Voltage staircase in a current-biased quantum-dot Josephson junction
VVoltage staircase in a current-biased quantum-dot Josephson junction
D. O. Oriekhov, Y. Cheipesh, and C. W. J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands (Dated: February 2021)We calculate the current-voltage ( I - V ) characteristic of a Josephson junction containing a resonantlevel in the weakly coupled regime (resonance width small compared to the superconducting gap).The phase φ across the junction becomes time dependent in response to a dc current bias. Rabioscillations in the Andreev levels produce a staircase I - V characteristic. The number of voltagesteps counts the number of Rabi oscillations per 2 π increment of φ , providing a way to probe thecoherence of the qubit in the absence of any external ac driving. The phenomenology is the same asthe “Majorana-induced DC Shapiro steps in topological Josephson junctions” of Phys. Rev. B ,140501(R) (2020) — but now for a non-topological Andreev qubit. I. INTRODUCTION
A single-mode weak link between superconductors sup-ports a two-level system with a spacing that is adjustablevia the superconducting phase difference [1, 2]. BecauseAndreev reflection is at the origin of the phase sensitivity,the levels are called Andreev levels. Although their exis-tence was implicit in early studies of the Josephson effect[3], the characteristic dependence ∝ (cid:113) − τ sin ( φ/
2) ofthe level spacing on the phase φ , with τ the transmis-sion probability, was only identified [4] with the adventof nanostructures. The present interest in quantum in-formation processing is driving theoretical [5, 6] and ex-perimental [7–10] studies of Andreev levels as qubits.To assess the coherence of the qubit one would use ac microwave radiation of the two-level system and performa time-resolved detection of the Rabi oscillations of thewave function [11]. In this work we will show how a dc current I dc and measurement of the time-averagedvoltage ¯ V can be used to detect Rabi oscillations of anAndreev qubit: The staircase dependence of ¯ V on I dc counts the number of Rabi oscillations per 2 π incrementof φ .Our study is motivated by Choi, Calzona, andTrauzettel’s report [12] of such a remarkable effect(dubbed “ dc Shapiro steps”) in a Majorana qubit —which is the building block of a topological quantumcomputer. As we will see, neither the unique topologicalproperties of a Majorana qubit (its non-Abelian braidingand fusion rules) nor its specific symmetry class (classD, with broken time-reversal and spin-rotation symme-try) are needed, a similar phenomenology can be foundin a non-topological Andreev qubit with preserved sym-metries (class CI).The outline of this paper is as follows. In the nextsection II we present the model of the weak link that wewill consider: a quantum dot connecting two supercon-ductors with a tunnel rate Γ small compared to the super-conducting gap ∆ . Such a Josephson junction has beenextensively studied [13–15] in the regime where Coulombcharging and the Kondo effect govern the charge transfer[17–19]. We will assume the charging energy is small andtreat the quasiparticles as noninteracting. FIG. 1. Current-biased, resistively-shunted Josephson junc-tion, formed out of two superconductors (phases φ L and φ R )separated by an insulator containing a quantum dot (tunnelrates Γ L and Γ R from the left and from the right). The su-perconducting phases become time dependent when a voltagedifference V develops in response to a dc current I dc . The dynamics of a current-biased, resistively shuntedquantum-dot Josephson junction is studied in Secs. IIIand IV. The voltage staircase is shown in Fig. 3 and theone-to-one relationship with the number of Rabi oscilla-tions is in Fig. 6. In the concluding section V we willexplain why the substitution of the quantum dot by aquantum point contact will remove the voltage staircase.
