Aug 26, 2015 - Bound quasiparticle states exist in the normal region of the SNS junc- .... roots are to be taken such that Reqe,h â¥ 0, Imqe â¥ 0, a...

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arXiv:1508.06348v1 [cond-mat.supr-con] 26 Aug 2015

Guangzhou 510641, People’s Republic of China

Abstract In this work, we investigated the nonadiabatic transport properties of the one-dimensional timedependent superconductor-normal metal-superconductor (SNS) Josephson junction biased by a current source and driven by a high-frequency-ac-gate-potential applied to the normal-metal layer. BCS superconductors are considered and treated with the time-dependent Bogoliubov-de Gennes equation. Using Floquet theory, we compute the transmission coeﬃcients and the Wigner-Smith delay times as a function of the incident energy and ﬁnd that they display resonances when one of the electron or hole Floquet wavevectors coincides with the bound quasiparticle state within the superconducting energy gap. The resonance varies with the phase diﬀerence between the two superconductors as a result of the bound quasiparticle level displacement. The supercurrent ﬂowing through the SNS junction is dramatically enhanced by the resonances. PACS numbers: 72.10.-d, 74.25.F-, 74.45.+c

*

Corresponding author. Electronic address: [email protected]

1

I.

INTRODUCTION

The scattering process often involves interferences between different quantum paths, in which constructive interference corresponds to resonance and destructive interference to antiresonance of the transmission. Asymmetric antiresonance with a minimum followed by a maximum is called a Fano resonance1,2 . There have been a great number of studies devoted to resonance and/or antiresonance in various quantum processes, such as scattered by an Anderson impurity3 , tunneling through a quantum dot4–6 , scattering from a donor impurity in an electron waveguide7,8 , transport in spin inversion devices9 , Mie scattering in plasmonic nanoparticles and metamaterials10 , and etc. In nonadiabatic quantum pumping, Floquet sidebands are formed by high-frequency oscillating potentials. In the case of a timedependent quantum well, Fano resonance occurs in the transmission spectrum11–14 , pumped shot noise12,13 , and Wigner-Smith delay times (WSDT)11,14 , when the energy or wavevector of one of the Floquet levels matches the quasibound level inside. To our knowledge, these resonant effects have not been discovered in the time-dependent superconductor-normal metal-superconductor (SNS) Josephson junction driven by a nonadiabatic electric potential applied to the normal region. By means of the Josephson effect, supercurrents can flow through the SNS junction biased by a current source15 . Bound quasiparticle states exist in the normal region of the SNS junction, each consisting of equal probabilities of particle and hole states. The Josephson effect is related to the Andreev reflection, the latter of which is defined by reflection of a particle into a hole at a pair-potential boundary with no change of current16 . It can be conjectured from previous studies that in nonadiabatic quantum pumping of the SNS junction, resonance is possible to occur in the supercurrent when one of the Floquet sidebands coincides with one of the bound quasiparticle states within the energy gap of the superconductor, which will be confirmed by the present theoretic work. Our approach is based on various development in methodology and concept recently. In classical mechanics, the duration time of a scattering process can be defined by the energy derivation of the action. In the absence of a time operator in quantum mechanics, the quantum analog of the duration time can be defined as the energy derivative of the quantum mechanical phase shift τ = ~dφ/dE during the scattering process. In a multi-channel or dynamic scattering process, the distribution of the WSDT τ1 , · · · , τN are the eigenvalues of 2

the N × N Wigner-Smith matrix Q(E) = −i~S † ∂S/∂E with S the scattering or Floquet scattering matrix14,17 and N = 4 (2nmax + 1) with nmax the maximal Floquet channel index in the present dynamic electron-hole system. The density of states (DOS) ρ (E) is directly related to Q(E) and the WSDT τn by ρ (E) = (2π~)−1 TrQ (E) = (2π~)−1

X

τn .

