Experimental demonstration of Aharonov-Casher interference in a Josephson junction circuit
I. M. Pop, B. Douçot, L. Ioffe, I. Protopopov, F. Lecocq, I. Matei, O. Buisson, W. Guichard
EExperimental demonstration of Aharonov-Casher interference in aJosephson junction circuit
I. M. Pop , B. Douçot , L. Ioffe , I. Protopopov , F. Lecocq , I. Matei , O. Buisson and W. Guichard Institut Néel, CNRS et Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589,Universités Paris 6 et 7, 4 place Jussieu, 75005 Paris, France Department of Physics and Astronomy, Rutgers University,136 Frelinghuysen Rd., Piscataway, NJ 08854, USA and L. D. Landau Institute for Theoretical Physics, Kosygin str. 2,Moscow 119334, Russia and Institut fuer Nanotechnologie,Karlsruher Institut fuer Technologie, 76021 Karlsruhe, Germany
A neutral quantum particle with magnetic moment encircling a static electric charge acquires aquantum mechanical phase (Aharonov-Casher effect). In superconducting electronics the neutralparticle becomes a fluxon that moves around superconducting islands connected by Josephson junc-tions. The full understanding of this effect in systems of many junctions is crucial for the designof novel quantum circuits. Here we present measurements and quantitative analysis of fluxon in-terference patterns in a six Josephson junction chain. In this multi-junction circuit the fluxon canencircle any combination of charges on five superconducting islands, resulting in a complex pattern.We compare the experimental results with predictions of a simplified model that treats fluxons as in-dependent excitations and with the results of the full diagonalization of the quantum problem. Ourresults demonstrate the accuracy of the fluxon interference description and the quantum coherenceof these arrays.
The formation of macroscopically large coherentstates in systems with many unquenched degrees offreedom tests our understanding of quantum mechan-ics and it is essential for quantum computation. Oneof the most striking consequences of such coherenceare interference patterns that are expected to ap-pear when a charged particle encircles a magnetic flux(Aharonov-Bohm effect[1]) or when a flux encirclesa charge (Aharonov-Casher effect[2]). The quantumcoherence implied by these effects is a fragile phe-nomenon which is easily destroyed by uncontrolled de-grees of freedom. In an ideal Josephson junction arraymost microscopic degrees of freedom are quenched byelectron pairing into Cooper pairs; the only remainingdegrees of freedom are the phase of the order parame-ter on each island or the charge conjugated to it. Ob-servation of the interference provides the evidence ofthe full control of the quantum system in the groundstate. In Josephson junction arrays it proves the ir-relevance of the uncontrolled degrees of freedom suchas two level systems, non-equilibrium quasi-particles,etc.Aharonov-Bohm (AB) effect in small and largeJosephson arrays is a very well established phe-nomenon: in the former it leads to critical currentoscillations in SQUIDs[3], in the latter it results ina complicated magnetic field dependence with manypeaks at commensurate fields[4–7]. The experimentalconfirmation of its dual, the Aharonov-Casher (AC)effect, is less clear. It was observed for small Joseph-son circuits where vortices moved in a ring encirclinga single charge[8]. However, large arrays studied in anumber of works show the appearance of the interme-diate “normal” phase of the arrays which is character-ized by a non-zero resistance[6, 9, 10]. Non-zero re-sistance implies that the fluxon motion is dissipative;this excludes quantum coherence. It is very importantto establish the presence or absence of this dissipationand its possible origin in well controlled medium size arrays. This is the main goal of our work.The duality of AB and AC effects can be illustratedby analyzing the quantum mechanical phase resultingfrom the braiding of particle with charge q and a neu-tral particle with magnetic moment (cid:126)µ . If the chargedparticle is at rest while the neutral one moves, theformer generates an electrical field (cid:126)E µ at the posi-tion (cid:126)R µ of the latter that gives the interaction energy I = [ (cid:126) µc × (cid:126)E µ ] · ˙ (cid:126)R µ . Conversely, the magnetic mo-ment at rest generates a vector potential (cid:126)A q at theposition (cid:126)R q of the moving charge, that gives the in-teraction energy I = qc (cid:126)A q · ˙ (cid:126)R q . In either case, theacquired phase is given by the time integral of theinteraction energy. In case of the moving charge,this phase is δφ AB = (cid:0) qhc (cid:1) ¸ (cid:126)A q · (cid:126)dR q (AB effect);in case of a moving magnetic moment this phase is δφ AC = hc ¸ (cid:16) (cid:126)µ × (cid:126)E µ (cid:17) · (cid:126)dR µ (AC effect). Experimen-tally the former was first observed 50 years ago asan electron interference pattern in magnetic field [11];the latter was measured with percent accuracy in neu-tron and atomic interferometry experiments [12, 13].In Lorentz invariant systems of neutral and chargedparticles the distinction between the two effects is im-possible. What seems as AB effect for the observerin the rest frame of the neutral particle becomes ACeffect for the observer in the rest frame of the chargedparticle. In solid state devices these effects are distin-guishable because the rest frame is fixed by the device,and therefore the observation of AB interference doesnot imply the one of AC and vice versa. Because un-controlled degrees of freedom turn out to be mostlyelectric charges (either background charges or non-equilibrium quasi-particles in superconductors), ex-perimentally it is difficult to remain in the rest frameof the charge and, consequently, the observation of theAC effect is much more challenging.A Josephson junction circuit can be described ei- a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r ther in terms of the superconducting phase or in termsof the charges of its islands. If the charging energy E C = e / (2 C ) is larger than the Josephson energy E J , Cooper pairs are almost localized. The dynam-ics of a Josephson junction array can be viewed asdue to the rare motion of these pairs. In the oppo-site limit E J (cid:29) E C the charge is delocalized and thearray dynamics can be viewed as due to rare phaseslips resulting in the motion of fluxons. The nature ofthe elementary excitations does not preclude howeverthe description of the circuit in the charge (or phase)basis. The elementary excitations only become morecomplicated objects for E J > E C in the charge basisor for E C > E J in the phase basis.In the following, we analyze the ground state prop-erties of a 6 Josephson junction chain as a functionof the gate voltage that induces polarization chargesfor E J /E C = 2 − . Because in this regime the in-dividual excitations are fluxons, the properties of thecircuit are due to the interference between phase-slipson different junctions. The interference pattern is dueto the charges induced on the array islands which isexactly AC effect. The difference between the longerchain (of six junctions) studied here and the previ-ous works[8, 30] is that fluxons can take one of thesix possible routes resulting in a much more compli-cated interference. In the following we compute theexpected properties of the array assuming that phaseslips are independent excitations. Because this as-sumption might be questioned for E J /E C = 2 − we have also performed the complete diagonalizationof the Hamiltonian. Finally we compare the resultsof both approaches to the measured data. Our mainconclusion is that the phase slip approximation pro-vides a semi-quantitative description of the data inthis regime and that the observed interference pat-tern evidences the quantum coherent dynamics of ourrelatively large circuits.Fig.1a shows an idealized view of our circuit: a su-perconducting ring containing five islands connectedby Josephson junctions. A gate voltage V G inducesthe charge frustration q i = C gi V G / (2 e ) + q i on the i -th island. Here q i are static offset charges. Thecouplings to the gate electrode C gi are not equal forall islands and induce a general charge configuration κ = ( q , q , q , q , q ) , expressed in units of e . Whitetraces in Fig.1a represent six possible paths for afluxon to cross the ring, through one of the six Joseph-son junctions. The probability of this event is given bythe quantum phase-slip amplitude υ j of a single junc-tion. υ j contains an AC phase-factor depending on theislands charges q i (see eq. (1) and (2)). The groundstate of the SQUID chain depends on the CoherentQuantum Phase-Slip (CQPS) amplitude that resultsfrom the macroscopic interference of six fluxons. TheCQPS amplitude, v ∗ , is obtained by summing up allphase-slip amplitudes on the junctions [15] (for com-putation details see the Supplementary Informationtext): v ∗ = (cid:88) j =1 v j and v j = v exp (cid:34) i π j − (cid:88) k =1 q k (cid:35) (1) VV C C C C (b) G
50 mK Φ S Φ C I Φ C I bias C qqqqq γ C=270pFL=8nH R O δ I = A C Figure 1: Schematic view of the experimental setup used toprobe the phase-slip interference in a chain of 6 Josephsonjunctions. In (a) we show an idealized view of the exper-imental design. The chain contains five small supercon-ducting islands connected to each other and to the leadsby identical Josephson junctions. The islands are coupledto a nearby gate electrode. In (b) we present the electri-cal scheme of the measurement. The six-SQUID chain isinserted in a superconducting loop. The flux Φ C createdby on-chip coils controls the phase difference γ over thechain. The independently controlled flux Φ S through theSQUID loops is used to tune in situ the Josephson cou-pling E J = E J cos ( π Φ S / Φ ) , where E J = 2 K and Φ isthe magnetic flux quantum. The charging of one SQUIDis E C = 660 mK . The phase difference over the read-outjunction is denoted by δ . The gate electrode couples to thecharge q i on island i via the capacitance C gi . The couplingto the central island C g is at least 10 times larger thanall other capacitances and determines the dominant gateeffect at low voltage. (c) shows the calculated ground andfirst excited state for the 6 junction chain as a function ofthe phase bias γ and the induced charge q on the centralisland, for charge configurations of the type (0 , , q , , . Here υ is the magnitude of the phase-slip amplitudefor a single Josephson junction. In the quasi-classicalapproximation, valid at E J (cid:29) E c , it is[16]: v = 8 (cid:114) E J E C π (cid:18) E J E C (cid:19) . e − (cid:113) EJEC (2)The first two energy levels of a Josephson junctionchain are shown in Fig. 1c. These energy levels havebeen calculated by diagonalizing the Matveev-Larkin-Glazman (MLG) tight-binding Hamiltonian [15]: H | m (cid:105) = E m | m (cid:105) − v ∗ [ | m − (cid:105) + | m + 1 (cid:105) ] (3)Here E m = E J N ( γ − πm ) is the energy of the | m (cid:105) state of the chain polarized at phase γ and m is thequantum variable that counts the number of vorticeshaving crossed the chain through one of the junctions.The model (3) makes two important assumptions: thequantum phase slips on different junctions lead to thesame quantum states and these tunneling processesare independent events. As it can be seen from eq.(1), the AC interference of CQPS is an intrinsically e periodic effect.In our sample, each junction of the chain is real-ized by a SQUID (see Fig. 1b) to enable tunableJosephson coupling E J . Consequently we can control in-situ the strength of the quantum phase slip am-plitude v through the magnetic flux Φ S . To measurethe CQPS effect on the ground state of a Josephsonjunction chain, we have shunted the chain by a largeread-out junction (see Fig. 1b, [17] and [18]). Theflux Φ C in the superconducting loop containing theread-out junction and the chain, enables the controlof the bias phase γ = Φ C − δ over the chain. δ is thephase difference on the read-out junction.We have measured the switching current (see theMethods section) of the entire Josephson junction cir-cuit containing both the chain and the read-out junc-tion. We start by presenting the gate-voltage depen-dence of the switching current (Fig. 2) for small vari-ations of the gate voltage (so that | V G | (cid:28) e/C g )and for two different ratios of E J /E C . As the cen-tral coupling C g is ∼ times larger than any othercoupling, the gate voltage only polarizes the