Aharonov-Bohm interferometry with a tunnel-coupled wire
A. Aharony, S. Takada, O. Entin-Wohlman, M. Yamamoto, S. Tarucha
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Aharonov-Bohm interferometry with atunnel-coupled wire
A. Aharony , , , S. Takada , O. Entin-Wohlman , , , M.Yamamoto , and S. Tarucha , Department of Physics, Ben Gurion University, Beer Sheva 84105, Israel Ilse Katz Center for Meso- and Nano-Scale Science and Technology, Ben GurionUniversity, Beer Sheva 84105, Israel Raymond and Beverly Sackler School of Physics and Astronomy, Tel AvivUniversity, Tel Aviv 69978, Israel Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo, 113-8656,Japan PRESTO-JST, Kawaguchi-shi, Saitama, 332-0012, Japan Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama, 351-0198,JapanE-mail: [email protected]
Abstract.
Recent experiments [M. Yamamoto et al. , Nature Nanotechnology , 247(2012)] used the transport of electrons through an Aharonov-Bohm interferometerand two coupled channels (at both ends of the interferometer) to demonstrate amanipulable flying qubit. Results included in-phase and anti-phase Aharonov-Bohm(AB) oscillations of the two outgoing currents as a function of the magnetic flux,for strong and weak inter-channel coupling, respectively. Here we present newexperimental results for a three terminal interferometer, with a tunnel couplingbetween the two outgoing wires. We show that in some limits, this system is aneven simpler realization of the ‘two-slit’ experiment. We also present a simple tight-binding theoretical model which imitates the experimental setup. For weak inter-channel coupling, the AB oscillations in the current which is reflected from the deviceare very small, and therefore the oscillations in the two outgoing currents must canceleach other, yielding the anti-phase behavior, independent of the length of the couplingregime. For strong inter-channel coupling, whose range depends on the asymmetrybetween the channels, and for a relatively long coupling distance, all except two of thewaves in the coupled channels become evanescent. For the remaining running wavesone has a very weak dependence of the ratio between the currents in the two channelson the magnetic flux, implying that these currents are in phase with each other. B interferometry with a tunnel-coupled wire
1. Introduction
In a recent paper, [1] some of us demonstrated a scalable flying qubit architecture in afour-terminal setup. Electrons were transported via an Aharonov-Bohm (AB) ring intotwo channel wires that have a tunable tunnel coupling between them. The superpositionof the two electron states between the two outgoing channels can be considered as a flyingqubit, which can be manipulated by the various gate voltages on the system. That paperalso exhibited an interesting variation of the relative phases of the AB oscillations in thecurrents in the two outgoing channels, as function of the coupling between them. In thepresent paper we present similar experimental results for a new (simpler) three-terminalsetup, and then present a simple theoretical model which reproduces them.The early experiments on two-terminal ‘closed’ AB interferometers [2] exhibiteda phase rigidity: The minima and maxima of the AB oscillations in the outgoingcurrent stayed at the same values of the magnetic flux through the interferometerring, irrespective of the details of a quantum dot which was placed on one arm of theinterferometer. At most, the phase of the oscillation jumped by π , interchanging theminima and maxima. This rigidity was due to the Onsager relation, by which unitarityand time reversal symmetry imply that the conductance through the interferometermust be an even function of the magnetic field. [3] One way to overcome this rigidity,and to measure the phase of the transmission amplitude through the quantum dot, wasto open the interferometer, allowing leaks of electrons out of the interferometer ringand thus breaking the unitarity condition needed for the Onsager relation. [4] Indeed,experiments with open interferometers [5] yielded a continuous shift in the oscillationphases. In a ‘two-slit’ geometry (as in Young’s classical diffraction experiment), theelectronic wave passes only once through each branch of the interferometer, and thenthis phase shift should be equal to the desired transmission phase. However, this ‘two-slit’ limit is achieved only when the electronic leaks are very large, and therefore theremaining visibility (i.e. the amplitude of the AB oscillations) is very small. [6]In the present paper we discuss an alternative way to cross between the ‘two-slit’and the ‘two-terminal’ limits, i.e. between the case in which the oscillation phase reflectsthe scattering phase through one branch of the interferometer and the case of full phaserigidity. This is achieved by having two outgoing wires, namely by our novel three-terminal setup, consisting of an AB ring and a coupled-wire. [7] A priori, the current ineach outgoing wire need not obey the Onsager phase rigidity, because electrons “leak”through the other outgoing wire. However, we show that the strength of the tunnel-coupling between the outgoing wires can cause the crossover between the above twolimits.Section 2 describes our experimental setup, as shown in Fig. 1. This is similarto that of Ref. [1], but now we use only three (and not four) terminals, one for theincoming current (on the left) and two for the outgoing coupled channels (on the right).Section 2 also presents results for the two outgoing currents, see Fig. 2. These resultsare also similar to those found in Ref. [1]. For strong inter-wire tunnel-coupling, the B interferometry with a tunnel-coupled wire
Figure 1.