(1)

n

The parametric conductance derivatives of a quantum dot was investigated by relating it to the distribution of the Wigner-Smith time-delay Matrix18 . And also the WSDT were investigated in the scattering by strong time-periodic driving fields11,14 . Experiments can yield information on the distribution of the WSDT. The driving field can be realized by applying an ultrahigh intensity laser or a local ac-signal top gate. In the research of the charge turnstile based on the superconducting hybrid structure, it was found that quantization of the current is affected by Andreev reflection and Cooper pair-electron cotunneling19 . Nonadiabatic effects in braidings of Majorana fermions in topological superconductors were studied using the time-dependent Bogoliubov-de Gennes (BdG) equations20 . The time-dependent BdG equation can be used to describe time-evolution of BCS superconductors under a parametrically time-dependent Hamiltonian20,21 . In this paper, we consider the transport properties of the time-dependent SNS Josephson junction biased by a current source and driven by a high-frequency-ac-gate-potential applied to the normal-metal layer. We use the Floquet scattering matrix method to solve the time-dependent BdG equation and relate the Floquet scattering matrix with the WSDT and the DOS. With the general relation between the Josephson supercurrent and the quasiparticle excitation spectrum22 as well as the bound quasiparticle states and the DOS of the continuous spectrum, the supercurrent is calculated.

II.

THEORETIC FORMALISM

We consider the nonadiabatic pumping properties in the one-dimensional time-dependent SNS Josephson junction biased by a current source and driven by a high-frequency-ac-gatepotential applied to the normal region. The time-dependent electric potential has the form of V1 cos ωt. Width of the normal region is L. The considered model is sketched in Fig. 1. The driving field can be realized by applying a local ac-signal top gate. Assuming the junction is located in the x-direction, the time-dependent BdG equation can be expressed 3

as20 i

u (x, t) d u (x, t) . = HBdG (t) dt v (x, t) v (x, t)

The time-dependent BdG Hamiltonian is HBdG (t) =

with h (x, t) = −

θ(x) is the step function.

h (x, t) †

∆ (x)

∆ (x) −h (x, t)

,

~2 ∂ 2 − EF + U (x, t) , 2m ∂x2

U (x, t) = θ (x) θ (L − x) V1 cos (ωt) , ∆ eφ/2 , x ≤ 0, 0 ∆ (x) = 0, 0 ≤ x ≤ L, ∆0 e−φ/2 , x ≥ L.

(2)

(3)

(4) (5)

(6)

In advance of the time-dependent treatment, we consider the bound quasiparticle states in the normal region within the energy gap of the superconductor. The quasiparticle energy ε is measured with respect to the Fermi energy EF . The current-flux normalized eigenfunctions of the static BdG equations are22 − + re ψS1,e + rh ψS1,h ,

x ≤ 0,

+ − + − aψN,e + bψN,e + cψN,h + dψN,h , 0 ≤ x ≤ L, + − te ψS2,e + th ψS2,h ,

with

(7)

x ≥ L,

1 ± ψN,e = (ke )−1/2 exp (±ike x) , 0 0 ± ψN,h = (kh )−1/2 exp (±ikh x) , 1 iη1e,h /2 e ± (2qe,h )−1/2 (ε2 /∆20 − 1)−1/4 exp (±iqe,h x) , ψS1,e,h = −iη1e,h /2 e

(8)

± while ψS2,e,h has η1e,h replaced by η2e,h and exp (±iqe,h x) replaced by exp [±iqe,h (x − L)]. h i1/2 1/2 1/2 1/2 Here, ke,h = (2m/~2 ) (EF + σe,h ε)1/2 , qe,h = (2m/~2 ) EF + σe,h (ε2 − ∆20 ) , η1e,h =

φ/2+σe,h arccos (ε/∆0 ), η2e,h = −φ/2+σe,h arccos (ε/∆0 ), σe = 1, and σh = −1. The square 4

roots are to be taken such that Reqe,h ≥ 0, Imqe ≥ 0, and Imqh ≤ 0. The function arccos t ∈ h i 1/2 [0, π/2] for 0 ≤ t ≤ 1, while arccos t = −i ln t + (t2 − 1) for t ≥ 1. Solvability of the continuity equations of the wave functions and their derivatives at x = 0 and x = L gives rise to the secular equation. Roots of ε of the secular equation are the bound quasiparticle energies Eb measured from the Fermi energy EF . These bound states consist of equal probabilities of particle and hole states. Numerical results of Eb as functions of ∆0 and φ are shown in Fig. 2. For a small superconductor energy gap ∆0 relative to the Fermi energy EF and a small normal-region width L considered here, only a single bound quasiparticle state is sustained in the normal region15 . The bound quasiparticle level is close to the gap surface and increases linearly with the gap width. It varies in an absolute trigonal function with the phase difference φ between the two superconductors. It approaches maximum at φ = 0 and 2π and approaches minimum at φ = π. These results reproduce those of Bardeen and Johnson15 in 1972. Now we use the Floquet scattering theory to solve the time-dependent BdG equation (2). Wave functions in the superconductor and normal regions can be written as +∞ ψS1,n , x ≤ 0, X e−iεn t ψN,n , 0 ≤ x ≤ L, ψ (x, t) = n=−∞ ψS2,n , x ≥ L,

