscope image of the flyingqubit sample. The outer metallic gates (light grey) definethe borders of the interferometer while the right tunnel gate(blue) allows for depletion of the centre AB island as well asfor adjustment of the tunnel barrier of the split-wire. Chang-ing the voltage of the side gate (red) allows to induce a phaseshift in the lower branch (see text). White squares representthe ohmic contacts. voltage V T is switched on, the sample goes through a se-ries of different regimes. Initially, the Aharonov-Bohmloop does not exist and one only observes universal con-ductance fluctuations. As V T is swept more negative, theAB region gets more depleted, yet the gate does not af-fect much the tunnel-coupled region. We refer to thisregime as the Strong Coupling Regime (SCR). Upon fur-ther increasing negative V T , one enters the regime of maininterest: the Weak Coupling Regime (WCR) where the2DEG is also partly depleted underneath the tunnel gate.In this regime, there is a finite coupling between the upand down channels in the wire region. Finally, upon fur-ther increase of V T , one enters a regime where the up-per and lower channels are decoupled. Below we list thedifferent experimental features when either scanning thetunnel gate through the different regimes or by scanningthe side gate voltage of V s (see Fig. 1) and which we at-tempt to reproduce with analytical as well as numericalapproaches. • (P1a) Magnetic field sweep in SCR: I u and I d show almost identical in-phaseoscillations with magnetic field • (P1b) Magnetic field sweep in WCR I u and I d show anti-phase oscillationswith magnetic field • (P2a) Side gate sweep in SCR:
AB oscillations show phase jumps for I u ( B ) and I d ( B ) • (P2b) Side gate sweep in WCR:
AB oscillations show a smooth phaseshift for I u and I d • (P3) Anti-phase oscillations in WCR: I u and I d show oscillations with respectto the tunnel gate voltage V T III. A MINIMUM (SCATTERING) THEORY OFTHE FLYING QUBIT.
In this section we develop a minimum scattering ap-proach which captures the main features of the exper-iment. In Fig. 2 we show a sketch of the actual de-vice used in Ref. 17. Here, the two left contacts of theoriginal device have been replaced by a single contact.This is a simplification which has no effect on the exper-imentally observed results. As we show below, the flyingqubit can be manipulated even in such a three-terminalconfiguration . The difference from the four-terminaldevice only appears as a shift of the AB oscillation phaseby π /2.The device consists of several distinct regions: the in-jecting region on the left, the central Aharonov-Bohm re-gion and the tunnel-coupled wire on the right which aremodelled by their respective scattering matrices, S inj , S ab and S tw . In this section, we assume for simplicitythat a single channel, labeled u p and d own, is propagat-ing inside each arm of the interferometer (this assump-tion will be relaxed in the numerics performed on themicroscopic model). FIG. 2: Schematic of the system used for the modelling.We divide the sample into several regions: the injection re-gion on the left, the central Aharonov-Bohm region and thetunnel-coupled wire on the right. The tunnel-coupled wire canfurther be split into three regions I, II and III: the two wiresare coupled only in region II and decoupled upon entering inregion I or III. a), b) correspond to two possible backscatteredelectron trajectories that could, in principle, contribute to thereflection amplitude.