The experimental setup for the interferometer (AB ring) plus the tunnel-coupled wire. The regions beneath the gate electrodes are depleted, so that theinterferometer paths are defined in between. The tunnel coupling between the upperand lower wires is modulated by the narrow tunneling gate V T . We apply the source-drain bias V sd on the left ohmic contact and measure the output currents of the tworight ohmic contacts. In the rest of this paper we develop a simple and minimal theoretical tight-bindingmodel, which imitates the experimental setup. Since the following sections are somewhattechnical, we first give a qualitative description of our model and of our results. In Sec. 3we formulate the tight binding equations which describe the scattering of electrons fromtwo quantum network models, which are constructed from one-dimensional wires. Inthese models, which are shown in Fig. 3, the AB loop is modeled by a triangle (ABC) ofone-dimensional wires, and the tunnel-coupling between the outgoing wires is modeledby many transverse wires, each having a tunneling energy V . These equations are solvedfor Fig. 3(a) in Sec. 4, and for Fig. 3(b) in Secs. 5 and 6. The amplitudes and therelative phases of the AB oscillations in the reflected and transmitted currents dependon V . At small V , the oscillations in the reflected current are very small, and thereforethe oscillations in the two outgoing currents must cancel each other, hence the “anti-phase” behavior. This phenomenon appears for both models in Fig. 3, independentof the length of the coupling between the outgoing wires. In contrast, the “in-phase”behavior appears only for a long region of tunnel-coupling between the wires, Fig. 3(b).For a very large V , all the wave functions on these wires become evanescent, due to a B interferometry with a tunnel-coupled wire V , in a range whichdepends on some anisotropy between the two wires, one finds only one “running” wavesolution on the coupled wires. For this single solution, the ratio of the wave amplitudes,and therefore also the ratio of the two outgoing currents, are practically independentof the magnetic flux. Therefore, both currents have the same flux dependence (up toa constant multiplicative factor). This explains the “in-phase” behavior. Section 7presents our conclusions.
2. Experiments
We employed an AB ring connected to a tunnel-coupled wire shown in Fig. 1. This three-terminal geometry is the simplest for realizing the two-slit experiment even comparedwith the four-terminal geometry employed in the previous work. [1]Our device isfabricated from a modulation doped AlGaAs/GaAs heterostructure (depth of 2DEG:125 nm, carrier density: 1 . × cm − , mobility: 2 × cm / Vs) using a standardSchottky gate technique. By varying the tunneling gate voltage V T we can modulatein-situ the tunnel coupling energy V . By applying a voltage to the gate V M the phaseacquired in one of the two paths can be varied. To observe the quantum interference,a low energy excitation current (excitation energy across the overall sample: 50 µ eV)is injected into the quantum wire on the left, and the output currents I ↑ and I ↓ aremeasured simultaneously by sweeping the magnetic field at each gate configuration. Allexperiments were performed using the dilution refrigerator with a base temperature of70 mK.When we apply a relatively small negative voltage on V T , we can deplete thecenter region of the ring to form an AB ring while keeping strong coupling betweenthe parallel quantum wires. This is because the gate electrode deposited to define thetunnel coupling is narrow. In such a strong coupling case, the two output currents I ↑ and I ↓ oscillate in-phase as shown in Fig. 2(a). Namely, the two output contactswork equally and the interferometer effectively works as a standard AB ring in a two-terminal setup. The total current oscillates with a period of h/eS , where S is thearea enclosed by the AB ring. This standard AB interference is subjected to phaserigidity, as would result from the Onsager law. Below we explain why this law appliesin this limit. The phase of the AB oscillation can thus only take the values 0 or π atzero magnetic field and as a consequence leads to phase jumps when the AB phase ismodulated by changing a voltage V M applied to a side gate of the AB ring as shown inFig. 2(c). The gate voltage irregularly shifts the phase of the AB oscillation, implyingthat the observed AB oscillation is not an ideal two-path interference, but a complicatedmulti-path interference.In contrast, our device can also be tuned into the weak coupling regime by applyinga large negative voltage on V T . When V T is properly tuned so that the coupled wireworks as a beam splitter to yield high visibility, and the potential change at the transition B interferometry with a tunnel-coupled wire -10-50510 I ( p A ) B (mT) -10010 I ( p A ) -50 -45 -40 -35 B (mT) -0.50-0.45-0.40 V M ( V ) B (mT) -0.44-0.42-0.40 V M ( V ) -50 -45 -40 -35 B (mT) -10-50510 I ( p A ) (a) (b)(c) (d) Figure 2. (a) Typical AB oscillation in the strong coupling regime. The black andred curves are the measured I ↑ and I ↓ respectively, after subtraction of a smoothedbackground. (b) Typical current oscillation in the weak coupling regime. Tunnelcoupling energy is roughly a few 100 µ eV. The oscillations are extracted by subtractingthe smoothed background. (c) Intensity plot of I ↑ as a function of the perpendicularmagnetic field B and side gate voltage V M in the strong coupling regime. (d) Intensityplot of I ↑ as a function of B and V M in the weak coupling regime. region between the AB ring and the coupled wire is small enough, the observed twooutput currents oscillate with opposite phases with almost the same amplitude (seeFig. 