middleisland. The values of the island charges result fromthe combined effect of the gate voltage and off-setcharges. In the particular case of our circuit, the lattervary randomly within a time scale of ∼ min in aver-age, enabling the measurement of a single charge con-figuration during a gate voltage scan that takes ∼ minutes. Thus, by repeating the same voltage sweep,we measure different charge configurations. The re-sults presented in Fig. 2 were post-selected from alarge set of data, of about runs by choosing thelargest observed switching current oscillations. Thelargest oscillations displayed in Fig. 2 correspond toa particularly simple case in which all charges, ex-cept the one on the middle island are close to zero: κ (cid:48) = (0 ± . , ± . , C g V g e , ± . , ± . . Noticethat the ab-initio probability to produce this chargeconfiguration is . = 0 . which translates into 6configurations out of 3000, so its observation supportsthe assumption of random charge distribution.The charge frustration on the middle island intro-duces a geometrical phase shift exp [ i πq ] betweenthe three CQPS occurring on the junctions at the leftside of the middle island and the three CQPS on thejunctions at the right of the middle island. This phaseshift between the different CQPS is graphically rep- v* → (a) v → v → v → v → v → v → (0, 0, 0, 0, 0) (0, 0, 0.25, 0, 0) (0, 0, 0.45, 0, 0) (0, 0, 0.875, 0, 0) - 2 - 1 0 1 2- 202468 - 2 - 1 0 1 2- 202468 ( c ) d Isw / i0 (%) E J / E C = 2 q ( 2 e ) ( d ) q ( 2 e ) E J / E C = 3 d Isw / i0 (%) ( e )
Isw (nA) F C d I S W ( b )
Isw (nA) F C d I S W - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 V G ( m V )- 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 V G ( m V ) Figure 2: Phase-slip interferences controlled by the po-larization charge on the middle island of a 6 Josephsonjunction chain, corresponding to the charge configurations κ (cid:48) = (0 ± . , ± . , C g V g e , ± . , ± . . (a) Schematicrepresentations of the total CQPS amplitude v ∗ (in red),obtained by summing up the 6 phase-slip probability am-plitudes v i (in black), represented as vectors in the com-plex plane. v ∗ is presented for four different charge con-figurations. (b) (c) The black diamonds represent themeasured variation of the switching current as a functionof the induced charge on the middle island, in the caseof E J /E C = 3 (b) and E J /E C = 2 (c). The y-axis isreported in units of the critical current of one junction, i . The red lines represent the theoretical calculations inthe independent phase slip approximation (MLG model)of CQPS interference. The chain was phase biased at aconstant phase: Φ C = 0 . π . The working points for themeasurements presented in (b) and (c) are presented inthe panels (d) and (e) at the right of each curve. Allcurves are shifted so that zero of the y axis correspondsto the switching current of the zero charge configuration (0 , , , , . The blue lines represent the correspondingcalculations using the diagonalization of the Hamiltonian(5). There is a reasonable agreement between the semi-classical MLG model of CQPS interference, numerical cal-culation and data. resented for several charge configurations in Fig. 2a.In this regime the total CQPS amplitude (eq. 1) as afunction of gate voltage becomes: | v ∗ | = 3 v (cid:113) πV g C g /e ) . (4)The phase-slip amplitude is expected to vanishcompletely v ∗ = 0 for the charge configuration (0 , , . , , while the maximum value v ∗ = 6 v is ob-tained for the charge configuration (0 , , , , . Thered line of Fig. 2 shows the corresponding theoreticalcalculation using the CQPS model (3),(4). Aroundthe charge configuration (0 , , . , , we expect acomplete suppression of the total phase-slip amplitude v ∗ (see Fig. 2a), hence an increase of the supercur-rent through the chain. For E J /E C = 3 , the changein the measured switching current due to the full sup-pression of CQPS is ∼ ∼ ofthe critical current of one SQUID, i . Increasing theCQPS-amplitude by decreasing the ratio E J /E C to avalue of , the oscillation amplitude of the switchingcurrent increases to ∼ nA which represents ∼ of i (see Fig. 2c).We now turn to the discussion of more complex in-terferences of CQPS realized by increasing the gatevoltage ( | V G | (cid:38) e/C g ) that leads to the polarizationof the islands next to the central island. In Fig. 3c weshow two interference patterns that were post-selectedfrom a total of 200 curves. Again, the selectioncriteria was the maximum observed switching cur-rent amplitude. In the following we show that thesemeasured patterns can be understood by consideringcharge configurations in which only q , q ≈ : κ (cid:48)(cid:48) =(0 ± . , q (0)2 + C g V g e , q (0)3 + C g V g e , q (0)4 + C g V g e , ± . .The corresponding ab-initio probability is . = 4% which translates into ∼ curves out of the measured .The charge frustration on the middle, the secondand the fourth islands introduces geometrical phaseshifts between the CQPS of the second, third and forthjunctions. Fig. 3a show the corresponding CQPS am-plitudes as vectors in the complex plane for severalcharge configurations. The resulting switching cur-rent oscillations δI SW , presented in Fig. 3b, show acomplex pattern, composed of a fast harmonic arisingfrom the strong C g coupling and a slower evolving en-velope due to the weaker C g and C g couplings (seeTable I). In Fig. 3c we show the measured interfer-ence patterns for two different phase biases Φ C of theJosephson junction chain; these biases were chosenclose to Φ C = π in order to maximize the responseof the chain. For the top curves in Fig. 3b and cwe polarized the chain at Φ C (cid:46) π so we expect theswitching current to increase when the phase slips aresuppressed. Similarly, for the bottom curves, where Φ C (cid:38) π , we expect the switching current to decreasewhen the chain becomes classical. The exact shapeof the oscillations envelope depends on the configura-tion of the offset charges q i . For the two calculatedcurves we have chosen the offset charges configura-tions q (0)2 , q (0)4 that give the best fit the experimentaldata. The exact values of the fit parameters are shownin table. I. Φ C q (2 e ) q (2 e ) C g ( aF ) C g ( aF ) C g ( aF )0 . π .