A general property of a 2DEG is the smoothness of theelectrostatic potential generated by the surface gates feltby the electrons in the 2DEG as it is situated approxi-mately 90 nm below the surface. Here we assume thatthe scattering is mostly forward, which is valid in a spe-cific configuration. Suppose the width of each wire is keptunchanged between the ring and coupled-wire region, thepotential change ∆ V of each wire at this transition re-gion is simply defined by the tunnel coupling. For small ∆ V (smooth potential change), the length scale of thetunnelling (cid:126) v x / ∆ V becomes much larger than that ofthe potential change at the transition region, which sup-presses tunnel-backscattering into the other arm. Sincethe potential is also smooth with respect to the Fermiwave length, intra-wire backscattering is also suppressed. A. Generalities
Let us first discuss the structure of the different scatter-ing matrices for the three distinct regions. S inj is ratherarbitrary. It is characterised by the amplitude for anelectron injected from the left to be transmitted into theupper a u and lower a d channel (which, we assume, doesnot depend on the injecting channel). The probability | S bs | for an electron to be backscattered into the inject-ing electrode is obtained from the current conservationlaw | S bs | + | a u | + | a d | = 1 . The Aharonov-Bohm re-gion is translation invariant along x , and therefore, nobackscattering occurs there and the upper/lower elec-trons are simply transmitted into their respective arms.The down transmission amplitude picks up an Aharonov-Bohm phase e i π Φ / Φ with respect to the upper one where Φ = h/e is the flux quantum and Φ = SB is the mag-netic flux through the Aharonov-Bohm region ( B : mag-netic field, S : effective surface of the AB ring). Next wedescribe the tunnel-coupled wire region. If the variationof the lateral confinement potential at the transition re-gion between the AB ring and the coupled wire is smooth,the electrons can only be transmitted and ring encirclingpaths due to backscattering can be ignored. Once theelectrons are scattered into the tunnel-coupled wire, thecorresponding transmission matrix can be parametrisedas t tw = (cid:18) t uu t ud t du t dd (cid:19) (1)with t tw t † tw = 1 (current conservation). Summing up allthe probability amplitudes for the different paths takenby the electrons, we arrive at a Mach-Zehnder like ex-pression t u = t uu a u + e i π Φ / Φ t ud a d (2) t d = t du a u + e i π Φ / Φ t dd a d (3)where t u/d is the total amplitude for an electron injectedfrom the left to be transmitted in the upper/lower right electrode. The corresponding currents are given by theLandauer formula , dI u/d dV b = 2 e h T u/d . (4)Here T u/d = | t u/d | is the total transmission probabilityfrom the left to the right upper/lower electrode.Before going further, let us discuss the conclusionsthat can be already drawn at this stage. First, prop-erties (P1a) and (P2a) are rather natural: in the strongcoupling limit, the system is essentially a two-terminal Aharonov-Bohm device where forward scattering in thewire makes the upper and lower current homogeneous(P1a). Onsager symmetry imposes that G ( B ) = G ( − B ) which leads to phase rigidity as the phase of the con-ductance can only take multiples of π at zero magneticfield, as observed in many experiments . Current-conservation imposes that the injected current I inj = I bs + I u + I d , hence property (P1b) simply translates into I bs being independent on B . From the sample geometrywe can also make some assumptions about the electrontrajectories that dominantly contribute to the transportproperties. Figure 2 shows two different backscatteredtrajectories that could potentially contribute to the elec-tronic transport. In particular, in order to observe anAharonov-Bohm effect on I bs , one needs both trajecto-ries. We have argued, however, that trajectory b) can beneglected due to the smoothness of the confining poten-tial that prevents backscattering at the interface betweenthe Aharonov-Bohm and tunnel wire region, hence onlya) contributes to the reflection amplitude and I bs is essen-tially independent of the magnetic field (P1b). Property(P2b) is also due to the absence of the backscatteringtrajectory b). It straightforwardly leads to a realizationof the two-slit experiment once the electron is injectedinto the AB ring. B. Scattering matrix of the tunnel-coupled wire