2(b)). In other words, the total outgoing current I tot = I ↑ + I ↓ has very smallAB oscillations. This result strongly suggests that backscattering does not contributeto the main oscillation. Furthermore, when the phase difference between I ↑ and I ↓ isexactly π , the phase of the oscillation evolves smoothly and linearly with V M withoutany jump (see Fig. 2(d)). These results are in contrast to what is observed for thestandard two-terminal AB interferometer. The observed interference does not sufferfrom the multi-path contribution that modulates the total current, but captures thephase difference between the two paths that linearly shifts with V M , suggesting therealization of a true two-path interference. In what follows, we show that the anti-phaseoscillation and the smooth phase shift are reproduced in a simple tight-binding modeland prove that the measured phase shift is the bare phase shift of the upper path.In the experiment, the visibility in both the in-phase and the anti-phase oscillationsis further decreased by the existence of many transmitting channels with different tunnelcouplings. However, as we show below, a model with a single channel in each wirecaptures the above mentioned observed features. B interferometry with a tunnel-coupled wire
3. Tight-binding models
We now construct tight-binding models which imitate the experimental setup and allowa systematic study of the effects of various parameters on the outgoing currents. Themodels contain single-level sites n , with site energies ǫ n , and nearest-neighbor hoppingmatrix elements J nm . The models are simple enough to be solved fully analytically, sothat we can capture clearly the properties of all eigenstates. The Hamiltonian is thuswritten as H = X n ǫ n | n ih n | − X h nm i ( J nm | n ih m | + h . c . ) , (1)where h nm i denotes a bond between the neighboring sites n and m . The Schr¨odingerequation for the electron’s wave function | Ψ i ≡ P m h m | Ψ i| m i ≡ P m ψ ( m ) | m i istherefore ( ǫ − ǫ n ) ψ ( n ) = − X m J nm ψ ( m ) , (2)where ǫ is the energy of the electron.The system is connected to three leads, one on the left hand side and two onthe right hand side (see Fig. 3). These leads are described by one-dimensionalchains, with zero site energies and constant nearest-neighbor hopping matrix elements J n,n +1 = J n,n − ≡ J . The left lead has n ≤
0, and the two outgoing leads have n ≥ n .Within each lead, the Schr¨odinger equation is ǫψ ( n ) = − J [ ψ ( n −
1) + ψ ( n + 1)] , (3)with the solutions ψ = e ± ikn and ǫ = − J cos( k ), where the dimensionless wave number k contains the lattice constant. In the following we look for a scattering solution, inwhich we set ψ in ( n ) = e ikn + re − ikn , n ≤ ,ψ ↑ out ( n ) = t ↑ e ik ( n − n ) , n ≥ n ,ψ ↓ out ( n ) = t ↓ e ik ( n − n ) , n ≥ n . (4)Here, r denotes the amplitude of the reflected wave, while t ↑ and t ↓ denote the amplitudesof the two outgoing waves. These amplitudes determine the two outgoing currents andthe reflected current, via the Landauer formula.[8] In the linear response limit (zerobias between the left and right leads) and at zero temperature, the ratios of the twooutgoing currents and of the reflected current to the incoming one are given by the twotransmission and one reflection coefficients, T ↑ ≡ | t ↑ | , T ↓ ≡ | t ↓ | , R = | r | , (5)with T ↑ + T ↓ + R = 1 . (6)We next start with a simple three-terminal interferometer, Fig. 3(a), for which wegive an explicit analytical solution. As we show, this simple case already captures much B interferometry with a tunnel-coupled wire A Ε u Ε d BC (b) A Ε u Ε d BC DF
Figure 3. (a) The simplest three-terminal model. (b) The model for the interferometerwith the tunneling-coupled wires. of the “anti-phase” behavior. We then add the tunneling-coupled wires, and show howthey generate the “in-phase” behavior for strong tunneling strength.
4. Three terminal AB interferometer
The simplest model for a three terminal interferometer is shown in Fig. 3(a). Theinterferometer is modeled by a triangle