65 25 410 421 . π .
12 0 . Table I: Fit parameters for the calculated QPS interferencepatterns presented in Fig. 3b
The qualitative behavior of the interference pat-tern agrees perfectly with the theoretical expectationsbased on the simple picture of addition of complex am-plitudes. Using the fitted values of q (0) i we evaluatethe gate voltages corresponding to the special chargeconfigurations shown in Fig. 3a and indicated themon the measured and calculated curves. For instance,point ( i ) corresponds to the configuration where allcharges are zeros, the same configuration that was re-alized in measurements displayed in Fig. 2a at q = 0 .Notice that in the measurements presented in Fig. 3its position is shifted along the x axis by the offsetcharges on the islands and . Because for arbitrary v* → (a) v → v → v → v → v → v → (0, 0, 0, 0, 0) (i) (0, 0, 0.5, 0, 0) (0, 0.5, 0.5, 0.85, 0)(0, 0.5, 0, 0.85, 0) (ii) (0, 0.33, 0.5, 0.33, 0)(0, 0.33, 0, 0.33, 0) (iii) (0, 0.1, 0.5, 0.9, 0)(0, 0.1, 0, 0.9, 0) (iv) - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0 2 5- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 52 . 0- 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 ( b ) q ( i v )( i i i )( i i ) F C = 0 . 9 pF C = 1 . 1 p q / ( 2 e ) = 0 . 0 0 , q / ( 2 e ) = 0 . 6 5 q / ( 2 e ) = - 0 . 1 2 , q / ( 2 e ) = 0 . 4 5 d Isw (nA) q ( i ) - 2 5 - 2 0 - 1 5 - 1 0 - 5 0 5 1 0 1 5 2 0 2 5- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 52 . 0- 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 ( c ) q E J / E C = 3 d Isw (nA) q ( i ) ( i i ) ( i i i ) ( i v ) F C = 1 . 1 pF C = 0 . 9 p Figure 3: CQPS interferences induced by the polariza-tion charge on the middle and the first two lateral is-lands (0 , q , q , q , of the 6 Josephson junction chainat E J /E C = 3 . (a) Schematic representations of the 6phase-slip probability amplitudes v i (in black) and the to-tal CQPS amplitude v ∗ (in red) as vectors in the com-plex plane, for several particularly chosen charge config-urations. (b) The calculated switching current oscilla-tion, δI SW , induced by the polarization charges for a largesweep of the gate voltage V G , at two different phase biases.The polarization charge on the central island q is shownon the lower x-axis and the charge on one of the lateralislands q is shown on the higher x-axis. (c) The measured δI SW over a large sweep of V G at the same phase-biases Φ C as in (b). The value of Φ C for each curve is shown onthe right side of the figure. values of q i the total CQPS amplitude is less than N v ,one expects that maximal oscillations as a function of q occur around point ( i ) (see Fig. 3a). Indeed, nextto this point, the switching current δI SW oscillationshave the largest amplitude. The chain goes from theperfectly coherent phase-slips regime at q = 0 , wherethe switching current is minimum (the zero level inFig. 2b and c), to the maximally dephased configu-ration at q = 0 . when the phase slips are canceled,the chain is almost classical and the critical currentis enhanced. Point ( ii ) corresponds to opposite limitin which the oscillations are strongly suppressed dueto interference induced by non-zero charges at q , q . Working point ( iii ) shows the situation when the to-tal CQPS amplitude is suppressed at q = 0 and itnever reaches the maximum N v for any value of q (see Fig. 3a). In this case we expect small oscillationsof δI SW that reach the maximum supercurrent for theclassical chain. The case of ( iv ) shows that it is notnecessary to have the CQPS perfectly aligned as in ( i ) in order to have a large amplitude of δI SW oscilla-tions. As expected, the δI SW oscillations around ( iv ) are comparable in amplitude to the ones around point ( i ) .The Aharonov-Casher interference of phase slips isexpected to be a e periodic effect. In general, ran-dom e quasi-particle poisoning is a severe problemfor the observation of the interference effect as the e contamination reduces the accessible charge intervalfrom [0 , e ] to [0 , e ] . As a consequence, the ampli-tude of the oscillations in the interference pattern issignificantly reduced by e quasi-particle poisoning[8].In our case we observe the quasi-particle poisoning bythe appearance of a e periodicity in the δI SW vs. V G oscillations for temperatures above T = 300 mK . Atbase temperature T = 50 mK we scan the full chargespace interval [0 , e ] enabling the observation of com-plete destructive CQPS interference (see Fig. 2). Fur-thermore, spectroscopy measurements of propagativemodes in long SQUID chains [19] independently con-firms the value of the gate capacitance C g (cid:39) pF ,corresponding to a e periodic δI SW vs. V G curve atlow temperature.As can be seen from the periodic dependence of themeasured switching current, the island charge config-uration does not change significantly during the mea-surement. Although each