Let us now focus on the tunnelling region and computethe transmission matrix t tw . As seen from Fig. 2, atthe entrance and exit of tunnel-coupled wire (region Iand III), the tunnel barrier is infinitively high and, as aconsequence, the two separated wires are fully decoupled,whereas in the central region (region II), the couplingis finite. We suppose that the transition between theregions happens smoothly.The eigenstate in region II for mode α takes the form, Ψ α ( x, y ) = Ψ α ( y ) e ik α x (5)and we consider the situation where only two modes canpropagate in the wire, hereafter labeled the symmetric | S II (cid:105) and anti-symmetric | A II (cid:105) mode. Indeed, Ψ α ( y ) cor-responds to the solution of the 1D Schrödinger equationalong the transverse direction y , [ − (cid:126) m ∂ ∂y + V ( y )]Ψ α ( y ) = E Ψ α ( y ) (6)with E = E F − (cid:126) m k α and its two solutions in the absenceof the tunnelling gate are respectively a symmetric andanti-symmetric function of y . The actual wave functionof these two eigenstates of the tunnel-coupled region aredisplayed in Fig. 4. The discussion of the evolution ofthese states when changing the tunnel barrier height ispostponed to the next subsection where we treat thisissue semi-analytically.Upon going from region I to II, | A II (cid:105) is essentially un-affected (the weight of Ψ A II ( y ) in this tunnelling region isvery small so the gate hardly affects this mode) while | S I (cid:105) is smoothly transformed into | S II (cid:105) whose wave functionis essentially Ψ S I ( y ) = | Ψ A I ( y ) | . The two modes | S I (cid:105) and | A I (cid:105) are degenerate and can also be rewritten ascombinations of the modes that propagate in the upper( |↑(cid:105) ) or lower ( |↓(cid:105) ) parts: | S I (cid:105) = |↑(cid:105) + |↓(cid:105) ; | A I (cid:105) = |↑(cid:105) − |↓(cid:105) (7)The transmission matrix t tw can now be obtained by thefollowing adiabatic argument: let us start with an elec-tron in mode |↑(cid:105) = ( | S I (cid:105) + | A I (cid:105) ) / . In the beginning ofregion II, the wave function has smoothly evolved into ( | S II (cid:105) + | A II (cid:105) ) / . Towards the end of region II, the wavefunction has picked up mode-dependent phases and reads ( e ik S L | S II (cid:105) + e ik A L | A II (cid:105) ) / where L is the total lengthof region II. Then, the wave function is smoothly trans-formed into ( e ik S L | S III (cid:105) + e ik A L | A III (cid:105) ) / which can bere-expressed as ( e ik S L [ |↑(cid:105) + |↓(cid:105) ] + e ik A L [ |↑(cid:105) − |↓(cid:105) ]) / . Wecan directly read from this expression the amplitude tobe transmitted in the up ( ( e ik S L + e ik A L ) / ) and down( ( e ik S L − e ik A L ) / ) channel. Repeating the procedure forspin down we arrive at, t tw = exp( i k S + k A L ) (cid:18) cos( k A − k S L ) i sin( k S − k A L ) i sin( k S − k A L ) cos( k A − k S L ) (cid:19) (8)Putting everything together, the Landauer formula fi-nally provides dI u/d dV b = 2 e h (cid:2) | a u | + | a d | ± | a u | − | a d | k A − k S ) L ] ± | a u a d | sin[( k A − k S ) L ] cos (2 π Φ / Φ + φ ) (cid:3) (9)Equation 9 now provides a general analytical descrip-tion of the interferometer. Changing the amplitudes a u and a d allows to control the symmetry of the injectedwave function. When injecting into one arm only, thelast term of equation 9 cancels and the system reducesto a simple split-wire. This simple analysis shows thatthe currents in the upper and lower branches oscillateanti-phase as a function of ∆ k = k A − k S . Varying ∆ k isequivalent to changing the tunnel gate voltage and henceexplains the experimentally observed oscillations with re-spect to V T [property (P3)]. In a similar way, for a given ∆ k the two output currents have opposite sign and willalso oscillate anti-phase as a function of magnetic field(P1b). C. Semi-analytical analysis