II. ANDREEV LEVEL HAMILTONIAN
We consider the Josephson junction shown in Fig. 1,consisting of a quantum dot in the normal state (N) cou-pled via a tunnel barrier to superconductors (S) at theleft and right, with pair potentials ∆ e iφ L and ∆ e iφ R .We focus on the weakly coupled regime, when the tunnelrates Γ L and Γ R through the barrier are small comparedto ∆ .We assume that the fully isolated quantum dot has asingle electronic energy level E within an energy rangeΓ = Γ L + Γ R from the Fermi energy µ . The normal-state conductance G N is then given by the Breit-Wignerformula G N = 2 e h τ BW , τ BW = Γ L Γ R ( E − µ ) + Γ . (2.1)Coupling of electrons and holes by Andreev reflection a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b FIG. 2. Andreev levels ± E A ( φ ) according to the full expres-sion (2.2) (solid curve) and in the weak-coupling approxima-tion (2.4) (dashed curve, parameters E = 0 . µ = 0,Γ L = Γ R = 0 . ). from the superconductor produces a pair of Andreev lev-els at energies ± E A ( φ ), dependent on the phase differ-ence φ = φ L − φ R between the left and right supercon-ductors.A simplifying assumption of our analysis is that theCoulomb charging energy U is small compared to Γ andcan be neglected. If U is larger than Γ but still smallerthan ∆ , the main effect of the charging energy is a shiftof the energy level of the dot, E (cid:55)→ E + U/
2. Provided E > U becomes larger than ∆ the supercurrent is reduced by afactor Γ / ∆ because tunneling of a Cooper pair into thequantum dot is suppressed [17–19].To describe the non-equilibrium dynamics of the junc-tion we seek the effective low-energy Hamiltonian of time-dependent Andreev levels. This requires information notonly on the eigenvalues but also on the eigenfunctions.In subsections II A and II B we summarize results fromRefs. 16, 20–22 for the time-independent situation, whichwe need as input for the dynamical study starting fromsubsection II C. A. Andreev levels
For arbitrary ratio of Γ and ∆ the energies of theAndreev levels are equal to the two real solutions ± E A of the equation [20, 21]Ω( E, φ ) + Γ E (cid:113) ∆ − E = 0 , (2.2)with Ω( E, φ ) = (∆ − E ) (cid:2) E − ( E − µ ) − Γ (cid:3) + ∆ Γ L Γ R sin ( φ/ . (2.3) In the weak-coupling regime Γ (cid:28) ∆ , assuming also | E − µ | (cid:28) ∆ , this reduces to E A = ∆ eff (cid:113) − τ BW sin ( φ/ , ∆ eff = (cid:113) ( E − µ ) + Γ , (2.4)no longer dependent on ∆ . The two Andreev levels havean avoided crossing at φ = π , separated by an energy δE = (cid:112) E − µ ) + (Γ L − Γ R ) , (2.5)see Fig. 2.The equilibrium supercurrent, at temperatures k B T (cid:28) Γ, is given by I eq ( φ ) = − e (cid:126) dE A dφ = e Γ L Γ R sin φ (cid:126) E A ( φ ) , (2.6)with critical current (maximal supercurrent) I c = e (cid:126) (cid:18)(cid:113) ( E − µ ) + Γ − (cid:113) ( E − µ ) + Γ − Γ L Γ R (cid:19) . (2.7)There is no contribution from the continuous spectrumin the weak-coupling regime [20]. B. Effective Hamiltonian: time-independent phase
For time-independent phases the effective low-energyHamiltonian in the weak-coupling regime Γ (cid:28) ∆ followsfrom second-order perturbation theory [16, 22], H = − (cid:0) e iφ L Γ L + e iφ R Γ R (cid:1) a †↑ a †↓ + H.c.+ ( E − µ )( a †↑ a ↑ + a †↓ a ↓ ) . (2.8)Here a ↑ and a ↓ are the fermionic annihilation operatorsof a spin-up or spin-down electron in the quantum dot.The corresponding Bogoliubov-De Gennes (BdG)Hamiltonian H is a 4 × a ↑ , − a †↓ , a ↓ , − a †↑ ) and Ψ † , H = Ψ † · H · Ψ + E − µ. (2.9)It is block-diagonal, so we only need to consider one 2 × H = (cid:18) E − µ e iφ L Γ L + e iφ R Γ R12 e − iφ L Γ L + e − iφ R Γ R µ − E (cid:19) . (2.10)One readily checks that the eigenvalues ± E A of H aregiven by Eq. (2.4). C. Effective Hamiltonian: time-dependent phase