ABC of single-level sites. Each corner of thistriangle is connected to a one-dimensional lead, as described above. On the upper andlower arms of the triangle (AB and AC) we place one-dimensional chains of sites, oflengths n u and n d , (in the figure, n u = 3, n d = 2) with uniform site energies ǫ u and ǫ d and with uniform nearest-neighbor hopping energies j u and j d . The vertical armrepresents the tunneling between the two channels, with tunneling energy V betweenthe sites B and C .Within the upper and lower arms, the solutions to Eqs. (2) are given by ψ ℓ ( n ) = [sin( k ℓ n ) ψ ℓ ( n ℓ ) + sin[ k ℓ ( n ℓ − n )] ψ ( A )] / sin( k ℓ n ℓ ) , (7)where ℓ = u, d , and the wave number k ℓ = arccos[( ǫ ℓ − ǫ ) / (2 j ℓ )] again contains theappropriate lattice constant. In the geometry of Fig. 3(a), we have ψ u (0) = ψ d (0) = ψ ( A ) = 1 + r , ψ u ( n u ) = ψ ( B ) = t ↑ and ψ d ( n d ) = ψ ( C ) = t ↓ . Gauge invariance allows usto place the AB phase on any bond around the interferometer loop.[9] Measuring thisflux Φ in units of the unit quantum flux times 2 π , we place the corresponding phase onthe bond BC, and the tight binding equations for the triangle become ǫt ↑ = − J e ik t ↑ − V e − i Φ t ↓ − j u ψ u ( n u − ,ǫt ↓ = − J e ik t ↓ − V e i Φ t ↑ − j d ψ d ( n d − ,ǫ (1 + r ) = − j u ψ u (1) − j d ψ d (1) − J ( e − ik + re ik ) . (8) B interferometry with a tunnel-coupled wire ψ u (1) = (1 + r ) y u + t ↑ x u and ψ u ( n u −
1) = (1 + r ) x u + t ↑ y u ,with similar expressions for the lower branch, with x ℓ = j ℓ sin( k ℓ ) / sin( k ℓ n ℓ ) , y ℓ = j ℓ sin[ k ℓ ( n ℓ − / sin( k ℓ n ℓ ). Finally, one finds the solutions t ↑ = [ x u ( J e − ik − y d ) + V x d e − i Φ ](1 + r ) /d ,t ↓ = [ x d ( J e − ik − y u ) + V x u e i Φ ](1 + r ) /d , r = − iJ sin( k ) d/D , (9)where d = ( J e − ik − y u )( J e − ik − y d ) − V ,D = ( J e − ik − y u − y d ) d − x u ( J e − ik − y d ) − x d ( J e − ik − y u ) − x u x d V cos Φ . (10)Interestingly, r depends on Φ only via the term with cos Φ in the denominator D .Therefore, R = | r | = 1 − T ↑ − T ↓ is an even function of the flux, as might be expectedfrom the Onsager relation.To imitate a flat density of states ( dǫ/dk ) in the external leads (within the presenttight-binding model), one usually chooses electron energies near the center of the band, ǫ = 0 or k = π/
2. For this energy, one has T ↑ = 4 J [ x u ( J + y d ) + x d V + 2 x u x d V ( J sin Φ − y d cos Φ)] / | D | ,T ↓ = 4 J [ x d ( J + y u ) + x u V − x u x d V ( J sin Φ + y u cos Φ)] / | D | , (11)The denominator has the general form D = Q − x u x d V cos Φ + iP , where P = J ( J − y u − y d − y u y d + x u + x d + V ) ≡ P + J V ,Q = (2 J − y u y d + V )( y u + y d ) + x u y d + x d y u ≡ Q + ( y u + y d ) V , (12)so that | D | = P + ( Q − x u x d V cos Φ) .For n u = n d = 1 one has y u = y d = 0, and therefore also Q = 0, and | D | = P + 4 x u x d V cos Φ, while both numerators in Eqs. (11) contain the term ± x u x d J V sin Φ. It turns out that in this special case the numerators determinethe locations of the maxima and minima of the two transmissions, and thereforethe two outgoing currents always have opposite phases, with maxima or minima atΦ = (1 / m ) π (with integer m ) for all the values of the various parameters. Examplesare shown in Fig. 4.In the experiment it is difficult to tune the interferometer exactly into the symmetriccase discussed above. Also, in the special case n u = n d = 1, the model contains nodependence on the gate voltages on the branches of the AB interferometer; for example,the site energy ǫ u is included in the model only for sites between A and B in Fig. 3,which requires n u >
1. To investigate the dependence of the results on the gate voltage V M (Fig. 1), which is represented by the site energy ǫ u , we thus studied the modelwith n u >
1. For a small tunnel coupling, V = 0 . J , typical results are shown in Fig.5(a). This figure was drawn for n u = 5, a “gate voltage” on the upper arm of theinterferometer ǫ u = . J and n d = 1 (so that x d = j d and y d = 0). As seen in the B interferometry with a tunnel-coupled wire V = .1 J - - FΠ (b) V = J - - FΠ Figure 4.
Typical results for the simple three-terminal interferometer, with k = π/ ǫ = 0), j u = J, j d = . J , n u = n d = 1. Red: T ↑ . Blue: T ↓ . Green: R . (a) V = 0 . J . (b) V = 3 J . figure, the oscillations in R have a very small amplitude. Since T ↑ + T ↓ = 1 − R , thismeans that the oscillations in the two outgoing transmissions must be in “anti-phase”,as indeed seen in the same figure.In the experiments, an anti-phase behavior was also always accompanied by asmooth shift of the maxima and minima of the outgoing currents with the gate voltage V M . To test for this, Fig. 5(b) shows the locations of the maxima and minima of T ↑ asfunction of the “gate voltage” ǫ u . Indeed, this variation is smooth, and there appear nojumps between maxima and minima. A similar graph for the extrema of T ↓ turns outto be very close to Fig. 5(b): the minima of T ↓ are very close to the maxima of T ↑ , andvice versa, for practically all the values of the gate voltage. A similar behavior is foundfor other values of n u . The only difference is that the number of extrema at a fixed flux(in the range − J < ǫ u < J ) is equal to ( n u − n u −
