Collective Mode and Its Interaction with Electromagnetic Field in Superconducting Tunnel Junction

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(1)Journal of the Physical Society of Japan. DRAFT. Collective Mode and Its Interaction with Electromagnetic Field in Superconducting Tunnel Junction Takanobu Jujo. ∗. Graduate School of Materials Science, Nara Institute of Science and Technology, Ikoma, Nara 630-0101, Japan (Received February 20, 2012). We investigate the properties of collective modes and their interaction with the electromagnetic field in a supercoducting tunnel junction. A microscopic calculation is performed by taking account of the tunneling and Coulomb interaction terms explicitly instead of using averaged terms as in previous works. The excitation energies of in- and out-of-phase collective modes are pushed up to finite values by the Coulomb interaction similarly to the plasma frequency in bulk superconductors, and the dispersion relation is also derived. The tunnel junction has different collective modes depending on the sites, and the absorption of microwave photons occurs at the interface between the superconductor and the insulating barrier. This dependence on sites is important in estimating the transition dipole moment, and the calculated result is much smaller than that obtained in phenomenological studies. KEYWORDS: tunnel junction, collective modes, superconductors, transition dipole, interface and bulk, gauge invariance. 1. Introduction A superconducting system is a candidate device for quantum information processing. This has the possibility of being realized by the circuit quantum electrodynamics, in which small Josephson junctions are treated as artificial atoms.1 (There are several types of superconducting quantum bit such as charge, flux, and phase qubits.2 ) The advantage of using this system is regarded as being that the transition dipole moment of the superconducting tunnel junction is 104 times larger than that of an atom.3 The experiment based on Cooper pair box has been performed,4 and the observation of vacuum Rabi splitting implies that the system is in the strong coupling limit. The basis for regarding a superconducting tunnel junction as an artificial atom lies in the competition between the Josephson coupling energy (EJ ) and the electrostatic ∗. E-mail address: jujo@ms.aist-nara.ac.jp 1/17.

(2) J. Phys. Soc. Jpn.. DRAFT. energy (EC ) at the mesoscopic scale.5 (In particular, the case of EJ /EC ≫ 1 is called the. transmon.6–8 ) This concept is usually studied with the use of a simplified Hamiltonian in which the phase and the particle number are treated as quantum mechanical operators. The microscopic basis of this model is considered to be given by integrating out fermions in the path-integral method.9 The interaction of a Josephson junction with a radiation field was also theoretically studied in ref. 10, though competition between the phase and the particle number was not taken into account [the tunneling term (EJ ) was neglected except for the interaction term with the external field]. In ref. 10, a uniform magnetic field was needed to observe coupling between the junction and the field. This is because the coupling term was not properly included. In this paper, we consider the problem of how a superconducting tunnel junction interacts with an electromagnetic field, and we estimate the coupling strength microscopically to determine the size of the transition dipole moment. To solve this problem,. it is necessary to clarify what the electromagnetic field interacts with and where it operates in the junction. It is shown that an out-of-phase collective mode of superconductors interacts with an electric field at the interfaces of the junction. In usual studies of the Josephson junction the dependence of the tunneling term on sites is not taken into account and the electrostatic potential is approximated with the use of a capacitance.9 These averaged terms are inappropriate for investigating the properties of collective modes. In §2 we give a formulation for a calculation of collective modes; the effective action. is obtained by integrating out the electrons and is later used to investigate the collective mode and its interaction with electromagnetic fields. In §3 we investigate the collective. modes of a Josephson junction and discuss how to separate the interface and bulk parts. Then the in- and out-of-phase modes between two superconductors are introduced, the spectra of these modes are calculated, and finally the gauge invariance of our formulation is shown. In §4, the response function is calculated and the transition dipole is estimated. We set ~ = c = 1 in this paper.. 2/17.