When the left and right superconductors are at dif-ferent voltages ± V /
2, the superconducting phase be-comes time dependent. We choose a gauge such that φ L ( t ) = φ ( t ) / φ R ( t ) = − φ ( t ) /
2, evolving in time ac-cording to the Josephson relation˙ φ ≡ dφ/dt = (2 e/ (cid:126) ) V. (2.11) The voltage bias imposes an electrical potential on thequantum dot, which shifts µ by an amount γeV with γ = (Γ L − Γ R ) / Γ.The time dependent BdG Hamiltonian then becomes H ( t ) = (cid:18) E − µ − (cid:126) γ ˙ φ ( t ) e iφ ( t ) / Γ L + e − iφ ( t ) / Γ R12 e − iφ ( t ) / Γ L + e iφ ( t ) / Γ R µ − E + (cid:126) γ ˙ φ ( t ) (cid:19) = (cid:2) E − µ − (cid:126) γ ˙ φ ( t ) (cid:3) σ z + Γ (cid:2) σ x cos φ ( t ) − γσ y sin φ ( t ) (cid:3) . (2.12)The Pauli matrices act on the electron-hole degree of free-dom. The corresponding current operator is given by I ( t ) = 2 e (cid:126) ∂∂φ H ( t ) = − e Γ2 (cid:126) (cid:2) σ x sin φ ( t ) + γσ y cos φ ( t ) (cid:3) . (2.13)Notice that the Hamiltonian (2.12) depends both on φ ( t ) and on ˙ φ ( t ), unless Γ L = Γ R . It is possible toremove the ˙ φ -dependence by a time-dependent unitarytransformation [23], but since this does not simplify oursubsequent calculations we will keep the form (2.12). III. VOLTAGE STAIRCASE
As shown in Fig. 1, a time-independent current bias I dc is driven partially through the Josephson junction,as a supercurrent I S ( t ), and partially through a parallelresistor R as a normal current I N ( t ) = V ( t ) /R . Substitu-tion of the Josephson relation (2.11) gives the differentialequation dφ ( t ) /dt = (2 eR/ (cid:126) )[ I dc − I S ( t )] . (3.1)Here we neglect the junction capacitance (overdampedregime of a resistively shunted Josephson junction) [24].The supercurrent is obtained from the expectationvalue I S ( t ) = (cid:104) Ψ( t ) | I ( t ) | Ψ( t ) (cid:105) , (3.2)where the current operator is given by Eq. (2.13) andthe wave function evolves according to the Schr¨odingerequation i (cid:126) ddt | Ψ( t ) (cid:105) = H ( t ) | Ψ( t ) (cid:105) . (3.3)As initial condition we take φ (0) = 0 and | Ψ(0) (cid:105) theeigenstate of the Andreev level at − E A for φ = 0. The dc current I dc is increased slowly from zero to some max-imal value and then slowly decreased back to zero. The I – V characteristic is obtained by averaging V ( t ) over amoving time window in which I dc is approximately con-stant.Results of this numerical integration are shown in Fig.3. We observe a staircase dependence of ¯ V on I dc . Thenonzero voltage appears at the critical current (2.7) forthe up-sweep and disappears at a slightly lower currentfor the down sweep. (A similar difference between switch-ing current and retrapping current was found for the Ma-jorana qubit [26].) The voltage steps at I dc > I c alsoshow hysteresis: the voltage jump up happens at larger dc current than the voltage jump down. IV. ANDREEV QUBIT DYNAMICS
The voltage staircase of Fig. 3 is a signature of Rabioscillations of the Andreev qubit formed by the two An-dreev levels in the Josephson junction, in much the sameway that the voltage steps of Ref. 12 were driven by Rabioscillations of a Majorana qubit. Let us investigate theAndreev qubit dynamics.