measurement point implies repeated switchings into the dissipative state ofthe junctions, where large numbers of quasi-particlesare excited, after the circuit relaxes back to the dis-sipationless state, the charge configuration is stableenough in order to enable the measurement of the in-terference pattern. We have directly measured the fre-quency of random charge jumps by repeating the samemeasurement several times and we observe a typicaltime of τ qp ∼ minutes between changes in the islandcharge configuration. This time interval is sufficientin our case as it enables the measurement of severalhundreds of experimental data points.It is well established that slow drift of chargeinduced by fluctuating TLS leads to δq (cid:28) e attime scales of minutes[20]. A significant chargedrift at this and smaller time scales is attributed tonon-equilibrium quasi-particles jumping between theislands[21]; the equilibration of these quasi-particlesis made difficult by their localization in subgapstates[22]. Surprisingly, the time scales of thesejumps might be dramatically different even in sim-ilar devices. Charge fluctuation times, similar tothe one observed here, have been reported previ-ously in small highly resistive Josephson islands[23],in small charge-phase qubits[17, 24] and in Cooper-pair transistors[25, 26]. However much shorter times, τ qp < s , were reported for the fluxonium circuit[27]and even shorter times, τ qp ∼ µs , were reported forlarger devices such as the transmon[28]. Reasonably good agreement between our measure-ments and a model of independent phase slips mightbe surprising given the modest values of E J /E C =2 − of our Josephson chains, because this approxima-tion is expected to be correct only for E J /E C (cid:29) [15].Josephson junction circuits with E J /E C ∼ are typ-ically analyzed using the charge basis description andonly a couple of charging states are needed for accu-rate results. At larger E J /E C the number of requiredstates grows, which makes the problem of numericaldiagonalization difficult, especially for large systems.From a practical point of view it is important to com-pare both approaches and their validity as a functionof E J /E C , because the calculation is many orders ofmagnitude faster in the phase slip approximation, inparticular for systems containing a large number ofjunctions[29].We now discuss the details and the validity of thediagonalization of the full Hamiltonian of the chain: H = e (cid:80) i,j (cid:2) C − (cid:3) ij ( Q i − q i ) ( Q j − q j ) ++ (cid:80) i =1 E J [1 − cos ( ϕ i − ϕ i − )] (5)where Q i is the charge (in units of e ) on the i-th island and ϕ i is the superconducting phase on theisland. C − is the matrix of inverse capacitance ofthe chain. The first sum in the expression (5) is thecharging energy for the islands of the chain and thesecond sum represents the Josephson couplings forall junctions in the chain. As the total phase differ-ence γ across the chain is fixed, we have ϕ = 0 and ϕ = γ . The charges Q i are multiples of the elemen-tary charge of a Cooper pair. As Q i and ϕ i are conju-gate variables, the chain’s wavefunction ψ ( ϕ , ..., ϕ ) is π periodic in ϕ i . From the Hamiltonian (5) onecan see that its energy spectrum is a periodic func-tion of the polarization charges q i : indeed, any mod-ification of the polarization charge by the charge of aCooper pair q i → q i + 1 can be absorbed by the uni-tary transformation exp( i ˆ ϕ i ) which changes Q i into Q i + 1 , while leaving ϕ i invariant. Therefore the su-percurrent through the chain I ( ϕ , ..., ϕ ) remains un-changed when the polarization charges change by amultiple of the charge of a Cooper pair.The blue curves in Fig. 2b and c show the calcu-lated switching current oscillations, from the numer-ical diagonalization of the total Hamiltonian (5) (fordetails see the Methods section), as a function of V G for the charge configurations κ (cid:48) . These calculationsagree reasonably well with the switching current cal-culated using the semi-classical CQPS approximation(the MLG model). The modulation of the critical cur-rent expected theoretically is somewhat larger thanthe data. This is due to the fact that in the experi-ment the random charges q i are not exactly zeros; thisresults in the interference that decreases the observedamplitude of the switching current modulation as wediscussed above for the charge configurations κ (cid:48)(cid:48) .In conclusion, we have presented a quantitativestudy of the Aharonov-Casher effect exhibited byfluxon motion in a multi-junction circuit. We com-pared the data with the expectations based on thediagonalization of the full Hamiltonian of the chain inthe charge basis. Our results show that the groundstate properties of a short Josephson junction chaincan be fully understood in terms of phase slip dynam-ics even in a parameter range that has been