In this subsection we would like to get some more phys-ical insight into the experimental system by calculatingthe precise dependence of ∆ k on the tunnel gate volt-age V T . This can be done by numerically solving theSchrödinger equation (Eq. 6) of our system . For thiswe first implement the electrostatic potential felt by theelectrons which are at a depth of 90 nm below the sur-face gates by following the approach of Ref. 30, where theelectrostatic potential created by a polygon surface gateis calculated by solving the Laplace equation (screeningeffects by the electrons in the 2DEG are however nottaken into account). The obtained electrostatic potentialprofile of the split-wire along the y-direction for differ-ent tunnel gate voltages is shown in Fig. 3. It resemblesqualitatively the one of the experimental situation of thedata we present later on and has been used to realize thenumerical simulation in section IV. As can be seen fromFig. 3 we explore in detail the crossover region betweenthe SCR and the WCR regime. FIG. 3: Electrostatic potential V ( y ) created by the electro-static gates defining the split-wire for different tunnel gatevoltages. The horizontal lines correspond to the quantizedenergies of the symmetric and anti-symmetric state for eachtunnel gate voltage. Assuming an infinitely long tunnel-coupled wire we canthen calculate numerically the actual wave function ofthe symmetric and anti-symmetric state as displayed inFig. 4. At zero tunnel gate voltage the weight of thesymmetric state is pinned in the centre of the split-wire,whereas the anti-symmetric wave function has its weightwithin each tunnel-coupled wire. When increasing thetunnel barrier, the symmetric and the anti-symmetricwave functions are displaced differently. However, uponfurther increasing the tunnel barrier, the symmetric andanti-symmetric wave function become similar and finallydegenerate when completely decoupled.This can easily be seen when plotting the correspond-ing eigenenergies of the symmetric and anti-symmetricstate as illustrated in panel a) of Fig. 5. At zero tunnelbarrier height the energy difference is simply the energyseparation of the two lowest energy states of the poten-tial created by the approximately harmonic confinementof the outer electrostatic gates of the split-wire, whereas
FIG. 4: Normalized wave function probability of the sym-metric (left) and anti-symmetric (right) state in the tunnel-coupled wire for different tunnel gate voltages. at large tunnel barrier height, the energy difference van-ishes as the two minima of the split-wire potential aredecoupled. The absolute energy values of the two statesmove to higher energy as the confinement potential isstronger due to the strong influence of the tunnel barrierand eventually cross the Fermi energy (in our case 11.4meV, see section IV).Similarly, the dispersion relation of our system as wellas the values for ∆ k = k A − k S as a function of tunnelgate voltage and magnetic field can be evaluated, whichis detailed in the appendix. At low energy, which is ofinterest here, we observe that the energy bands for thesymmetric and anti-symmetric states are affected by thetunnel coupling as well as the magnetic field. For zeromagnetic field, ∆ k is decreasing when increasing the neg-ative tunnel gate voltage (Fig. 5b). This is expected sincethe symmetric and anti-symmetric state become degener-ate. The contrary is observed when applying a magneticfield. The influence of the magnetic field is to displace thewave functions with respect to the centre of the tunnel-coupled wire (see appendix). As a consequence, ∆ k fora given tunnel gate voltage is increasing with magneticfield. This can also be seen in the field dependence of ∆ kfor fixed tunnel gate voltages (Fig. 5c). The stronger thetunnel barrier, the stronger is the increase in ∆ k. For acompletely decoupled wire the slope of ∆ k with respectto magnetic field finally saturates. As a consequence, theelectrons will pick up an additional phase difference whentraversing the tunnel-coupled wire. This will eventuallylead to a change in oscillation period of the magneto con-ductance oscillations. We will come back to this point inSection IV.Having numerically determined the values of ∆ k fordifferent tunnel barrier heights, we can then computethe current in the upper (lower) branch using equation9. The corresponding conductance versus V T trace isshown in Fig.6. We clearly see that the two output cur-rents oscillate in anti-phase with respect to the tunnelgate voltage V T as observed in the experiment. At zerotunnel gate voltage the two output currents are equalsince