A. Adiabatic evolution
In the adiabatic regime of a slow driving, (cid:126) ˙ φ (cid:28) δE ,transitions between the Andreev levels can be neglectedand the phase evolves in time as an overdamped classicalparticle, ˙ φ + dU A /dφ = 0 , (4.1)moving in the “washboard potential” [24] U A ( φ ) = − (2 eR/ (cid:126) ) (cid:2) φI dc + (2 e/ (cid:126) ) E A ( φ ) (cid:3) , (4.2)plotted in Fig. 4. FIG. 3. Current-voltage characteristic of the quantum-dotJosephson junction, for two different parameter sets [25]. Theblue curve is for increasing dc current, the red curve for de-creasing current. The Andreev levels in Fig. 2 correspondto the parameters in panel a). The critical current (2.7) isindicated by the black arrow. The time dependence of the phase resulting from inte-gration of Eq. (4.1) is shown in panel a) of Fig. 5. Panelb) tracks the adiabatic dynamics of the Andreev qubit,by plotting the Bloch sphere coordinates R = ( X, Y, Z ),with R α ( t ) = (cid:104) Ψ( t ) | σ α | Ψ( t ) (cid:105) . The qubit dynamics is4 π -periodic in φ , because the Hamiltonian (2.12) is 4 π -periodic: When φ is increased by 2 π one has H (cid:55)→ σ z H σ z ,so on the Bloch sphere the qubit is rotated by π aroundthe z -axis ( X (cid:55)→ − X , Y (cid:55)→ − Y ). B. Pulsed Rabi oscillations
Panels c) and d) of Fig. 5 show the full non-adiabaticdynamics, obtained by integration of Eq. (3.3) for thesame parameter set as in panels a) and b). Transitionsbetween the Andreev levels produce pronounced Rabi os-cillations of the qubit, also visible as small oscillations in φ ( t ).Because the supercurrent carried by the two Andreevlevels ± E A has the opposite sign, the inter-level transi-tions reduce I S , thereby increasing I N = I dc − I S andhence ¯ V . This is evident from Fig. 5c, which shows that FIG. 4. Washboard potential (4.2) that governs the time de-pendence of the superconducting phase in the adiabatic limit.The curve is plotted for the junction parameters of Figs. 2and 3a, at a value of I dc slightly above the critical current I c . the first 2 π increment of φ , without interlevel transitions,takes a time δt ≈ (cid:126) / ∆ , while the second 2 π incre-ment, with Rabi oscillations, only takes a time δt = 700.The average voltage ¯ V (cid:39) π/δt is therefore increased bya factor 10 / φ crosses (2 n − π and increases rapidly to 2 nπ ,which is the steepest part of the washboard potential (seeFig. 4).To estimate the Rabi frequency we substitute Ψ( t ) = (cid:0) u ( t ) e iφ ( t ) / , v ( t ) e − iφ ( t ) / (cid:1) in the Schr¨odinger equation(3.3) and make the rotating wave approximation, dis-carding rapidly oscillating terms ∝ e iφ ( t ) : i (cid:126) ˙ u ( t ) = [ E − µ + eV ( t )] u ( t ) + Γ v ( t ) ,i (cid:126) ˙ v ( t ) = − [ E − µ + eV ( t )] v ( t ) + Γ u ( t ) . (4.3)(We have set Γ L = Γ R for simplicity.) If we further ne-glect the slow time dependence of the voltage, we obtainoscillations ∝ sin ω R t of the Bloch vector components X, Y, Z with Rabi frequency (cid:126) ω R = (cid:113) ( E − µ + eV ) + (Γ / . (4.4)The oscillations in Fig. 5d near t = 1000 × (cid:126) / ∆ have aperiod of 35 (cid:126) / ∆ , while T R = π/ω R = 40 (cid:126) / ∆ if we set V = RI dc , in reasonable agreement. C. Voltage steps count Rabi oscillations