1) levels). For small V , these results can also be obtained analytically.Expanding T ↑ and T ↓ to linear order in V yields T ↑ ≈ J P + Q [ A u + 2 x u x d V ( J sin Φ + C u cos Φ)] ,T ↓ ≈ J P + Q [ A d + 2 x u x d V ( − J sin Φ + C d cos Φ)] , (13)where A u = x u ( J + y d ), A d = x d ( J + y u ), C u = 2 A u Q / ( P + Q ) − y d , C d =2 A d Q / ( P + Q ) − y u , while P and Q are the values of P and Q at V = 0.The last terms in the transmissions (13) can be written as J sin Φ + C u cos Φ = q J + C u cos(Φ − β u ) and − J sin Φ + C d cos Φ = − q J + C d cos(Φ − β d ), with β u = arccos( C u / q J + C u ) , β d = arccos( − C d / q J + C d ) . (14)Since C u and C d have a smooth dependence on the “gate voltages” ǫ u and ǫ d , this impliesa smooth dependence of the observed phase shifts on these energies. In fact, we findnumerically that C u is close to − C d , and therefore β u ≈ β d , again consistent with theanti-phase behavior. B interferometry with a tunnel-coupled wire β ’s are small, they are linearin the the bare phase shifts on the upper arm, e.g. β u ∝ n u k u − mπ , with m an integer.As a result, they are also linear in ǫ u . In this linear regime the phase shift β u is directlyproportional to the shift in the optical path of the electron wave function on the upperbranch. A measurement of this phase shift yields this bare phase shift, as in the two-slitinterferometer! We have thus demonstrated that our system can be used for measuringphase shifts. Unlike the open two-terminals interferometer, used for the same purposeby Shuster et al. [5], our system does not necessarily have a small visibility (i.e. asmall amplitude of the AB oscillations). In practice, the visibilities in our system aredecreased by the existence of many channels with different tunnel coupling.(a) V = .1 J - - FΠ (b) V = .1 J - - ΒΠ- - Ε u J Figure 5. (a) Same as Fig. 4(a), but with n u = 5 and ǫ u = . J . (b) The locations ofthe maxima (red) and minima (blue) in T ↑ (denoted by β ) as functions of the “gatevoltage” ǫ u . The above anti-phase behavior, and the smooth variation of the phases of bothoutgoing currents with the gate voltage, appear only for small V . For large V , bothof the outgoing currents are small, proportional to 1 /V , and the reflection R is closeto 1. The visibility is even smaller, of order 1 /V . In this limit, the details of the ABoscillations depend on the parameters of the device. For example, if the hopping energythrough the lower branch of the interferometer is large (e.g. for a large j d , or near aresonance of the dots on this branch, when n d > | x d | , | y d | ≫
1) we find that I ↓ ≪ I ↑ ≪ R ≃
1. In this case, one has I ↑ ≃ − R , and therefore the minima of I ↑ follow the maxima of R . Since R is even in Φ, these extrema are ”phase-locked”at integer multiples of π , just as for the unitary two-terminal interferometer. If oneignores the small current in the lower outgoing wire then the system indeed behaveslike the two-terminal interferometer. Technically, one can see this from Eq. (11): allthe Φ − dependent terms there change sign when x u changes sign, which happen aftercrossing resonances in the upper branch. In contrast, the phases of the small I ↓ are notlimited by the Onsager restrictions, since most of the current ”leaks” though the upperwire. Therefore, from the point of view of the lower wire, the interferometer is ‘open’[2, 6], and the phase of I ↓ varies smoothly with the ”gate-voltage” ǫ u . An example of themaxima and minima of both currents in such a case is shown in Fig. 6(a,b). This simpleseparation between the ”two-terminal” (for the upper wire) and the ”two-slit” (for thelower wire) behaviors disappears when the two outgoing currents are comparable, see B interferometry with a tunnel-coupled wire V , thebehavior for large V does not reproduce the “in-phase” behavior observed experimentallyfor the tunnel-coupled wires. Therefore, we now turn to model the latter system.(a) (cid:1) (cid:1) ΒΠ(cid:1) (cid:1) Ε u J (b) (cid:1) (cid:1) ΒΠ(cid:1) (cid:1) Ε u J (c) (cid:1) (cid:1) ΒΠ(cid:1) (cid:1) Ε u J (d) (cid:1) (cid:1) ΒΠ(cid:1) (cid:1) Ε u J Figure 6.
Locations of maxima (red) and minima (blue) for V = 10 J . The otherparameters are the same as in Fig. 5. (a) T ↑ for j d = 100 J . (b) T ↓ for j d = 100 J , (c) T ↑ for j d = J . (d) T ↓ for j d = J .
5. The tunnel-coupled wires
Our model for the tunnel-coupled wires is shown in Fig. 3(b). The triangle ABC on theleft still represents the AB interferometer, with a flux Φ penetrating it. The sequenceof N rectangular loops within the rectangle BDFC represents the two coupled wires,BD and CF. Each such loop has a flux φ (in the same units) through it, and eachvertical bond represents the tunneling matrix element V between the wires. Since theconfinement of each wire is strong and the transverse spreading of the wave functionis much smaller than other length scales of the sample, this simple representation ofreplacing each quantum wire with a single chain is usually sufficient. In most of thefollowing calculations we imitate the experiment and assume that the area of BDFCis equal to one third of the area of the interferometer loop ABC, and therefore we use φ = Φ / (3 N ). To imitate the continuous tunnel wires one would like N to be very large,and therefore φ is very small. In practice we perform calculations at several large valuesof N , and ensure that the results do not vary much with N . The two horizontal lineson the right hand side represent the two