the upper and lower channels are strongly coupled.Increasing the negative tunnel gate voltage induces anti- FIG. 5: a) energy dependence of the symmetric (green) andanti-symmetric (blue) state as a function of tunnel gate volt-age for B = ∆ k for different magnetic fields. c) magnetic field dependence of ∆ k for different tunnel gate voltages. d) surface area increasewith respect to the AB ring calculated using the results of c). phase oscillations until the tunnel gate completely sepa-rates the two channels (P3). This demonstrates that thetuning of the tunnel gate allows to reach a fully electri-cal control of the repartition of the output currents intothe upper/lower branch. When the two output currentsare equal, the tunnel-coupled wire behaves like a perfectbeam splitter. IV. MICROSCOPIC THEORY: MODEL ANDSIMULATION
In the preceding section we have been able to under-stand the underlying physics of the Aharonov-Bohm in-terferometer coupled to a tunnel-coupled wire by means
FIG. 6: Conductance of the upper/lower channel as a func-tion of tunnel gate voltage as given by Eq.(9). The mappingbetween V T and ∆ k was performed numerically. of a simplified analytical model (complemented with anumerical calculation of the mapping between V T and ∆ k = k A − k S ). Even though the analytical model pro-vides basic concepts for designing the flying qubit device,it relies on the assumption that encircling paths inducedby backscattering are fully suppressed. In the following,we make use of a detailed microscopic model to confirmthat we can indeed suppress the encircling paths for theweak tunnel coupling by correctly choosing the deviceconfiguration. We show that the main experimental fea-tures (P1)-(P3) are very well reproduced with the simu-lations. Neglecting screening and Coulomb interactions,our potential calculations do not allow to discuss precisequantitative agreement between experiments and simu-lations. Interestingly however, we find that for a certainparameter range, both the conductance and the visibilityof the oscillation can be tuned close to what is observedin the experiments. A. Microscopic model
In the following numerical simulations, the sample ismodelled by a simple single-band Schrödinger equationthat includes the confining potential V ( x, y ) due to thegate structure as well as an uniform magnetic field B. m [ i (cid:126) (cid:126) ∇ − e (cid:126)A ( x, y )] ψ ( x, y ) + V ( x, y ) ψ ( x, y ) = Eψ ( x, y ) (10)For the actual simulations, the model is discretized on asquare lattice with lattice constant a and we introducethe wave function ψ n x ,n y ≡ ψ ( n x a, n y a ) . The discretizedSchrödinger equation reads, − e − iφn y ψ n x +1 ,n y − e + iφn y ψ n x − ,n y − ψ n x ,n y +1 − ψ n x ,n y − + V n x ,n y ψ n x ,n y = ( E/γ + 2) ψ n x ,n y (11)where γ = (cid:126) / (2 ma ) and φ = eBa / (cid:126) . The numeri-cal calculations of the differential conductances are per-formed with the Kwant code . Kwant is a general pur- pose library designed for quantum transport. The cal-culations are done within the standard framework of theLandauer-Büttiker approach which is also equivalent,in this context, to the non-equilibrium Green’s functionformalism (NEGF) .The system that we used for the simulations is com-posed of approximately 800 ×
100 lattice sites as shownin Fig. 7a. We have taken a sufficiently large numberof lattice sites such that no influence of the discretisa-tion on the transport properties is observed and we canhence safely assume that we are in the continuum limit.The separation of the tunnel-coupled wire on the rightside has been implemented by a smooth transition to-wards the ohmic contacts. These contacts are mimickedby semi-infinite wires following the standard approach inNEGF. In order to convert the tight-binding parametersinto experimental units , we fix the Fermi energy of thesystem to 11.4 meV corresponding to an electron den-sity of n s = 3.2 · cm − . Using a lattice constant of a = 5 nm we define gates and distances in real spaceunits.In order to provide a realistic electrostatic potential as-sociated with the different electrostatic gates of the sam-ple, we follow the approach of Ref. 30 as briefly mentionedin section II. The two-dimensional potential at a depth of90 nm below the surface obtained with this approach isplotted in