(3) J. Phys. Soc. Jpn.. DRAFT. 2. Formulation We consider a system in which two superconductors are coupled by a thin insulating barrier. The action for a tunnel junction is written as Z Z X ∂ ∂ L R R L S = dτ ψ¯iσ (τ ) ψiσ (τ ) + ψ¯iσ (τ ) ψiσ (τ ) + dτ [H0L (τ )+H0R (τ )+HC (τ )+HT (τ )]. ∂τ ∂τ iσ H0L,R is the Hamiltonian for the superconductor on the left and right sides of the junction. and is written as X X X L A ¯L L L L L L L L ¯L H0L (τ ) = −t [eA ψ¯iσ (τ )ψiσ (τ )−g ψ¯i↑ (τ )ψ¯i↓ (τ )ψi↓ (τ )ψi↑ (τ )]. ij ψiσ (τ )ψjσ (τ )+eji ψjσ (τ )ψiσ (τ )]−µ <i,j>σ. Here, eA ij = exp[ie. iσ. R ri rj. i. dr·A(r)] introduces the electromagnetic field, and t and g represent. the transfer term and superconducting interaction, respectively. The tunneling term and Coulomb interaction term are respectively written as follows: X R A ¯R L ¯L HT (τ ) = t′ [eA LR ψi′ σ (τ )ψi′ σ (τ ) + eRL ψi′ σ (τ )ψi′ σ (τ )]i′ =(x,y,z=1) , i′ σ. and. X e2 X L R 2 [δni (τ )Vij δnLj (τ ) + δnR (τ )V δn (τ )] + e δnLi (τ )Vij′ δnR ij j (τ ). i j 2 i,j i,j p P L L 2 2 2 ¯L Here, δnq i (τ ) = σ ψiσ (τ )ψiσ (τ ) − n0 , Vij = 1/ (xi − xj ) + (yi − yj ) + (zi − zj ) , Vij′ = 1/ (xi − xj )2 + (yi − yj )2 + d˜2ij , and d˜ij := |Zi − Zj | = d + (ziL − 1)a + (zjR − 1)a; HC (τ ) =. d is the width of the insulating barrier and a is the lattice constant. (We consider that. the dielectric constant of the insulator is effectively included in d, and this constant is not written explicitly.) The disposition of the junction is taken to be that in Fig. 1. (The large letter Z denotes the coordinate of this system, and z R,L denotes a label for each superconductor.) The z-axis is taken to be perpendicular to the interface. Both superconductors are attached to the insulator at z1 = 1; ziL and ziR are oppositely labelled in the z-axis, and then Z = −d/2 and d/2 correspond to z1L = 1 and z1R = 1, respectively. (Hereafter, we mainly use z as a label of 1 ≤ z ≤ Nz , and the corresponding. relation of Fig. 1 should be referred to in the case that z is used as a coordinate of the system.) HT and HC are different from those in previous works (for example, ref. 9) in. the point that the spatial variations of the tunneling and Coulomb effect are taken into account. These are important in studying the dispersion of collective modes. Taking account of the finiteness in the z-direction, the wave function of electrons is. 3/17.

(4) J. Phys. Soc. Jpn.. zLi. DRAFT. i=1 Nz. zR i. Nz. Z. d SC(L) Fig. 1.. SC(R). The right-hand and left-hand superconductors [SC(R) and SC(L)] are located at d/2 ≤ Z ≤. d/2 + (Nz − 1)a and −d/2 ≥ Z ≥ −d/2 − (Nz − 1)a, respectively. These coordinates are labeled as 1 ≤ ziR,L ≤ Nz for 1 ≤ i ≤ Nz .. written as11 L ψiσ. 1 = N. r. N X N X Nz X 2 L eikx x+iky y sin(kζ z)ψkσ . Nz + 1 n =1 n =1 x. y. ζ=1. We consider the periodic boundary condition for the superconducting plane parallel to the interface (the number of sites is N 2 ), and kζ = πζ/(Nz + 1). The Coulomb interaction term is written as X 1 XX δnq,z Vq;z,z ′ δn−q,z ′ , δni Vi−j δnj = 2 N ′ q i,j z,z P −iqx xi −iqy yi where δnq,z = δni . The discrete summation is approximated by inxi ,yi e. tegration, and then Vq;z,z ′ = p 2 qx + qy2 ).. ′ 2π −¯ e qa|z−z | a2 q¯. ′ and Vq;z,z ′ =. ′ 2π −¯ e q[d+(z−1)a+(z −1)a] a2 q¯. (¯ q =. With these setups, we can derive an effective action by using the Hubbard-. Stratonovich transformation and integrating out the electrons. The derivation is complicated because of the finiteness in the z-direction and is given in Appendix A. Then the result is written as follows. ϕL−q,z′ (−ωl ) −1 QL00 0 1X ′ (ωl ) q,zz L R Seff = { ϕq,z (ωl ) ϕq,z (ωl ) Vˆq;z,z ′ + ϕR (−ωl ) 0 QR00 (ωl ) 2 l,q −q,z ′ q,zz ′ QLµν (ωl ) 0 LR q,zz ′ 1 −1 µL µR ˜ ˜ + Qq (ωl )δz=z ′ =1 −1 1 δµ=ν=z + Aq,z (ωl ) Aq,z (ωl ) 0 QRµν (ω ) q,zz ′. +. R ϕL q,z (ωl ) ϕq,z (ωl ). . QL0µ (ωl ) q,zz ′. 0. 0. QR0µ (ωl ) q,zz ′. ˜νL ′ (−ωl ) A −q,z ˜νR ′ (−ωl ) A. l. . ˜µL (−ωl ) A −q,z ′ ˜µR (−ωl ) A −q,z ′. . +. ˜µL ˜µR A q,z (ωl ) Aq,z (ωl ). . −q,z. QLµ0 (ωl ) q,zz ′. 0. 0. QRµ0 (ωl ) q,zz ′. (1). The definitions of Q are given in Appendix B. The summation is taken over 1 ≤ z, z ′ ≤ 4/17. . ϕL (−ωl ) −q,z ′ (−ωl ) ϕR −q,z ′. }..