outgoing leads, as before. The tight bindingequations presented in the previous subsection are now supplemented by the equations B interferometry with a tunnel-coupled wire ψ ↑ ( n )and by ψ ↓ ( n ), respectively ( n = 0 , , , ..., N ). Choosing the same gauge as before,the Schr¨odinger equations for an electron with energy ǫ on these sites (0 < n < N ), are( ǫ − ǫ ↑ ) ψ ↑ ( n ) = − j ↑ [ ψ ↑ ( n + 1) + ψ ↑ ( n − − V e − iφ n ψ ↓ ( n ) , ( ǫ − ǫ ↓ ) ψ ↓ ( n ) = − j ↓ [ ψ ↓ ( n + 1) + ψ ↓ ( n − − V e iφ n ψ ↑ ( n ) , (15)where ǫ ↑ and ǫ ↓ are the (constant) site energies, which model gate voltages applied toeach wire separately, while j ↑ and j ↓ represent the corresponding hopping energies. Also, φ n = Φ + nφ . We shall later return to the boundary conditions, ψ ↑ (0) = ψ ( B ) = ψ u ( n u ) , ψ ↓ (0) = ψ ( C ) = ψ d ( n d ) ,ψ ↑ ( N ) = ψ ( D ) = t ↑ , ψ ↓ ( N ) = ψ ( F ) = t ↓ . (16)We first discuss the solution of the above tight-binding equations within the ladder.A wave-like solution of these equations can be found by setting ψ ↑ ( n ) = e i ( Kn − φ n / u ↑ , ψ ↓ ( n ) = e i ( Kn + φ n / u ↓ (where again the dimensionless wave number K contains thelattice constant along the ladder). The amplitudes u ↑ , ↓ must then obey the linearequations [ ǫ − ǫ ↑ + 2 j ↑ cos( K − φ/ u ↑ + V u ↓ = 0 ,V u ↑ + [ ǫ − ǫ ↓ + 2 j ↓ cos( K + φ/ u ↓ = 0 . (17)Therefore, the wave-numbers K are the solutions of the determinant equation,[ ǫ − ǫ ↑ + 2 j ↑ cos( K + − φ/ ǫ − ǫ ↓ + 2 j ↓ cos( K + φ/ − V = 0 , (18)i.e. 4 j ↑ j ↓ cos K + 2[( ǫ − ǫ ↑ ) j ↓ + ( ǫ − ǫ ↓ ) j ↑ ] cos( φ/
2) cos K + ( ǫ − ǫ ↑ )( ǫ − ǫ ↓ ) − V − j ↑ j ↓ sin ( φ/ ǫ − ǫ ↑ ) j ↓ − ( ǫ − ǫ ↓ ) j ↑ ] sin( φ/
2) sin
K . (19)As stated, we need results for large N and therefore for small φ . At φ = 0 wehave a quadratic equation, with two solutions, cos K = c ± ≡ [( ǫ ↑ − ǫ ) /j ↑ + ( ǫ ↓ − ǫ ) /j ↓ ± q [( ǫ ↑ − ǫ ) /j ↑ − ( ǫ ↓ − ǫ ) /j ↓ ] + 4 V / ( j ↑ j ↓ )] /
4. For small V , the two solutionsfor cos K remain in the range − < c ± <
1, and therefore each of them correspondsto waves running in opposite directions, with wave numbers K , = ± arccos[ c + ] and K , = ± arccos[ c − ]. However, as V increases one of | c ± | crosses the value 1 when V = ( ǫ − ǫ ↑ ± j ↑ )( ǫ − ǫ ↓ ± j ↓ ). Above these values of V , K , and/or K , becomecomplex, and the corresponding waves become evanescent. Figure 7 shows an exampleof regions in the ǫ ↑ − V plane, for the special parameters j ↑ = j ↓ = J , and ǫ = ǫ ↓ = 0.The numbers on the diagram (4, 2 or 0) indicate the number of “running” solutions,with real K ’s, in each region.For a large but finite N we need to solve Eq. (19) for a finite small φ . In practice wedo that by searching a solution for K close to each of the four solutions found at φ = 0.For each value of φ this yields four waves, with wave numbers K , K , K , K [an B interferometry with a tunnel-coupled wire K ℓ and then choose the right signs of sin K ℓ which satisfiy Eq.(19)]. As V increases at fixed | ǫ ↑ − ǫ | < j ↑ , at first all the four K ’s are real, then twoof them become complex and then the other two also become complex. As we shall seebelow, each of these regimes ends up with a different qualitative behavior of the twooutgoing currents. Ε - - Figure 7.
Regions in the ǫ ↑ − V plane (both in units of J ) with 4, 2 and no runningsolutions; the numbers of running solutions are indicated. Other parameters are φ = 0, j ↑ = j ↓ = J , ǫ = ǫ ↓ = 0. For each of the four K ’s, the corresponding amplitudes of the wave functions obeythe relation u ↓ ℓ = − u ↑ ℓ [ ǫ − ǫ ↑ + 2 j ↑ cos( K ℓ − φ/ /V . (20)The above solutions represent the eigenstates of the infinite periodic ladder. For theinfinite ladder, one cannot accept the evanescent solutions, which increase to infinityfor n → ∞ or for n → −∞ . Therefore, one considers only real values of K , and oneends up with two energy bands with energies ǫ ( K ) which are given by the solution ofthe quadratic equation (19) in ǫ . For the simple case presented in Fig. 7, j ↑ = j ↓ = J , ǫ = ǫ ↓ = 0 and φ = 0, these two bands are given by ǫ = [ ǫ ↑ − J cos K ± q ǫ ↑ + 4 V ] / ǫ are found as the intersections of thehorizontal line at that energy with these two functions (see dashed lines in the figure). At ǫ ↑ = V = 0 the two bands coincide, and therefore every ǫ corresponds to two degeneraterunning waves, namely four waves. However, non-zero values of ǫ ↑ and/or of V split thetwo bands, and then there exist energy ranges in which there appear only two runningsolutions, or even gaps for which there are no running solutions. In these regions theremaining waves are evanescent, which are forbidden for the infinite ladder but allowedin the finite ladder. As we show below, one needs to be in these regions (and thereforeto have some asymmetry between the wires) in order to obtain the in-phase behavior ofthe outgoing currents. B interferometry with a tunnel-coupled wire - - Π- - Ε J Figure 8.
The two energy bands for the infinite ladder, for φ = 0, j ↑ = j ↓ = J , ǫ ↓ = 0and ǫ ↑ = V = J . The dashed lines are at energies with four running solutions ( ǫ = J )and two running solutions ( ǫ = 2 J ).