Fig. 7b. For convenience, we also separate theright tunnel-coupled wire from the middle island, suchthat we can sweep its voltage independently. This allowsus to investigate the influence of the tunnel-coupled wireon the AB oscillation frequency without affecting the de-pleted AB area. When performing the simulations, weadjust the gate voltages for the simulations by the fol-lowing procedure: First, we fix the energy to match theFermi energy and then sweep simultaneously all the outergates to obtain the desired conductance similar to theone of the experiment. Afterwards, we sweep the desiredparameter ( V T , V S or B ) and record the two output cur-rents. B. Comparison between numerics and experiment
In the following, we first address the issue of the mag-neto oscillations in the strong coupling regime (P1a &P2a). We apply a finite gate voltage to the centre islandin order to form an Aharanov-Bohm ring and then sweepthe magnetic field as well as the side gate voltage V S . Thesimulated data is confronted with the experimental datain Fig. 8. For all simulations we set the total conductance(transmitted as well as backscattered signal) to approxi-mately five channels, similar to the experimental condi-tions. In this regime, we can safely assume that electroninteractions can be efficiently screened and the Landauer-Büttiker approach is valid. In this two-terminal setup,the upper and lower current oscillate in-phase and sev-eral phase jumps are observed when sweeping the sidegate voltage V s as imposed by the phase rigidity. One FIG. 7: Top: lattice site model of the sample (see Fig.1). Forclarity we only display few lattice sites. In the actual samplethe lattice grid was much finer e.g. 800 ×
100 lattice sites.Bottom: electrostatic potential felt by the electrons in the2DEG, about 90 nm below the surface. also remarks the symmetry with respect to magnetic fieldas imposed by the Onsager relations. Let us note thatthe values for the side gate voltage are much smaller forthe simulated data to induce a phase jump. This dif-ference is simply due to electron screening, which is nottaken into account in the simulations. One can evaluatea scaling factor of about 20-30 by comparing the pinch-off voltages of the individual gates between experimentand simulations.
FIG. 8: Magneto conductance oscillations in the strong cou-pling regime after subtraction of a smooth background. Left:simulations, right: experimental data. Top: magneto conduc-tance oscillations for small tunnel gate voltage V T . The blue(green) curve corresponds to the current in the upper (lower)contact. Bottom: 2D colour plot of the magneto conductanceoscillations of the total transmitted current (upper - lower)as a function of side gate voltage V s . Phase jumps are clearlyobserved in the simulated as well as experimental data. The more interesting regime, however, is the weak cou- pling regime when a finite tunnel coupling is induced bythe right tunnel-coupled wire. In this case one observesthe appearance of anti-phase oscillations when impos-ing a finite gate voltage on the tunnel gate as shown inFig. 9. The simulated data (left panel) reproduces nicelythe experimentally observed anti-phase oscillations (rightpanel). From the simulations we find that anti-phaseoscillations appear even by imposing only a very smalltunnel barrier. For the present simulations V T has beenset to -8.3 mV where we are basically in a single wireregime with a very flat bottom for the electrostatic po-tential. We associate the appearance of the anti-phaseoscillations to a situation where the potential change atthe transition region between the AB ring and the tunnel-coupled wire is such that the symmetric modes of the ABring can be smoothly coupled to the symmetric and anti-symmetric modes within the tunnel-coupled wire whichfinally leads to anti-phase oscillations. It should hence bepossible to induce anti-phase oscillation in a single wirewhen shaping carefully the potential landscape. In sucha "peculiar" single wire regime, one is also able to inducea smooth shift of the AB oscillations when sweeping theside gate voltage V S . This is shown in the bottom panelsof Fig. 9. Features (P1b & P2b) are hence nicely re-produced by the simulations. The absolute amplitude ofthe conductance oscillations is very sensitive to the sidegate