(5) J. Phys. Soc. Jpn.. DRAFT. Rz z Nz and µ, ν = 0, x, y, z. Here, A˜zL Aq,z ′ (ωl )dz ′ − 2ei ΦLq,z (ωl ) (with the use of q,z (ωl ) = i ˜zL ˜zL ˜zL ˜zR this A˜z , uzL q,z (ωl ) = −e[Aq,z+1 (ωl ) − Aq,z (ωl )] and wq (ωl ) = −e[Aq,z=1 (ωl ) − Aq,z=1 (ωl )]).. We discuss the collective modes and the response to electromagnetic fields on the basis of this effective action. 3. Collective Modes 3.1 Interface and bulk. As preparation for investigating the collective mode, we divide the above action into the bulk and interface parts. We make two approximations for the z-dependence of Qz,z ′ to perform this calculation. The first approximation is that we consider the case in which the interface (surface) part is on z = 1 (Nz ) and the bulk part is on 2 ≤ z ≤ Nz − 1. instead of considering a gradual change between the two parts. This corresponds to the approximation that Qq;z,z is independent of z for 2 ≤ z ≤ Nz − 1, which holds. approximately except for around z ≃ 2 or Nz − 1.12 The second approximation is that. Qq;z,z ′ 6= 0 only for |z − z ′ | ≤ 1.12 The assumption that Qzz q;1,z = 0 for z ≥ 3 indicates P zz zz that Qzz q;1,1 + Qq;1,2 = 0 because of the general relation z ′ Qq;z,z ′ = 0. In the same. zz zz way, Qzz q;z,z−1 + Qq;z,z + Qq;z,z+1 = 0 for z ≥ 2. This relation is used in the following. calculation.. By using the above approximations, we give an outline of the calculation in a symbolical way. We introduce thefollowing orthogonal matrix (t Uˆ Uˆ = ˆ1) to extract the  1 t~0 0   ~0 sˆ ~0. Here, sˆ is an (Nz − 2) × (Nz − 2) matrix with compointerface part: Uˆ =    t~ 0 0 1 q nents siζ := Nz2−1 sin(zi kζ ); zi = i−1 (2 ≤ i ≤ Nz −1) and kζ = π(ζ−1) (2 ≤ ζ ≤ Nz −1). Nz −1   a1 b~1 0   PNz −1 t~ ′ t ~ , ˆ = siζ siζ ′ = δζζ ′ .) We consider the following type of matrix: m ( i=2 b m ˆ b 1 N   ~ 0 bN aN where b~1 = (b1 , 0, · · · , 0), b~N = (0, · · · , 0, bN ), and (m ˆ ′ )ij = a′ δi,j + b′ δi,j±1 . The diagonalization of this matrix is partly achieved with the use of Uˆ ; Nz X. ~m ~ = tφ ~ Uˆ −1 Uˆ m ~ =φz=1 a1 φz=1 + ˆφ ˆ Uˆ −1 Uˆ φ φz mz,z ′ φz ′ =: t φ. z,z ′ =1. X. [φz=1˜b1ζ φ˜ζ + φ˜ζ ˜b1ζ φz=1 ] +. ζ. +. X. Here, φ˜ζ =. i=2. siζ φzi a ˜ζ = a′ + b′ 2coskζ , ˜b1ζ = b1 s2ζ , and ˜bNz ζ = bN sNz −1ζ . 5/17. φ˜ζ a ˜ζ φ˜ζ. ζ. [φz=Nz ˜bNz ζ φ˜ζ + φ˜ζ ˜bNz ζ φz=Nz ] + φz=Nz aN φz=Nz .. ζ. PNz −1. X.