(9)

with εn = ε + n~ω, ε an energy within the static continuous quasiparticle spectrum above the energy gap, n an integer varying from −∞ to +∞, and + − − + ψS1,n = aln ψS1,e,n + bln ψS1,h,n + cln ψS1,e,n + dln ψS1,h,n , +∞ P + − − + ψN,n = am ψN,e,m Jn−m (V1 /~ω) , + bm ψN,h,m + cm ψN,e,m + dm ψN,h,m m=−∞ − ψS2,n = arn ψS2,e,n l/r

(10)

+ + − + brn ψS2,h,n + crn ψS2,e,n + drn ψS2,h,n .

l/r

Here, an and bn are the probability amplitudes of the incoming electron and hole waves from the left/right superconducting electrode of the nth Floquet channel with energy εn , l/r

l/r

respectively, while cn and dn are those of the outgoing electron and hole waves. And we

5

have

while

1 ± ψN,e,n = (ke,n )−1/2 exp (±ike,n x) , 0 0 ± ψN,h,n = (kh,n )−1/2 exp (±ikh,n x) , 1 iη1e,h;n /2 e ± (2qe,h;n )−1/2 (ε2 /∆20 − 1)−1/4 exp (±iqe,h;n x) , ψS1,e,h;n = e−iη1e,h;n /2 ± ψS2,e,h;n

has

η1e,h;n

replaced

by

by

exp [±iqe,h;n (x − L)] with ke,h;n = i1/2 h 1/2 1/2 EF + σe,h (ε2n − ∆20 ) , η1e,h;n (2m/~2 ) η2e,h;n = −φ/2 + σe,h arccos (εn /∆0 ).

η2e,h;n

and

(2m/~2 ) =

1/2

exp (±iqe,h;n x)

(EF + σe,h εn )1/2 ,

(11)

replaced qe,h;n

φ/2 + σe,h arccos (εn /∆0 ),

= and

Jn (x) are the nth-order first kind Bessel func-

tions. By continuity of ψ and ∂ψ/∂x at the normal-superconducting interfaces, the electron-hole Floquet scattering matrix expressed as

cln

alm

ee eh ′ eh rnm rnm t′ ee nm t nm

alm

bl X r he r hh t′ he t′ hh bl dl X nm nm nm nm m m n Snm , = = ′ eh r ee eh ′ ee r cr a r t t r a m nm nm m nm m nm m n he hh ′ he ′ hh r r brm tnm tnm r nm r nm bm dn

(12)

can be obtained by matrix algebra (see the Appendix). We define the total electron and hole reflection and transmission coefficients as Ree = Tee =

+∞ P

n=−∞ +∞ P

n=−∞

ee 2 |r0n | , Reh =

2 |tee 0n | ,

Teh =

+∞ P

eh 2 r , 0n

n=−∞ +∞ P

n=−∞

2 teh 0n .

(13)

The summation is over all propagating modes and with a cutoff of nmax in numerical treatment. The Wigner-Smith matrix can be obtained by Q (ε) = −i~S † ∂S/∂ε. The DOS ρ (ε) is directly related to Q (ε) and the WSDT τn by Eq. (1). We can also define the total P WSDT as τWS = τn . Biased by a current source flowing from the S1 to the S2 supern

conducting electrodes and driven by a high-frequency-ac-potential in the normal region, the supercurrent flowing from the S1 to the S2 region can be calculated by the general relation22 I = I1 + I2 , 6

(14)

with 2e dEb tanh (Eb /2kB T ) , ~ dφ Z ∞ ∂ρ 2e dε ln [2 cosh (ε/2kB T )] . I2 = − 2kB T ~ ∂φ ∆0 I1 = −

III.