tradition-ally described in terms of charge dynamics. The mea-surements also show that the polarization charges onthe islands of the chain can be controlled with suf-ficient precision and they are stable enough to en-able the observation of the chain’s collective behav-ior at the time scale of minutes. We believe thatour results will provide a starting point to reconsiderthe physics of large Josephson junction arrays, longJosephson chains and their possible applications tothe frequency-to-current conversion device or a topo-logically protected qubit.We would like to thank B. Pannetier, Daniel Es-teve, Frank Hekking, Gianluca Rastelli and ChristophSchenke for fruitful discussions. The research hasbeen supported by the European STREP MIDAS andthe French ANR QUANTJO. L. Ioffe acknowledgessupport from NSF ECS-0608842, ARO 56446-PH-QCand DARPA HR0011-09-1-0009. I. Protopopov ac-knowledges support from the Alexander von Hum-boldt Foundation and the DFG Center for FunctionalNanostructures. Methods
Switching current measurements
The switchingcurrent was determined from the switching probabilityat %. We apply ∼ bias-current pulses of am-plitude I bias and measure the switching probability asthe ratio between the number of switching events andthe total number of pulses. The measured switchingcurrent corresponds to the escape process out of thetotal potential energy containing the contributions ofthe read-out junction and the chain. We can calcu-late this escape process and therefore deduce the ef-fect of quantum phase-slips on the ground state of thechain [18]. From the escape rate, we can deduce theescape probability P SW as a function of the currentbias I bias and infer the theoretical switching current(at P SW = 50% ) for each biasing point (Φ C , V G ) . Numerical diagonalization of the exact chain Hamil-tonian
The Hamiltonian (5) gives the exact descrip-tion of the Josephson circuit (provided that all en-ergy scales remain small compared to Cooper gap ∆ and no other degrees of freedom are involved. Forthe purposes of numerical diagonalization one has tolimit the number of charging states on each island.This approximation can be easily tested by compar-ing the results of diagonalization for different numberof allowed charging states. For the problem here with E J /E C = 2 − it is sufficient to keep 7 charging statesto get the results with − accuracy. Supplementary information: The hopping term inthe Matveev-Larkin-Glazman theory of quantumfluctuations
Here we present the detailed derivation of the hop-ping term v ∗ of the MLG model in the charge frus-trated chain. Similar calculations have been per-formed for the Josephson chain [15] and for slightlydifferent Josephson circuits [30, 31]. To calculate thehopping term we need to find the classical trajecto-ries connecting states before and after one phase-slipevent. There are N such trajectories, each of themcorresponding to the phase slip occurring on a par-ticular junction in the chain. In a semi classical ap-proximation, the contribution of the phase slip in thejunction i to the hopping term is governed by theimaginary-time action S i on the corresponding tra-jectory: v i = Ae − S i (6)The prefactor A accounts for the contribution of thenon-classical trajectories close to the classical one thatdefines S i .In order to calculate the actions S i , we need to de-rive the complete Lagrangian for the Josephson chain.The electrostatic effects in the Josephson chain are de-scribed by the following Hamiltonian: H C = 12 (cid:88) i,j (cid:2) C − (cid:3) ij ( Q i − q i ) ( Q j − q j ) (7)The polarization charges q i = C gi V g e are controlledby the gate voltage. We would like to mention thatin our experimental setup we have added screeninglines to the central gate, in order to obtain a couplingto the central island at least 10 times larger than thecouplings to the rest of the chain: C g (cid:39) ∗ C g , C g (cid:39) ∗ C g , C g .Since the charges Q i and the phases of the islands ϕ i are canonical conjugate variables, the equation ofmotion for the phase reads: ˙ ϕ i = ∂H C ∂Q i = (cid:88) j (cid:2) C − (cid:3) ij ( Q j − q j ) = ⇒ Q i = (cid:88) j C ij ˙ ϕ j + q i (8)Using eq. (8) we can rewrite the charging Hamilto-nian (7) in the phase notation: H C = 12 (cid:88) i,j C ij ˙ ϕ i ˙ ϕ j (9)The charge part of the Lagrangian for the Josephsonjunction chain reads: L C = (cid:88) i Q i ˙ ϕ i − H C (10)Following formula (10) and using the expressions (8)and (9) we get for the charge Lagrangian the followingexpression: L C = (cid:88) ij C ij ˙ ϕ i ˙ ϕ j + (cid:88) i q i ˙ ϕ i − (cid:88) ij C ij ˙ ϕ i ˙ ϕ j L C = 12 (cid:88) ij C ij ˙ ϕ i ˙ ϕ j + (cid:88) i q i ˙ ϕ i (11)The capacitance matrix C ij contains the values ofall coupling between the islands. However, in real-ity, due to the geometry of the sample, the capaci-tance between first neighbors is orders of magnitudelarger