(6) J. Phys. Soc. Jpn.. DRAFT. Here we discuss the two approximations made above. The first approximation should be revised in order to take account of the gradual change between the interface and bulk parts. This effect is included by the replacement of the above a1 and b1 with matrices (the bulk part a ˜ζ remains unchanged). This requires a calculation of matrices with dimensions larger than 2 × 2 to evaluate the first determinant on the right-hand side. of eq. (4). However, this modification does not change the q-dependence discussed in P §3.3 because it is derived from the term with ζ in eq. (4) (the interaction between. the interface and bulk parts). As for the second approximation, in the case of Qz,z ′ 6= 0 ˜ 1ζ = Q1,2 s2ζ and Q ˜ ζ = Qz,z + Qz,z+12coskζ are replaced by Q ˜ 1ζ = for |z − z ′ | ≥ 2, Q P PNz −1 ˜ ˆ z=2 Q1,z szζ and Qζ = Qz,z + n Qz,z+n 2cos(nkζ ), respectively, using the above U . This replacement does not alter the following discussion qualitatively.. 3.2 In- and out-of-phase modes √1 ( 1 1 ), we define the in- and out-of-phase modes as ϕ± (ωl ) := q,z 2 1 −1 1 R µ± µL µR ˜ ˜ ˜ ϕq,z (ωl )] and Aq,z (ωl ) := √2 [Aq,z (ωl ) ± Aq,z (ωl )]. Hereafter, we consider. Using the matrix √1 [ϕL (ωl ) q,z 2. ±. the case that the two superconductors are the same (|∆L0 | = |∆R 0 | = |∆0 |), and then. Rµν µν QLµν q,zz ′ (ωl ) = Qq,zz ′ (ωl ) = Qq,zz ′ (ωl ). (The static phases of the two superconductors can iφR be different; ∆L0 = |∆0 |eiφL and ∆R .) With this assumption the effective 0 = |∆0 |e. + − action given by eq. (1) is separated into in-phase (Seff ) and out-of-phase (Seff ) parts; + − Seff = Seff + Seff , 1X ± ±µν ± ± ± 00 ˜ν± ˜µ± {ϕq,z (ωl )[vq;z,z Seff = ′ + Qq,zz ′ (ωl )]ϕ−q,z ′ (−ωl ) + Aq,z (ωl )Qq,zz ′ (ωl )A−q,z ′ (−ωl ) 2 l,q 0µ µ0 ± ˜µ± ˜µ± +ϕ± q,z (ωl )Qq,zz ′ (ωl )A−q,z ′ (−ωl ) + Aq,z (ωl )Qq,zz ′ (ωl )ϕ−q,z ′ (−ωl )}.. (2) µν µν ± LR Here, Q+µν Q−µν q,zz ′ (ωl ) = Qq,zz ′ (ωl ) and q,zz ′ (ωl ) = Qq,zz ′ (ωl )+2Qq (ωl )δz=z ′ =1 δµ=ν=z . vq;1,1 = h i q(d+a) ¯ qa a2 q¯ 1+e−2¯ a2 q¯ a2 q¯ ± 1−e−2¯ −e−qd 1 ± qd )(1−e−2¯ qa ) ± 1−e−2¯ qd , vq;z,z |1<z<Nz = qa , vq;Nz ,Nz = qa , and 2π (1−e−2¯ 2π 1−e−2¯ 2π 1−e−2¯. ± ± vq;z,z−1 = vq;z−1,z =. ¯ a2 q¯ −e−qa qa 2π 1−e−2¯. ± (1 < z ≤ Nz ), and otherwise vq;z,z ′ = 0.. 6/17.

(7) J. Phys. Soc. Jpn.. DRAFT. 3.3 Calculation of collective modes 0 Firstly, we set A = 0 to discuss the collective modes, and then the action is Seff = P P Nz 1 t~ t~ ~ ˆ φφ Φq,z , and q,zz ′ φ−q,z ′ ; φq,z = ϕq,z z,z ′ =1 φq,z m q 2   −i 0b 0z 00 (−Qq,zz ′ qb + Qq,zz ′ ) vq,zz ′ + Qq,zz ′ 2e .  m ˆ φφ q,zz ′ = −i −i 2 a0 ab z0 az zb zz (q Q + Qq,zz ′ ) ( 2e ) (−qa Qq,zz ′ qb + qa Qq,zz ′ − Qq,zz ′ qb + Qq,zz ′ ) 2e a q,zz ′ (3). Here, we omit the suffix ±, and differences of the matrix elements exist only in vq;11 0 ˆ and Qzz q;11 (ωl ). By applying the above transformation matrix U to Seff , we can extract the interface part from the bulk part. The collective modes are found by calculating the determinant of the matrix m ˆ φφ , which is given by # N −1 " z Y X φφ Y φφ φφ −1 φφ φφ ˆ˜ φφ ). ˆ ˆ ˆ Det(m m ˜ q,zζ (m ˜ q,ζ ) m ˜ q,zζ Det m ˆ q,zz − Det(m ˆq ) = q,ζ z=1,Nz. (4). ζ=2. ζ. We neglect the coupling between z = 1 and z = Nz , which can be shown to be small ˆ˜ and m with the use of sNz −1ζ = (−1)ζ s2ζ . The relation between the components of m ˆ is the same as that in §3.1.. Firstly, we consider the collective mode of the bulk part. In this case, there is no ˆ˜ φφ is written difference between the in- and out-of-phase modes. The determinant of m q,ζ. as follows. 