(15) (16)

NUMERICAL RESULTS AND INTERPRETATIONS

In our numerical treatment, experimentally realistic parameters are used. EF = 20 meV, ∆0 = 2 to 3 meV, ~ω = 5 to 6 meV, V1 = 1 meV, and nmax = 1 > V1 /~ω. Numerical results of the total electron and hole reflection and transmission coefficients defined in Eq. (13) are shown in Fig. 3. Sharp resonances occur in Ree , Reh , and Teh ; sharp antiresonances occur in Tee , when one of the electron or hole Floquet sidebands coincides with the bound quasiparticle state Eb by εe − ~ω = Eb in the electron channel and ~ω − εh = Eb in the hole channel within an accuracy of 0.001 meV. Eb = 1.99519 and 1.99487 meV can be obtained by solution of the secular equation for φ = 0 and 0.025 in radian, respectively. Current conservation secures unitarity of the Floquet scattering matrix S. It can be seen from Fig. 3 that Ree + Reh + Tee + Teh = 1, which confirms unitarity of the S matrix. From Fig. 2 it can be seen that Eb decreases as φ in creases for φ < π. As φ increases from 0 to 0.025 in radian, resonance from the electron Floquet channel occurs at a larger energy and resonance from the hole channel occurs at a smaller energy according to the relation εe − ~ω = Eb and ~ω − εh = Eb . It can also be seen that the resonance peaks of electrons are broader than that of the holes. This is because that the virtual mass of the hole is larger than that of the electron and the characteristic time of the former is longer than the latter, which is prominently seen in Fig. 4. In the range ∆0 < ε < ~ω + ∆0 we considered, ke,h;0, ke,h;±1, qe,h;0, and qe,h;±1 are all real in the normal region; ke,h;0, ke,h;+1, qe,h;0, and qe,h;+1 are real in the superconducting regions; ke,h;−1 is imaginary and qe,h;−1 complex in the two superconducting regions evanescent to ±∞. The incident quasiparticle state transfers through the propagating quasiparticle Floquet sideband and interferes with the direct transmission giving rise to a resonance. Sharp resonances also occur in the τWS (see Fig. 4) featuring the characteristics of the Floquet scattering matrix. The resonant τWS of the hole is larger than that of the hole because of larger virtual hole mass, which is natural in electron-hole systems. Our numerical

7

results also show that τWS outside of the resonant peak is 10−14 to 10−13 s, a normal WSDT in nanoscale tunneling. Values of the τWS at resonance are as high as 10−7 s, which is a dramatic enhancement in comparison with the normal 10−14 s and possible to be observed in experiment. The energy gap of BCS superconductors ∆0 is approximately related to the temperature by ∆0 = 3.2kB Tc (1 − T /Tc )1/2 with Tc the critical temperature23 . From Fig. 2 it can be seen that the bound quasiparticle energy increases linearly with ∆0 for fixed EF , L, and φ. It can also be seen from Figs. 3 and 4 that resonances in the reflection and transmission coefficients and τWS have extremely sharp peaks with their widths smaller than 0.0002 meV in a wide energy window of 5 meV. The peak energy is εe − ~ω = Eb and ~ω − εh = Eb within an accuracy of 0.001 meV. These results based on the time-dependent SNS Josephson junction suggest a potential way to detect the bound quasiparticle state energy, the superconducting energy gap, and hence the superconducting transition temperature. Supercurrent of the considered device are calculated by Eqs. (14) to (16). The numerical results are given in Fig. 5. The supercurrent flowing through a static Josephson SNS junction is15 in the order of meV · e/~. It can be seen from Fig. 5 that the supercurrent is dramatically enhanced in magnitude by the nonadiabatic driving force. This can be interpreted by the change in the continuous DOS above ∆0 as a result of the Floquet states formed. As demonstrated in Eq. (1), the DOS is proportional to the sum of the eigenvalues of the Wigner-Smith matrix Q (ε) = −i~S † ∂S/∂ε with S the Floquet scattering matrix. The eigenvalues of the Wigner-Smith matrix corresponds to the WSDT. The Floquet sidebands are formed in the high-frequency driven transport processes. The sidebands contribute to the dynamic DOS and enhance the supercurrent. From Figs. 3 and 4, it can be seen that when one of the Floquet sidebands coincides with the bound quasiparticle state, sharp resonances occur in the transmission and WSDT spectrum, which further enhances the supercurrent. Enhancement by the resonances is vital as the direction of the supercurrent is reversed against the original current bias. Energies of the peaks in τWS and hence in ρ corresponding to the hole-Floquet-channel resonance decreases with φ while energies of the peaks corresponding to the electron-Floquet-channel resonance increases with φ governed by the relations εe − ~ω = Eb and ~ω − εh = Eb as Eb decreases with φ for φ < π. I1 in Eq. (15) is proportional to −dEb /dφ and is positive for φ < π and negative for φ > π. I2 in Eq. (16) is proportional to the integral of −∂ρ/∂φ over ε. Contribution of the hole-channel 8