voltage and can vary between (0.01 - 0.1) × e /h for the investigated gate voltage scan. Let us note thatthe smoothness of the phase shift is sensitive to symme-try of the gate voltages applied to the individual gates.For instance, if we induce an asymmetry of two equiv-alent gates the phase shift becomes more irregular andthe anti-phase oscillations are not perfectly anti-phaseany more. This is also observed in the experiments.The most interesting observation of the experiment iscertainly the conductance oscillations with respect to thetunnel gate voltage in the WCR. This allows to partitionthe output current into the upper/lower channels andhence tune the tunnel-coupled wire into a beam split-ter regime. In this case the left tunnel gate is fully de-pleted to inject the current only into the lower branchof the AB interferometer. In Fig. 10 we show the sim-ulated as well as the experimental data. While at verysmall tunnel gate voltage (SCR) the two output currentsare basically equal, we observe anti-phase oscillations forboth data sets when approaching the WCR. For stronglynegative gate voltage the tunnel barrier splits the tunnel-coupled wire into two independent wires. Again the cor-respondence between experiment and simulation is quiteremarkable. Let us note, however, that in the experimentfor a gate voltage regime below the 2D pinch-off ( V T ≈ -0.3 to -0.5 V) the oscillations are suppressed. This is mostprobably due to density alignment of one of the subbandscaused by electron-electron interactions , which setsthe corresponding channel into off-resonance. Such in-fluences of the Coulomb interaction is not taken into ac-count in our model.Analysing the magneto conductance oscillations as a FIG. 9: Magneto conductance oscillations in the crossover re-gion between the SCR and WCR after subtraction of a smoothbackground: left simulations, right experimental data. Top:magneto conductance oscillations for V T = -8.3 mV. The blue(green) curve corresponds to the current in the upper (lower)contact. Bottom: 2D colour plot of the magneto conductanceoscillations of the total transmitted current (upper - lower)as a function of side gate voltage. For both data sets one ob-serves a smooth phase shift of the AB oscillations as a functionof side gate voltage V s .FIG. 10: Conductance as a function of tunnel gate voltageafter subtraction of a smooth background: left simulations,right experimental data. The blue (green) curve correspondsto the current in the upper (lower) contact. function of tunnel gate voltage V T , we observe in thenumerical simulations a change in the magneto conduc-tance oscillation frequency when passing from the SCRto the WCR as displayed in Fig. 11. For the SCR ( V T = 0V) we observe in-phase oscillations as expected andthe oscillation period corresponds simply to the surfacearea enclosed by the AB loop. When increasing the tun-nel barrier height (decrease of tunnel gate voltage) oneclearly observes an increase of the number of periods fora given magnetic field scan. This implies that the effec-tive AB surface area increases. We emphasize this bytaking the fast Fourier transform (FFT) of the magnetooscillations and by plotting the FFT peak position as afunction of the tunnel gate voltage (bottom panel). One clearly sees an increase of the Fourier peak as a functionof tunnel gate voltage as indicated by the dashed line inFig. 11. FIG. 11: Simulations of the evolution of the AB conductanceoscillations for different values of the tunnel gate voltage V T .A smooth background has been subtracted from the data.Bottom right panel: frequency of the magneto oscillations(obtained by FFT of the other panels) as a function of V T . To explain this observation, the electron has to pickup an AB phase over a significant distance in the tunnel-coupled wire. We associate this effect to the presenceof the magnetic field which displaces the wave functionwith respect to the center of the tunnel-coupled wire dueto the Lorentz force. As a consequence, the electronswill acquire an additional Aharonov-Bohm phase whichis proportional to (cid:126) e ∂ ∆ k∂B L , in agreement with the semianalytical results of Fig. 5 (bottom panel). This canbe interpreted as a surface area increase and explainsthe observed change in the magneto oscillation period inthe simulations