2 ˜ 00 ˜ zz ˜ iz ˜ ij ˜ 0z ˜ i0 ˆ˜ φφ )/˜ 4e2 Det(m q,ζ vq,ζ = q0 Qq,ζ + 2q0 qi Qq,ζ + qi qj Qq,ζ − 2q0 Qq,ζ − 2qi Qq,ζ − Qq,ζ ˜ 00 Q ˜ ij ˜ i0 ˜ 0j ˜ 00 ˜ iz ˜ i0 ˜ 0z ˜ 00 ˜ zz ˜ 0z ˜ z0 + qi (Q vq,ζ . q,ζ q,ζ − Qq,ζ Qq,ζ )qj − 2qi (Qq,ζ Qq,ζ − Qq,ζ Qq,ζ ) − (Qq,ζ Qq,ζ − Qq,ζ Qq,ζ ) /˜. (5). ˜ µν := Qµν + 2coskζ Qµν Here, Q and v˜q,ζ := vq;z,z + 2coskζ vq;z,z+1 = q;z,z+1 q;z,z q,ζ h i ¯ a2 q¯ 2e−qa a3 2 −¯ qa + (1 − coskζ ) 1−e−qa . For small q and kζ , v˜q,ζ ≃ 4π [¯ q + (kζ /a)2 ]; ¯ ) 1 − e ¯ 2π(1+e−qq. this corresponds to the inverse of the Fourier transform of the Coulomb interaction in 0z the three-dimensional system. With the use of Q0z q;z,z +2Qq;z,z+1 = 0 as mentioned above, ˜ 0z = Q0z +2coskζ Q0z ˜ zz = (1−coskζ )Qzz in the same Q = (1−coskζ )Q0z , and Q q,ζ. q;z,z. q;z,z+1. way. It is also shown that. ˜ 0i Q q,ζ. q;z,z. ∝ q0 qi and. ˜ az Q q,ζ. q,ζ. q;z,z. ∝ qa . By using these relations, eq.(5) = 0. ˜ 00 [(iq0 )2 − ω 2 ] = 0 in the limit of q¯, kζ → 0 [ω 2 := is written as Q q,z b b. 2 zz ˜ ij 4π qi Qq,z qj −kζ Qq;z,z /2 ]. 2 a (¯ q a) +kζ2. This equation has a solution at q0 6= 0. Therefore, our formulation of the bulk part is confirmed to obtain a well-known collective mode, which is pushed up to the plasma frequency (ωb ) by the Coulomb interaction.13 Next, we consider the collective mode at the interface. The determinant of m ˆ φφ q,11 −. 7/17.

(8) J. Phys. Soc. Jpn.. P. ζ. DRAFT. ˆ˜ φφ (m ˆ φφ −1 ˆ˜ φφ is written as follows. m q,1ζ ˜ q,ζ ) m q,1ζ X φφ ± ˆ˜ (m ˆ φφ −1 ˆ˜ φφ /(¯ 4e2 Det m ˆ φφ vq,1 + a′ F0 + v¯q,z F ′ ) m q,11 − q,1ζ ˜ q,ζ ) m q,1ζ ζ. ′ ± ′ ¯ 00 ¯ i0 ¯ ij ¯ ±zz ¯ 00 = q02 Q ¯q,z F ′ ) vq,1 +Q F + 2q0 qi QF + qi QF qj + [(−Qq,1 + d F0 )(¯ F + a F0 + v. (6). ′ ′ 2 ± ¯ 0b ¯ ij ¯ 00 ¯ i0 ¯ 0j − 2Q vq,1 + a′ F0 + v¯q,z F ′ ). F qb b F0 − (b F0 ) + qi (QF QF − QF QF )qj ]/(¯. (The derivation of this equation is given in Appendix C.) The collective mode at the interface is found by solving the equation Dq± := eq.(6) = 0. We investigate Dq± for small q to see how the Nambu-Goldstone mode is changed by + the Coulomb interaction. In the case of the in-phase mode v¯q,1 ≃ ad¯ q /4π (for d ≫ a) ¯ +zz = 0, then and Q q,1 " # + + ′ zz zz ¯ ij qj v ¯ + a F −F Q −F Q q Q v ¯ + v F 0 0 0 i q;z,z+1 0 q,1 q;z,z+1 q;z,z+1 q,z q,1 ¯ 00 q02 + + + + + . Dq+ ≃ Q F + + ¯ 00 v¯q,1 + a′ F0 v¯q,1 + vq;z,z+1F0 Q v¯q,1 + vq;z,z+1F0 v¯q,1 + vq;z,z+1F0 F r ¯ ij qi Q q a)2 q,z qj /(¯ ′ ′ Here, F0 ≃ −F0 q¯a and F0 := . In the case of the out-of-phase mode Qzz q;z,z+1. − LR ¯ −zz v¯q=0,1 = a2 /2πd and Q q,1 = 2Qq , then " ( # ) 00 v + Q 1 1 1 1 q;z,z+1 ¯ −zz q;z,z+1 ¯ 00 q 2 − Q ¯ −zz + F0′ Q q¯a . Dq− ≃ Q + Qzz q,1 F 0 q,1 q;z,z+1 − + ¯ 00 − ¯ 00 − + ¯ 00 v¯q,1 QF v¯q,1 QF v¯q,1 QF + In both modes the excitation energy is pushed up to a finite value [ωq=0 := r q F′ − 4π zz ¯ −zz −1 + ¯100 ] by the effect of the Coulomb Qq;z,z+1 1+aF0 ′ /d and ωq=0 := −Q q,1 d Q v¯ 0. F. q,1. interaction as in the bulk case. By comparing. + ωq=0. with ωb , it can be seen that the. excitation energy of the in-phase mode is reduced by a factor of a/d from the plasma frequency (a numerical calculation indicates that F0′ is on the order of 1). The physical − meaning of ωq=0 is discussed in §4.. Both excitation energies have a linear dependence on q. [The excitation energy is. ± written as (ωq± )2 ≃ (ωq=0 )2 + c± q¯a for small q.] This dependence has been reported. previously for the two-dimensional system, though the excitation energy is zero at q = 0.14 In a tunnel junction, a difference arises in the finite value of ωq± at q = 0. If we ± consider a single superconductor by setting d → ∞ (t′2 d → 0), then ωq=0 = 0 and this. reproduces the single two-dimensional case.. The dependence of the collective modes on q is different from the result of ref. 15. This is because the formulation in ref. 15 phenomenologically introduces the electrostatic energy to represent the Coulomb interaction and then it neglects the q-dependence in the superconducting plane. If we consider the left and right superconductors as exact 8/17.