peak in ρ to I2 is negative for φ < π and positive for φ > π, while contribution of the electron-channel peak in ρ to I2 is reversed. Since the hole-channel peak in ρ is higher and sharper than the electron-channel, contribution of the former to I2 overweighs that of the latter, giving rise to the sign reversal of the supercurrent. As the resonances are extremely strong, the supercurrent in the nonadiabatic process is several orders larger than the static supercurrent. Since −∂ρ/∂φ is nonzero even when ρ is small at the energies away from the resonant peaks, the supercurrent varies in an irregular pattern as it is an integral result of −∂ρ/∂φ. For φ > π, Eb increases with φ. The supercurrent is positive or negatively small as a combined result of the resonant peaks and the integral of −∂ρ/∂φ over the large range of energy. The resonant peaks are sharper for stronger driving forces. As a result the supercurrent for ~ω = 6 meV is larger than ~ω = 5 meV for φ < π.

IV.

CONCLUSIONS

In conclusion, resonances are observed in the nonadiabatic transmission and WSDT spectra of the time-dependent Josephson SNS junction as a result of quantum path interferences. In the picture of Floquet scattering, the electron and hole Floquet sidebands are formed. When one of the electron or hole Floquet sidebands coincides with the bound quasiparticle state within the energy gap of the superconductors at εe − ~ω = Eb and ~ω − εh = Eb , resonances occur in Ree , Reh , Teh , and τWS ; antiresonances occur in Tee . The resonances are extremely sharp suggesting a potential experimental determination of Eb . Using the WignerSmith matrix, the dynamic DOS and supercurrent biased by a current source are obtained. Sharp resonances in the dynamic DOS consisting of Floquet modes greatly enhance the supercurrent in comparison with the condition without a pumping force.

V.

ACKNOWLEDGEMENTS

This project was supported by the National Natural Science Foundation of China (No. 11004063) and the Fundamental Research Funds for the Central Universities, SCUT (No. 2014ZG0044).

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VI.

APPENDIX: DERIVATION OF THE FLOQUET SCATTERING MATRIX

Continuity equations of ψ and ∂ψ/∂x [The Floquet state wave functions are defined in Eqs. (9) to (11)] at the normal-superconducting interfaces x = 0 and x = L can be expressed in matrix form as follows. e l h l e l h l M1u a + M1u b + M1u c + M1u d = MJe a + MJe c,

(17)

e l h l e l h l M1d a + M1d b + M1d c + M1d d = MJh b + MJh d,

(18)

e e e r h r e r h r M2+ a + M2− c = M2u a + M2u b + M2u c + M2u d,

(19)

h h e r h r e r h r M2− b + M2+ d = M2d a + M2d b + M2d c + M2d d,

(20)

e h e h e e M4u1+ al + M4u1− bl + M4u1− cl + M4u1+ dl = M5+ a + M5− c,

(21)

e h e h h h M4d1+ al + M4d1− bl + M4d1− cl + M4d1+ dl = M5− b + M5+ d,

(22)

e e e h e h M8+ a + M8− c = M4u2− ar + M4u2+ br + M4u2+ cr + M4u2− dr ,

(23)

h h e h e h M8− b + M8+ d = M4d2− ar + M4d2+ br + M4d2+ cr + M4d2− dr .