The key discovery of Ref. 12 is that steps in the time-averaged voltage track the change in the number of Rabioscillations of the Majorana qubit per 2 π increment ofthe superconducting phase. Fig. 6 shows the same corre-spondence for the Andreev qubit.If we estimate the duration δt of a 2 π phase incre-ment by the product of the number N of Rabi oscilla-tions and the Rabi period T R , we obtain the estimate FIG. 5. Time dependence of the superconducting phase (top row) and of the Bloch sphere coordinates of the Andreev qubit(bottom row), in the adiabatic limit (left column) and in the non-adiabatic regime in which transitions between the Andreevlevels produce Rabi oscillations of the qubit (right column). The junction parameters are those of Fig. 3a, at I dc = 0 . e ∆ / (cid:126) .The wave function was initialized as an eigenstate of the lowest Andreev level − E A (0) at t = 0. (2 e/ (cid:126) ) ¯ V = 2 π/δt (cid:39) ω R /N . A stepwise decrease of N with increasing I dc would then produce a stepwise in-crease of ¯ V . This argument is suggestive, but does notexplain the sharpness of the steps. We have no quantita-tive analytical derivation for why the steps are as sharpas they appear in the numerics. V. DISCUSSION
Two lessons learned from this study are: 1) Rabi os-cillations of an Andreev qubit can be counted “one-by-one” without either requiring time-resolved detection or ac driving; 2) The voltage staircase phenomenology ofRef. 12 does not need a topological Majorana qubit — itexists in a conventional Andreev qubit.We worked in the weak-coupling regime Γ (cid:28) ∆ be-cause it simplifies the calculations, but also for a physicsreason: The voltage staircase is suppressed when Γ be-comes larger than ∆ , due to a well-known decoherencemechanism [27, 28]: Equilibration of the Andreev levels ± E A ( φ ) with the continuous spectrum at | E | > ∆ when φ crosses an integer multiple of 2 π . Let us discuss this ina bit more detail.For Γ (cid:29) ∆ the Andreev levels are given by E A = ∆ (cid:113) − τ BW sin ( φ/ , (5.1)according to Eq. (2.2), with τ BW the Breit-Wigner trans-mission probability (2.1). The difference with the weak-coupling result (2.4) is that the reduced gap ∆ eff hasbeen replaced by the true gap ∆ . This means the An-dreev level merges with the superconducting continuum whenever φ = 0 modulo 2 π . As the phase evolves in timein response to the current bias, each 2 π phase incrementwill restart from an equilibrium distribution.Now if we examine Fig. 5, panels c) and d), we see thatthe Rabi oscillations are pulsed by the rapid increase ofthe phase in the ( π, π ) interval, and only fully develop inthe (2 π, π ) interval. Equilibration at φ = 2 π will restartthe cycle from t = 0, suppressing the Rabi oscillationsand hence the voltage staircase.For the same reason a superconducting quantum pointcontact will not show the voltage staircase: its Andreevlevels also reconnect with the superconducting contin-uum at φ = 0 modulo 2 π .This argument points to one difference in the Majoranaversus Andreev phenomenology of the voltage staircase:A topological Josephson junction needs to be magneticin order to prevent the equilibration of the Majoranamodes with the continuum at φ = 0 modulo 2 π [29]. Ina non-topological quantum-dot Josephson junction thiscan achieved without breaking time-reversal symmetry.As a topic for further research, it would be worthwhileto see if the voltage staircase can be used to count thenumber of Rabi oscillations over multiple 2 π phase incre-ments, since that would provide additional informationon the coherence time of the qubit. This could involvethe constructive interference of Landau-Zener transitionsat φ = π, π, . . . [30]. FIG. 6. Top panel: portion of the I – V characteristic from Fig. 3a, with red dotted lines into the the bottom panels to showhow the voltage steps line up with the change in the number N of Rabi oscillations of the qubit in a 2 π phase increment δφ . ACKNOWLEDGMENTS
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