(9) J. Phys. Soc. Jpn.. DRAFT. two-dimensional planes, then (ωq− )2 shows q 2 -dependence as in ref. 16 (in which the calculation focuses on a layered superconductor). It is the finiteness in the z-direction that changes the q-dependence. 3.4 Gauge invariance In this subsection, we consider a finite electromagnetic field (A 6= 0) and obtain the effective action for A by integrating out φ. By using the orthogonal matrix Uˆ defined above, the effective action from eq. (2) is rewritten as N. z 1XX t~ t~ t~ ~ q,i m ~ ~ ~ ~ [t φ ˆ φφ ˆ φA ˆ Aφ ˆ AA Seff = q,ij A−q,j ]. (7) q,ij φ−q,j + φq,i m q,ij A−q,j + Aq,i m q,ij φ−q,j + Aq,i m 2 q i,j=1 PNz + Here, the summation includes both the interface and bulk parts; i=1 fi = f1 PNz −1 t~ Φq,z=1(Nz ) , ζ=2 fζ + fNz . The components of the vectors are φq,1(Nz ) = ϕq,z=1(Nz ) P R Nz −1 t~ ~ q,2≤ζ≤Nz −1 = Aq,1(Nz ) = Aaq,z=1(Nz ) i z=1(Nz ) Azq,z ′ dz ′ , t φ s ϕ Φ zζ q,z q,z , z=2 P Rz ~ q,2≤ζ≤Nz −1 = Nz −1 szζ Aa and t A ˆ q,ij |i6=j = ˆ0 except i Azq,z ′ dz ′ (a = 0, x, y). m z=2 q,z. for i(j) = 1  and 2 ≤ j(i) ≤ Nz − 1, or i(j) = Nz and 2 ≤ j(i) ≤ Nz − 1. The matrices m ˆ  −i ˜ 00 ˜ 0b qb + Q ˜ 0z ) v˜q,ij + Q (−Q q,ij q,ij q,ij 2e  , m are m ˆ φφ = ˆ AA q,ij = q,ij −i −i a0 z0 2 ab az zb zz ˜ ˜ ˜ ˜ ˜ ˜ (q Q + Q ) ( ) (−q Q q + q Q − Q q + Q ) a a b a b q,ij q,ij q,ij q,ij q,ij q,ij 2e 2e     ab az 0b 0z ˜ ˜ ˜ ˜ Q Qq,ij Qq,ij Qq,ij  q,ij , m , and m ˆ φA =  ˆ Aφ q,ij q,ij = −i −i zb zz ab zb az zz ˜ ˜ ˜ ˜ ˜ ˜ Qq,ij Qq,ij (q Q + Q ) (q Q + Q ) a a q,ij q,ij q,ij q,ij 2e  2e  −i ˜ ab qb + Q ˜ az ) ˜ a0 (−Q Q q,ij q,ij  ˜ q,11 = Qq;z=1,z ′=1 , Q ˜ q,Nz Nz = Qq;z=Nz ,z ′=Nz ,  q,ij 2e . Here, Q −i z0 zb zz ˜ ˜ ˜ Q (−Q qb + Q ) q,ij. 2e. q,ij. q,ij. ˜ q,ζζ = Qq;z,z + 2cos(kζ a)Qq;z,z+1 (2 ≤ ζ ≤ Nz ), Q ˜ q,1ζ = Q ˜ q,ζ1 = s2ζ Qq;z=1,z ′ =2 , and Q ˜ to Q. ˜ q,Nz ζ = Q ˜ q,ζNz = sNz −1ζ Qq;z=Nz ,z ′=Nz −1 . v˜ has the same relation to v as Q Q ~ q,i from eq. (7), the effective action for the vector potential is After integrating out φ P P z t~ ˆ ~ written as SA = 21 q N i,j=1 Aq,i Kq,ij A−q,j . (The coupling between Az=1 and Az=Nz is ˆ q,ll = M ˆ AA − negligible because sNz −1ζ = (−1)ζ s2ζ , and we omit this coupling.) Here, K ˆ Aφ (M ˆ φφ )−1 M ˆ φA , M q,ll q,ll q,ll. ˆ q,lζ = N ˆ AA − M ˆ Aφ (M ˆ φφ )−1 N ˆ φA , K ˆ q,ζl K q,lζ q,ll q,ll q,lζ ˆ AA ′ δζζ ′ − N ˆ Aφ (M ˆ φφ )−1 N ˆ φA ′ ˆ q,ζζ ′ = N (l = 1, Nz ), and K q,11 q,ζζ q,ζ1 q,1ζ P Xφ φ φY XY XY −1 XY XY ˆ ˆ [M ˆ q,ij − ζ m ˆ q,iζ (m ˆ q,ζζ ) m ˆ q,ζj , N ˆ q,ij − q,ij = m q,ij = m. q,ll Aφ φφ AA −1 ˆ − Nˆ (M ˆ ) M ˆ φA , =N q,ζl q,ζl q,ll q,ll Aφ φφ φA −1 ˆ ˆ ˆ −N q,ζNz (Mq,Nz Nz ) Nq,Nz ζ ′ . m ˆ Xφ ˆ φq,ζζ )−1 m ˆ φY q,iζ (m q,ζj (X, Y =. A, φ; i, j = 1, ζ, Nz ; 2 ≤ ζ ≤ Nz − 1).]. We check the gauge invariance of this effective action SA . The gauge transformation. is generally given by Aµr (τ ) → Aµr (τ ) − ∂µ Ξr (τ ). For the case of a tunnel junction, this is Rz z ′ → tA ~ q,z = Aa ~ rewritten as t A −iΞ i A ′ dz 1 . [Aαq,z (ωl ) → Aαq,z (ωl )− q,z q,z qa q,z. q,z. 9/17.