(24)

l/r

l/r

l/r

l/r

al/r , bl/r , cl/r , and dl/r are column vectors made up of elements an , bn , cn , and dn , respectively. Elements of the coefficient matrices are defined as follows: e,h M1u

e,h M1d

−1/4 = exp (iη1e,h;n/2) (2qe,h;n )−1/2 ε2n ∆20 − 1 δn,m ,

(25)

−1/4 = exp (−iη1e,h;n /2) (2qe,h;n )−1/2 ε2n ∆20 − 1 δn,m , MJe,h = Jn−m (V1 /~ω) (ke,h;m)−1/2 ,

(26)

nm

nm

nm

−1/4 e,h M2u = exp (iη2e,h;n/2) (2qe,h;n )−1/2 ε2n ∆20 − 1 δn,m , nm −1/4 e,h e,h −1/2 2 = exp (−iη2e,h;n /2) 2qsn εn ∆20 − 1 δn,m , M2d nm e,h = exp (±ike,h;m L) (ke,h;m)−1/2 Jn−m (V1 /~ω) , M2± nm −1/4 e,h δn,m , = ±iqe,h;n exp (iη1e,h;n /2) (2qe,h;n)−1/2 ε2n ∆20 − 1 M4u1± nm −1/4 e,h = ±iqe,h;n exp (−iη1e,h;n /2) (2qe,h;n )−1/2 ε2n ∆20 − 1 M4d1± δn,m , nm e,h M5± = ±ike,h;m (ke,h;m)−1/2 Jn−m (V1 /~ω) , nm −1/4 e,h δn,m , M4u2± = ±iqe,h;n exp (iη2e,h;n /2) (2qe,h;n)−1/2 ε2n ∆20 − 1 nm

10

(27) (28) (29) (30) (31) (32) (33) (34)

−1/4 e,h = ±iqe,h;n exp (−iη2e,h;n /2) (2qe,h;n )−1/2 ε2n ∆20 − 1 δn,m , M4d2± nm e,h M8± = ±ike,h;m exp (±ike,h;m L) (ke,h;m)−1/2 Jn−m (V1 /~ω) . nm

(35) (36)

The matrix equations (17) to (24) can be transformed into larger matrix equations as

e M1u

h M1u

e M1u

h M1u

Me h e h M1d M1d M1d 1d h e h Me 4u1+ M4u1− M4u1− M4u1+ e h e h M4d1+ M4d1− M4d1− M4d1+ | {z Ml

and

e M2+

0

e M2−

0

a

al

bl cl dl }

MJ

0

MJ

0

e M1u

h M1u

e M1u

h M1u

Mr

ar

br cr dr

(37)

M 01

M 02

a

0 M 0 M J J b = , e Me c 0 M 0 5− 5+ h h 0 M5− 0 M5+ d | {z }

0 Mh h e h e h 0 M2+ M1d M1d M1d b M1d 2− = e h e h Me c Me 0 M 0 8− 8+ 4u2− M4u2+ M4u2+ M4u2− h h e h e h 0 M8− 0 M8+ d M4d2− M4d2+ M4d2+ M4d2− | {z } {z |

Then we have

al

bl −1 −1 = (Mr) · (M02) · (M01) · (Ml) · cl dl

ar

br cr dr }

.

(38)

.

(39)

By defining Mltor = (Mr)−1 · (M02) · (M01)−1 · (Ml) , Waa Wab Wac Wad W W W W bb bc bd ba Mltor = , Wca Wcb Wcc Wcd Wda Wdb Wdc Wdd −Wac −Wad 0 0 −W −W 0 0 bc bd Mcd = , −Wcc −Wcd 1 0 −Wdc −Wdd 0 1

11

(40)

(41)

(42)

Waa Wab −1 0

W W 0 −1 bb ba Mab = , Wca Wcb 0 0 Wda Wdb 0 0

(43)

we can obtain the electron-hole Floquet scattering matrix as S = (Mcd )−1 · Mab .

(44)

In Eqs. (42) and (43), “0” is a zero (2nmax + 1) × (2nmax + 1) matrix and “1” is a unitary (2nmax + 1) × (2nmax + 1) matrix with nmax the maximal Floquet channel index.

12

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C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991).