6. Interferometer with tunnel-coupled wires
We now return to the model of Fig. 3(b). Using Eq. (20), the general solution for the N + 1 sites ( n = 0 , , , · · · , N ) on the N squares of the ladder is written as a linearcombination of the four solutions, ψ ↑ ( n ) = X ℓ =1 A ℓ e i ( K ℓ n − φ n / ,ψ ↓ ( n ) = − X ℓ =1 [ ǫ − ǫ ↑ + 2 j ↑ cos( K ℓ − φ/ A ℓ e i ( K ℓ n + φ n / /V , (21)with the four yet unknown amplitudes { A ℓ } .Using the analogs of Eqs. (4) on the leads, we now have t ↑ , ↓ ≡ ψ ↑ , ↓ ( N ) [see Eqs.(16)]. The Schr¨odinger equations at these two end points (near the outgoing leads) arethus ( ǫ − ǫ ↑ ) ψ ↑ ( N ) = − j ↑ ψ ↑ ( N − − J t ↑ e ik − V e − iφ N ψ ↓ ( N ) , ( ǫ − ǫ ↓ ) ψ ↓ ( N ) = − j ↓ ψ ↓ ( N − − J t ↓ e ik − V e iφ N ψ ↑ ( N ) . (22)Similarly, Eqs. (8) are now replaced by the equations for the wave functions at the threecorners of the AB interferometer, ǫ (1 + r ) = − J ( e − ik + re ik ) − j u ψ u (1) − j d ψ d (1) , ( ǫ − ǫ ↑ ) ψ ↑ (0) = − j u ψ u ( n u − − j ↑ ψ ↑ (1) − V e − i Φ ψ ↓ (0) , ( ǫ − ǫ ↓ ) ψ ↓ (0) = − j d ψ d ( n d − − j ↓ ψ ↓ (1) − V e i Φ ψ ↑ (0) . (23)Equations (22) and (23), together with Eqs. (21) and (16), now reduce to five linearequations in the five unknowns { A ℓ } and r . Their solution yields the reflection coefficient R = | r | and the two transmission coefficients T ↑ = | t ↑ | and T ↓ = | t ↓ | . B interferometry with a tunnel-coupled wire φ = Φ / (3 N ). Below we plot results as a function of the fluxthrough the AB loop, namely Φ. For reasonable values of Φ this implies relatively smallvalues of φ . For the results presented below we used N = 1001 and N = 401 (thenumerical solutions of the five linear equations become difficult for large N and large V ,when the factors e iKN vary by many orders of magnitude) . These results change onlyslightly (quantitatively but not qualitatively) when we used other (large) values of N .In order to observe the dependence of the results on the gate voltage on the upperarm of the interferometer, we again need to have n u >
1. Below we present typicalresults with n u = 5. The results do not change qualitatively for a wide range of theother parameters. For small V we always find “anti-phase” behavior, even when the twowires are symmetric. As explained below, to see the ”in-phase” behavior we need to haveonly two ‘running’ waves, and this happens only with some asymmetry between the wires(see Fig. 7). Indeed, with some such anisotropy and and for appropriately chosen valuesof large V (explained below) we find “in-phase” behavior. This behavior requires thecoupled wires, and did not appear in the simpler three-terminal interferometer presentedin Sec. IV.We start with weak coupling, V = 0 . J . Figure 9(a) shows results for a smallcoupling V , for the parameters as indicated. Due to the anisotropy between the twobranches of the interferometer, j u = J, j d = . J , T ↑ is much larger than T ↓ . However,when we shift each of the transmissions by their average value, as shown in Fig. 9(b), it isobvious that the AB oscillations of the two transmissions exhibit the same kind of “anti-phase” behavior as we already saw in the previous section (and as seen experimentally).The main reason for the “anti-phase” behavior can be attributed to the weak oscillationsin R [which are practically zero in Fig. 9(b)]. The relation T ↑ + T ↓ = 1 − R then requiresthat the oscillating terms in the two transmissions must have opposite signs, so theycancel each other in the sum.Figure 9(c) shows the variation of the minima and maxima of T ↑ with ǫ ↑ . Similarto Fig. 5(b), these maxima and minima move smoothly with the gate voltage, withoutany jumps. These maxima and minima again reflect the energy levels of the upper armAB of the interferometer. Therefore, we conclude that here also the oscillations in thetwo transmissions reflect the basic “two-slit” interference around the AB loop.We now turn to large V . Figure 10 is similar to Fig. 9, except that V = 1 . J and N = 401. As can be seen from Figs. 10(a) and (b), the two outgoing transmissionsare practically identical to each other, and therefore they are fully in phase with eachother. Furthermore, they both are symmetric under Φ ↔ − Φ. As required by therelation T ↑ + T ↓ = 1 − R , the reflection coefficient R exhibits opposite oscillations,with a double amplitude [see Fig. 10(b)]. The maxima and minima of T ↑ , shown inFig. 10(c), exhibit full “phase locking”; they remain at integer multiples of π . (Forsome values of ǫ ↑ one observes additional red points between these integer values; theseindicate the splitting of the maxima into pairs of maxima, due to higher integer Fouriercomponents. One always finds only integer Fourier components, but with amplitudeswhich vary with the parameters). This phase locking is similar to that observed in the B interferometry with a tunnel-coupled wire - -
15 15 30 FΠ (b) - - FΠ- D T (c) - - ΒΠ- - Ε u J Figure 9.