(10) J. Phys. Soc. Jpn.. DRAFT. iqα Ξq,z (ωl ) (α = 0, x, y), and Azq,z (ωl ) → Azq,z (ωl ) − ∂Ξq,z (ωl )/∂z.] By performing this. transformation for A in SA , the coefficients of Ξq,z are shown to be zero by calculation. Therefore, the above action SA is invariant under the gauge transformation.. 4. Response Function R d/2+(z−1)a z ′ √i We consider A¯z− q,z := 2 −d/2−(z−1)a Aq,z ′ dz as an external field, which is a predominant component in the insulating barrier used as a cavity.10, 17 [A˜z− = A¯z− − i Φ− q,z. from the definitions below eq. (1) and in §3.2, and then the suffix z of L. q,z. A¯z− q,z. 2e. q,z. is equal to. R. z = z as a label in Fig. 1.] This field couples only to the out-of-phase mode as shown in eq. (2). The action is written as 1 X X t ~ − φφ− ~ ~ q,z m ~ ¯z− AzAz ¯z− ¯z− ¯z− t ~ Azφ′ φ ~ φAz [φ m ˆ ′ φ−q,z ′ + t φ SA = q,zz ′ A−q,z + Aq,z m q,zz −q,z ′ + Aq,z mq,zz ′ A−q,z ′ ]. 2 q z,z ′ q,z q,zz   0z Qq,zz ′ ˜ −zz ~ φAz′ =  , and ˆ AA Here, m ˆ φφ− q,zz ′ = Qq,zz ′ , m q,zz q,zz ′ is given in eq. (3), m −zz −i az + Qq,zz ′ ) (q Q 2e a q,zz ′ t Azφ −zz i m ~ q,zz ′ = Qz0 (Qzb q,ij qb − Qq,ij ) . The electric field is considered to exist only in q,zz ′ 2e ¯z− the junction part z1L ≤ z ≤ z1R , and then A¯z− q,z = Aq,1 for all z in the above SA . With P P −zz LR the use of z ′ Qaz q,zz ′ = 0 and z ′ Qq,zz ′ = 2Qq δz=1 , SA is rewritten as 1 X X t ~ − φφ− ~ − − ¯z− ¯z− − ¯z− LR ¯z− [ φq,z m ˆ q,zz ′ φ−q,z ′ + (−iQLR SA = q /e)(Φq,1 A−q,1 + Aq,1 Φ−q,1 ) + Aq,1 2Qq A−q,1 ]. 2 q ′ z,z. − By eliminating ϕ− q,z and Φq,z6=1 with the use of an orthogonal matrix as in the previous. section, the effective action is 1X − 1 −i LR − ¯z− ΦΦ − z− − z− LR ¯z− ¯ ¯ SA = Φq,1 m Φ + 2Q (Φq,1 A−q,1 + Aq,1 Φ−q,1 ) + Aq,1 2Qq A−q,1 . 2 q (2e)2 q,11 −q,1 2e q. LR ′ ′ ¯ ab ¯ a0 ¯ 0b Here, mΦΦ + d′ F0 + d0 F ′ − (qa Q q,11 = qa Qq,1 qb − 2Qq q,1 + b F0 + b0 F )(Qq,1 qb + − ¯ 00 + a′ F0 + a0 F ′ ); definitions of these quantities are b′ F0 + b0 F ′ )/(¯ vq,1 + Q q,1. given in the previous section and Appendix C. By using Dq− in §3.3, ¯ 00 + a′ F0 + v¯q,z F ′ ). Then mΦΦ ≃ = Dq− (¯ v − + a′ F0 + v¯q,z F ′ )/(¯ v− + Q mΦΦ q,11 q,11 −1 nq,1 q,1 h F i o 00 v +Q q;z,z+1 q;z,z+1 1 1 1 1 1 1 zz 2 LR ′ LR + Qq;z,z+1 v¯− + Q¯ 00 q¯a + Q¯ 00 + Q¯ 00 + F0 2Qq q0 − 2Qq ¯ 00 v¯− v¯− v¯− Q q,1. F. q,1. F. q,1. F. for small q. Finally, Φ− q,1 is eliminated, and then " # LR 2 (2Q ) 1 X ¯z− q 2QLR SA = A A¯z− q + −q,1 . 2 q q,1 mΦΦ q,11. 10/17. q,1. F.