then the stray capacitance between second or-der neighbors. This means that we can safely workwithin the so called nearest neighbor capacitance ap-proximation, and the matrix C ij only gets non zerocontributions for the elements closest to the main di-agonal: C − C ... − C C − C ... − C C ... ... ... ... ... − C − C C (12)Where C is the capacitance of one junction in thechain.Using the approximation (12) the expression of thecharge Lagrangian is simplified and it reads: L C = 12 (cid:88) i C ( ˙ ϕ i − ˙ ϕ i − ) + (cid:88) i q i ˙ ϕ i (13)Introducing the phase differences on the junctions θ i = ϕ i +1 − ϕ i and including the Josephson energy,we derive the complete Lagrangian of the chain: L = (cid:88) i (cid:34) ˙( θ i ) E C − E J cos θ i (cid:35) − (cid:88) i p i ˙ θ i , p i = i − (cid:88) j =1 q i (14)We can see that the Lagrangian (14) has two com-ponents which have very different physical conse-quences. The first sum that we call L is inde-pendent on the frustration charges q i . It gives acontribution to the real part of the phase-slip am-plitude v i , that is given by the Bloch band width v = 16 (cid:113) E J E C π (cid:16) E J E C (cid:17) . e − (cid:113) EJEC . For identical junc-tions in the chain, v is independent on the path chosenby the phase slip. The second sum of the Lagrangian(14), which we call δ L , has the form of a total timederivative. Hence, this term does not change the clas-sical equations of motion and the real part of the clas-sical action on a single trajectory. However, δ L givesthe tunneling amplitude along each path its own phasefactor. This phenomenon is mathematically equiva-lent to the AB effect for the phase variable ϕ i which is π periodic. Changing p i amounts to changing the pe-riodic boundary conditions ψ ( ϕ i + 2 π ) = e i πp i ψ ( ϕ i ) for the phase, in analogy to the motion of a chargedparticle on a circle, threaded by a flux tube.When a phase-slip occurs on junction i , the otherphase differences θ j are changed by: ∆ θ j = − πN + 2 πδ ij (15)Thus, the contribution to the phase-slip action fromthe j-th junction in the presence of charge frustrationreads: δS j = − i ˆ δ L dt = − i (cid:88) k p k ∆ θ k = − πip j − πiN (cid:88) k p k (16)Since the last term in the expression above does notdepend on k , it only adds an overall phase term forall phase slip trajectories, thus has no physical effecton the interference pattern and it can be dropped.Replacing this result in the formula (6), we get themathematical expression for the charge frustration de-phasing factor in the phase slip probability amplitudeof the j-th junction: δv j = e i πp j (17)So the phase slip probability amplitude on the j-th junction v j reads: v j = v exp (cid:34) i π j − (cid:88) k =1 q k (cid:35) (18)In other words, the absolute value of the probabilityamplitude for the QPS is the same as in the absence ofcharge frustration, but the geometric phase differencebetween the QPS is proportional to the total chargeon the islands between the junctions. Finally, the fullhopping term between the states | m (cid:105) and | m + 1 (cid:105) inthe presence of charge frustration is the sum of phaseslip amplitudes v i in all six junctions: v ∗ = (cid:88) i =1 v i (19)At zero gate voltage, the expression (19) reducesto v ∗ = N v that was used in the previous section tosolve the tight binding MLG Hamiltonian and calcu-late the expected switching current. Non-zero gatevoltage directly affects the interference of QPS bychanging the geometrical Aharonov-Casher phase dif-ference between phase slips in different junctions andthus provides a direct test for the quantum nature ofthe chain’s ground state. [1] Y. Aharonov and D. Bohm, “Significance of electro-magnetic potentials in the quantum theory,”
PhysicalReview , vol. 115, no. 3, pp. 485–491, 1959.[2] Y. Aharonov and A. Casher, “Topological quantumeffects for neutral particles,”
Physical Review Letters ,vol. 53, no. 4, pp. 319–321, 1984.[3] J. Clarke and A. I. Braginski,
The SQUID Handbook:Fundamentals and Technology of SQUIDs and SQUIDSystems , 1st ed. Wiley-VCH, August 2004.[4] R. F. Voss and R. A. Webb, “Phase coherence in aweakly coupled array of 20 000 nb josephson junc-tions,”
Physical Review B , vol. 25, no. 5, pp. 3446 –3449, Mar. 1982.[5] R. F. Voss, R. A. Webb, G. Grinstein, and P. M. Horn,“Magnetic field behavior of a josephson-junction ar-ray: Two-dimensional flux transport on a periodicsubstrate,”
Physical Review Letters , vol. 51, no. 8, pp.690 – 693, Aug. 1983.[6] C. D. Chen, P. Delsing, D. B. Haviland, Y. Harada,and T. Claeson, “Flux flow and vortex tunneling intwo-dimensional arrays of small josephson junctions,”
Physical Review B , vol. 54, no. 6, pp. 9449–9457, Oct.1996.[7] R. Fazio and H. van der Zant, “Quantum phase tran-sitions and vortex dynamics in superconducting net-works,”