when going from the SCR to the WCR.Note however, that the Lorentz force makes the symmet-ric and anti-symmetric states more localized in eitherof the two wires (see appendix) and induces an imbal-ance of the coupling of these states to the upper andlower wire. As a consequence, the visibility decreaseswith increasing surface area. In addition, increasing thelength of the tunnel-coupled wire enhances this surfacearea increase almost linearly. These effects could readilybe tested with the experimental set-up of ref. . Natu-rally, it would also be interesting to implement electroninteractions into the numerical simulations to allow forbetter quantitative agreement between theory and exper-iment as well as the possibility to study other effects suchas decoherence . V. CONCLUSION
We have presented a minimum scattering theory aswell as realistic simulations of an Aharonov-Bohm inter-ferometer connected to two tunnel-coupled wires, a solidstate implementation of a flying qubit. While our simpli-fied model could account for most experimental observa-tions by assuming suppression of backscattered inducedloop trajectories, our numerical simulations of the ac-tual experimental system with realistic parameters canreproduce the majority of the experimentally observedfeatures as well as suppression of multiple loops in theAB ring. These simulations are important in particu-lar for the understanding of rather subtle, geometry re-lated aspects. The good agreement between experimentand theory shows, that the physics of the flying qubit iswell described within the Landauer-Büttiker scatteringformalism. In addition to the interpretation of the ex-periments, the sort of simulations performed with Kwantcould be a useful tool for quantum device design and sig-nal optimisation.
VI. ACKNOWLEDGEMENTS
C.B. would like to thank M. Büttiker, L. Glazman,T. Jonckheere, T. Kato, T. Martin, C. Texier and R.Withney for useful discussions. T.B. acknowledges fi-nancial support from the Nanoscience Foundation Greno-ble. S. Takada acknowledges support from JSPS Re-search Fellowships for Young Scientists. M.Y. acknowl-edges financial support by Grant-in-Aid for Young Scien-tists A (no. 23684019). S. Tarucha acknowledges finan-cial support by MEXT KAKENHI "Quantum Cybernet-ics", MEXT project for Developing Innovation Systems,and JST Strategic International Cooperative Program.X.W. acknowledges financial support from the ERC grantMesoQMC. C.B. acknowledges partial financial supportfrom the French National Agency (ANR) in the frame ofits program BLANC FLYELEC project no anr-12-BS10-001 and from CNRS (DREI) - JSPS (PRC0677).
VII. APPENDIX
Dispersion relation:
For each set of experimentalparameters, e.g. tunnel gate voltage and magnetic field,we calculate the dispersion relations for the symmetricand anti-symmetric state as shown in Fig. 12 for V T =0 V and B = 0 T. From this we can extract the wavevectors in propagation direction ˆ x for each mode at theFermi energy E F and hence ∆ k . By taking k F = k y + k x we can also compute the eigenenergies of these two statesdue to confinement. Magnetic field dependence:
At zero magnetic field,the symmetric and anti-symmetric state are degenerateat high tunnel gate voltage. However, when a magneticfield is applied, the situation becomes rather subtle. To
FIG. 12: Dispersion relations for the symmetric (anti-symmetric) state in blue (red). The wave vectors in prop-agation direction have been extracted at the Fermi energy asindicated by the dashed lines. understand the underlying physics, we consider the two-dimensional Schrödinger equation (10).The vector potential can be expressed within theLandau-gauge (cid:126)A = − By ˆ e x , (12)which leads after separation of variables to [ (cid:126) k x + 2 e (cid:126) Byk x + e B y − (cid:126) ∆ + V ( y )]Ψ( y )= E Ψ( y )2 m. (13)We can now identify three spatially dependent terms:V(y) denotes the electrostatic potential in the tunnel-coupled wire created by the surface gates. The quadraticterm increases the parabolic confinement symmetrically,whereas the second term induces a tilt in the potentiallandscape which is linear in y, B and k x . This leads toa energy difference of the symmetric and anti-symmetricstate and hence to a finite ∆ k as depicted in Fig. 13. 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