23

J. Bardeen, L. N. Cooper, and J. R. Schrieﬀer, Phys. Rev. 108, 1175 (1957).

13

Floquet Scattering ÑΩ Eb D0

S1

N

S2

L

V1 cosΩ t FIG. 1: Model for the one-dimensional time-dependent Josephson SNS junction. Width of the normal region is L. Order parameters of S1 and S2 regions have the same absolute value of ∆0 and phase diﬀerence of φ. When ∆0 and the Fermi energy EF is small, only one bound quasiparticle state with the energy of Eb is conﬁned in the normal region consisting of equal probabilities of particle and hole states. A high-frequency-ac-gate potential of V1 cos ωt is applied to the normal region. Driven by the time-dependent potential, Floquet sidebands are formed with energy spacing ~ω. When the continuous spectrum above the energy gap has the electron or hole Floquet sideband falling within the energy gap and coincident with the bound state, resonance or antiresonance occurs in the transmission coeﬃcients and the Wigner-Smith delay times.

14

( 0.0

0.5

) 1.0

1.5

2.0

2.0 3.0

E

b

1.5

2.5

E

0.5

1.5

b

2.0

E (meV)

1.0

b

E (meV)

b

0.0 1.0

1.5

2.0

0

2.5

1.0 3.0

(meV)

FIG. 2: Bound quasiparticle energy Eb measured from the Fermi energy EF . Its variation as functions of ∆0 and φ corresponds to black and red symbols, respectively. Corresponding coordinates are in the same colors as the symbols. The parameters are EF = 20 meV, L = 10 ˚ A, φ = 0 for variation as a function of ∆0 and ∆0 =2 meV for variation as a function of φ, respectively.

15

0.25

0.20 0.20

0.25

0.15

0.20

0.15 e

h

0.15

e

0.15

0.10 0.05 0.05 0.00

0.05

0.00 3.0048

(a)

6.9948

(b)

3.0048

6.9948

0.00

0.10

eh

R

0.10

R

ee

h 0.10

0.20

0.05

0.00 0.25

1.0

0.25

1.0

0.9

0.9

0.7

0.7

0.15

e

h

h

0.6

0.15

0.10

e

0.5

0.3

3.0048

0.5

0.10

eh

0.05

0.4

0.6

0.20

0.20

(d)

T

ee

0.8

T

(c)

0.8

0.00

6.9948 3.0048

0.05

6.9948

=0 0.4

0.00

=0.025

0.3 3

4

5

6

7

8

(meV)

FIG. 3:

3

4

5

6

7

8

(meV)

Numerical results of the total reﬂection and transmission coeﬃcients (a) Ree , (b) Reh ,

(c) Tee , and (d) Teh as a function of the incident quasiparticle energy ε for diﬀerent φ. Sharp resonances occur in Ree , Reh , and Teh and sharp antiresonances occur in Tee , when one of the electron or hole Floquet sidebands coincides with the bound quasiparticle state Eb by εe − ~ω = Eb in the electron channel and ~ω − εh = Eb in the hole channel within an accuracy of 0.001 meV. Insets are zoom-in of the two resonant peaks. The parameters are EF = 20 meV, L = 10 ˚ A, ∆0 = 2 meV, ~ω = 5 meV, V1 = 1 meV, and nmax = 1. Eb = 1.99519 and 1.99487 meV for φ = 0 and 0.025 in radian, respectively.

16

1.2 1.2 1.0

1.0

h

0.6

0.8

-7

s)

0.8

WS

(10

0.4

0.6

e

0.2 0.0

0.4

3.0045

3.0048

6.9945

6.9948

6.9951

=0

0.2

=0.0025 0.0 3

4

5

6

7

8

(meV)

FIG. 4: Numerical results of the total WSDT τWS as a function of the incident quasiparticle energy ε for diﬀerent φ. Sharp resonances occur when one of the electron or hole Floquet sidebands coincides with the bound quasiparticle state Eb by εe − ~ω = Eb in the electron channel and ~ω − εh = Eb in the hole channel within an accuracy of 0.001 meV. Insets are zoom-in of the two resonant peaks. The parameters are the same as Fig. 3.

17

0.2

I (10

5

meV e/

)

0.0 -0.2 -0.4 -0.6

=5 meV -0.8

=6 meV -1.0 0.0

0.5

1.0

1.5

2.0

( )

FIG. 5:

Supercurrent as a function of φ for diﬀerent driving frequencies. The parameters are

EF = 20 meV, L = 10 ˚ A, ∆0 = 2 meV, V1 = 1 meV, and nmax = 1. We assume Tc = 30 K and hence T = 28.24 K.

18