Physics Reports-review Section of PhysicsLetters , vol. 355, no. 4, pp. 235–334, Dec. 2001.[8] W. J. Elion, J. J. Wachters, L. L. Sohn, and J. E.Mooij, “Observation of the aharonov-casher effect forvortices in josephson-junction arrays,”
Physical Re-view Letters , vol. 71, no. 14, pp. 2311–2314, Oct. 1993.[9] H. S. J. Van der Zant, F. C. Fritschy, T. P. Orlando,and J. E. Mooji, “Vortex dynamics in 2-dimensionalunderdamped, classical josephson-junction arrays,”
Physical Review B , vol. 47, no. 1, pp. 295–304, 1993.[10] E. Serret, “Etude de reseaux de nanojunction joseph-son: competition entre le champ magnetique etla geometrie,” Ph.D. dissertation, CRTBT, CNRS-Grenoble, 2002.[11] R. G. Chambers, “Shift of an electron interference pat-tern by enclosed magnetic flux,”
Physical Review Let-ters , vol. 5, no. 1, pp. 3–5, 1960.[12] A. Cimmino, G. I. Opat, A. G. Klein, H. Kaiser, S. A.Werner, M. Arif, and R. Clothier, “Observation ofthe topological aharonov-casher phase-shift by neu-tron interferometry,”
Physical Review Letters , vol. 63,no. 4, pp. 380–383, Jul. 1989.[13] K. Sangster, E. A. Hinds, S. M. Barnett, and E. Riis,“Measurement of the aharonov-casher phase in anatomic system,”
Physical Review Letters , vol. 71,no. 22, pp. 3641–3644, Nov. 1993.[14] J. R. Friedman and D. V. Averin, “Aharonov-casher-effect suppression of macroscopic tunneling of mag-netic flux,”
Physical Review Letters , vol. 88, no. 5, p.050403, Feb. 2002.[15] K. A. Matveev, A. I. Larkin, and L. I. Glazman, “Per-sistent current in superconducting nanorings,”
Physi-cal Review Letters , vol. 89, no. 9, p. 096802, 2002.[16] K. K. Likharev and A. B. Zorin, “Theory of the bloch-wave oscillations in small josephson-junctions,”
Jour-nal of Low Temperature Physics , vol. 59, no. 3-4, pp.347–382, 1985.[17] D. Vion, A. Aassime, A. Cottet, P. Joyez, H. Poth-ier, C. Urbina, D. Esteve, and M. H. Devoret, “Ma-nipulating the quantum state of an electrical circuit,”
Science , vol. 296, no. 5569, pp. 886–889, May 2002.[18] I. M. Pop, I. Protopopov, F. Lecocq, Z. Peng, B. Pan-netier, O. Buisson, and W. Guichard, “Measurementof the effect of quantum phase slips in a josephsonjunction chain,”
Nature Physics , vol. 6, no. 8, pp. 589–592, 2010.[19] I. M. Pop, “Quantum phase-slips in josephsonjunction chains,” Ph.D. dissertation, Universite deGrenoble, 2011, http://tel.archives-ouvertes.fr/tel-00572891/fr/.[20] N. M. Zimmerman, W. H. Huber, B. Simonds,E. Hourdakis, A. Fujiwara, Y. Ono, Y. Takahashi,H. Inokawa, M. Furlan, and M. W. Keller,“Why the long-term charge offset drift in sisingle-electron tunneling transistors is much smaller(better) than in metal-based ones: Two-levelfluctuator stability,”
Journal of Applied Physics , vol.104, no. 3, p. 033710, 2008. [Online]. Available:http://link.aip.org/link/?JAP/104/033710/1[21] J. M. Martinis, M. Ansmann, and J. Aumentado,“Energy decay in superconducting josephson-junctionqubits from nonequilibrium quasiparticle excitations,”
Phys. Rev. Lett. , vol. 103, no. 9, p. 097002, Aug 2009.[22] L. Faoro, A. Kitaev, and L. B. Ioffe, “Quasiparti-cle poisoning and josephson current fluctuations in-duced by kondo impurities,”
Phys. Rev. Lett. , vol. 101,no. 24, p. 247002, Dec 2008.[23] V. F. Maisi, Y. A. Pashkin, S. Kafanov, J. S. Tsai,and J. P. Pekola, “Parallel pumping of electrons,”
NewJournal of Physics , vol. 11, p. 113057, Nov. 2009.[24] A. Fay, E. Hoskinson, F. Lecocq, L. P. Levy, F. W. J.Hekking, W. Guichard, and O. Buisson, “Strong tun-able coupling between a superconducting charge andphase qubit,”
Physical Review Letters , vol. 100, no. 18,2008.[25] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M. H.Devoret, “2-electron quantization of the charge on asuperconductor,”
Nature , vol. 365, no. 6445, pp. 422–424, Sep. 1993.[26] E. Bibow, P. Lafarge, and L. P. Levy, “Resonantcooper pair tunneling through a double-island qubit,”
Physical Review Letters , vol. 88, no. 1, 2002.[27] V. E. Manucharyan, N. Masluk, A. Kamal, J. Koch,L. I. Glazman, and M. H. Devoret, “Evidence for co-herent quantum phase-slips across a josephson junc-tion array,” arXiv:1012.1928v1 , 2010.[28] J. A. Schreier, A. A. Houck, J. Koch, D. I. Schuster,B. R. Johnson, J. M. Chow, J. M. Gambetta, J. Ma-jer, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J.Schoelkopf, “Suppressing charge noise decoherence insuperconducting charge qubits,”
Phys. Rev. B , vol. 77,no. 18, p. 180502, May 2008.[29] W. Guichard and F. W. J. Hekking, “Phase-chargeduality in josephson junction circuits: Role of inertiaand effect of microwave irradiation,”
Physical ReviewB , vol. 81, no. 6, p. 064508, Feb. 2010.[30] J. R. Friedman and D. V. Averin, “Aharonov-casher-effect suppression of macroscopic tunneling of mag-netic flux,”
Physical Review Letters , vol. 88, no. 5, p.050403, Feb. 2002.[31] D. A. Ivanov, L. B. Ioffe, V. B. Geshkenbein, andG. Blatter, “Interference effects in isolated josephsonjunction arrays with geometric symmetries,”