Typical results for the tunneling-coupled wires, for weak coupling: N =1001, φ = Φ / (3 N ), ǫ ↑ = . J , ǫ ↓ = 0, j u = j ↑ = j ↓ = J, j d = . J , n u = 5 , n d = 1 , ǫ = 0and V = 0 . J . (a) Transmissions T ↑ (red) and T ↓ (blue) and reflection R (green),for ǫ u = . J . (b) An enlarged version of (a), showing T ↑ − h T ↑ i , T ↓ − h T ↓ i and R − h R i . (c) Maxima (red) and minima (blue) of T ↑ versus ǫ u [all other parametersare the same as in (a)]. two-terminal interferometer. [2] Apart from being locked, maxima occasionally turninto minima and vice versa. This also appeared in Fig. 6(a), and must arise from thesame origin: crossing of resonances on the branch AB of the interferometer. All of thesefeatures are very similar to those seen experimentally.For large V , electrons are strongly reflected from the points B and C of the structure.This explains the large value of R , and the small values of the transmitted currents.These features also appeared for the three terminal interferometer, without the tunnelcoupling between the outgoing wires. To understand the other features observed inFig. 10, we note that the point ǫ ↑ = 0 . J, V = 1 . V is just above the lower line inFig. 7 (it is difficult to solve the linear equations for larger V , when the imaginarypart of the evanescent wave numbers are large and the corresponding wave functionsgo beyond the accuracy limits of the computer. This is why we stay close to this line,and also why we only present numbers for N = 401). Within the region between thetwo lines in Fig. 7, two of the four solutions within the “ladder” decay exponentially,and we are left with only two running waves, which result from one of the solutions c + or c − . For ǫ ↑ >
0, the first evanescent waves (as we cross the lower line in Fig.7) are associated with c + becoming larger than 1. For the parameters in that figure,this happens at V = 2 J (2 J − ǫ ↑ ), namely at V ≈ . J . At that point one has c − = ǫ ↑ / (2 J ) − ≈ − .
95, and the two remaining running waves have very close values
B interferometry with a tunnel-coupled wire - -
15 15 30 FΠ (b) - - FΠ D T (c) - - ΒΠ- - Ε J Figure 10.
Typical results for the tunneling-coupled wires, for strong coupling: N = 401, V = 1 . J . All the other parameters are the same as in Fig. 9. (a)Transmissions T ↑ (red) and T ↓ (blue) and reflection R (green), for ǫ u = . J . (b) Anenlarged version of (a), showing T ↑ − h T ↑ i , T ↓ − h T ↓ i and R − h R i . (c) Maxima (red)and minima (blue) of T ↑ versus ǫ u (all other parameters are the same as in (a)). of | K | . At the same neighborhood, Eq. (20) yields u ↓ /u ↑ = − [ ǫ − ǫ ↑ + 2 j ↑ cos( K − φ/ /V ≈ − [ ǫ − ǫ ↑ + 2 j ↑ c − )] /V , (24)where we neglected the small φ . Since this ratio applies to both the running solutions,and since the evanescent solutions can be neglected, we conclude that the same ratioapplies to their linear combinations, ψ ↑ ( n ) and ψ ↓ ( n ). In particular, T ↓ /T ↑ = | t ↓ /t ↑ | ≈ | ǫ − ǫ ↑ + 2 j ↑ c − | /V , (25) practically independent of Φ (at least for fluxes which are not too large). Since this ratiois independent of Φ, the two transmissions are proportional to each other, and thereforethey are exactly in phase . Deviations will occur only at large fluxes, when φ becomessignificant.For the parameters used in Fig. 10, this equation yields T ↓ /T ↑ ≈ / [1 − ǫ ↑ / (2 J )] ≈ .
05. This explains the practical overlap of the two transmissions. For other parametersthe ratio need not be so close to unity, but the in-phase behavior will persist whenever wehave two evanescent waves. The Onsager relation requires that the full current throughthe system should be an even function of the flux. This implies that both R and T ↑ + T ↓ should be such even functions. Indeed, all our calculations show that R is even in Φ.Once we demonstrated that the two transmissions are proportional to each other, eachof them must be proportional to their sum, and therefore each of them separately must B interferometry with a tunnel-coupled wire
7. Conclusion
We have demonstrated a novel two-slit experiment, using an AB ring connected to atunnel-coupled wire in a three-terminal setup. Out simple tight binding model capturesmost of the observed behavior of the currents in the three terminal interferometer.For small tunnel-coupling V , one always has the anti-phase behavior. In that limit,the dependence of the the phase of the Aharonov-Bohm oscillations in the outgoingcurrents on the gate voltage acting on one arm of the interferometer follows the phaseof the electronic wave function on that arm. This three terminal interferometer thusbehaves like the two-slit or open interferometer. For large V one needs to tune theinter-wire coupling to a regime where one has only two running wave solutions. In thatregime, the ratio between the two outgoing currents is practically flux independent, andtherefore they are in phase, with phase rigidity. Although our model does not include allthe details of the interferometer, such as finite widths of the quantum wires, influenceof the multiple transport channels and accompanying electron-electron interaction, thesimple and analytically solvable model provides the guiding principles for realizing a‘two-slit’ experiment and for a reliable phase measurement in the three-terminal setup. Acknowledgments
Work at Ben Gurion University was supported by the Israel Science Foundation. S.Takada acknowledges support from JSPS Research Fellowships for Young Scientists.M.Y. acknowledges financial support by Grant-in-Aid for Young Scientists A (no.23684019). S.Tarucha acknowledges financial support by MEXT KAKENHHI QuantumCybernetics”, MEXT project for Developing Innovation Systems, and JST StrategicInternational Cooperative Program. The high quality 2DEG wafer was provided by A.D. Wieck.
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