(11) J. Phys. Soc. Jpn.. DRAFT. If Azq,z does not depend on z (for −d/2 ≤ z ≤ d/2), then A¯z− q,1 =. √i. 2. dAzq =. √id E z . 2q0 q. Then. z the response function under a uniform electric field (Eq=0 ) is given by. e2 d2 ǫJ . (iq0 )2 − ǫJ ǫC − 2 2 ¯ 00 ). Here, ǫJ = −QLR vq,1 + 1/Q q /e and ǫC = 2e (1/¯ F Kq=0 =. (8). The second term in the expression for ǫC indicates the screening effect of the. Coulomb interaction by the dielectric function at the interface. This effect is shown to be small by the numerical calculation of Q, and thus we neglect it hereafter. If we consider ǫC /N 2 (N 2 is the number of sites in a superconducting plane of area (Na)2 ), then ǫC /N 2 = e2 /[(Na)2 /(4πd)]. This corresponds to the electrostatic energy with (Na)2 /(4πd) as its capacitance. From Appendix B, ǫJ = 4t′2 |∆0 |2 cos(φL − 2 P sin2 kζ sin2 kζ ′ φR ) N12 Nz2+1 k,ζ,ζ ′ Ekζ E ′ (Ekζ +E ′ ) , and this expression indicates that ǫJ is the kζ. kζ. Josephson coupling energy per site. By defining EJ = N 2 ǫJ and EC = ǫC /N 2 , our result reproduces that of a quantized harmonic oscillator in which the Josephson cou-. pling energy competes with the electrostatic energy and the interval between discrete √ energy levels is EJ EC .5 √ From eq. (8), Kq shows that the absorption of a microwave photon occurs at EJ EC . i q h 2 ǫJ 1 1 √ √ The expression Kq=0 = (ed) − indicates that the transition 2 ǫC iq0 − ǫJ ǫC iq0 + ǫJ ǫC 1/4 . Then the transition dipole of the system as a whole is dipole per site is √ed2 ǫǫCJ 1/4 EJ ed √ . For instance we set d = 2 nm to estimate numerical values of this moment, 2 EC and then this is found to be smaller than 100 × eaB (aB is the Bohr radius) even in the. large limit of EJ /EC ≃ 100 such as the transmon.6 This value is much smaller than. that estimated in ref. 3 (104 × eaB ). A conceivable reason for this difference is that a microscopic derivation was not performed in their study but this large values were. derived from a macroscopic quantity such as the gap between the center conductor and the ground plane3 or the size of the superconducting island.6 Therefore it is possible that their estimation does not reflect an intrinsic transition dipole of a tunnel junction. 5. Summary and Discussion In this paper, we investigated the interaction between a superconducting tunnel junction and an electromagnetic field. In particular, we considered the absorption of a microwave photon by an out-of-phase collective mode. It was clarified where the absorption occurs, and the interaction strength was estimated. The transition dipole. 11/17.

(12) J. Phys. Soc. Jpn.. DRAFT. moment was found to be small compared with that of previous works. A possible reason for this difference is that the object with which an electric field interacts and its location were not specified in their studies. The distinction between the interface and bulk parts is important in investigating this interaction because these parts have different types of collective modes. In previous works, macroscopic quantities such as EJ and EC were used at the start of the calculation. In our study, we derived these quantities from a microscopic Hamiltonian; for example, the electrostatic energy was obtained microscopically from the Coulomb interaction between two superconductors. In §4, the response function was calculated under a uniform field (Eq=0 ). Here, we. estimate the coefficient of q¯a in mΦΦ q,11 to inspect the response to Eq6=0 . If we write the ¯ b , then ǫJ ǫC ≃ d |Q¯LR | . Then |Q¯LR | ≃ 10−10 for plasma frequency ωb in §3.3 as ωb2 = 4π Q a ωb a Qb Qb √ −4 LR ′ 2 ¯ ǫJ ǫC = 10 eV, ωb = 3 eV, and d/a = 10. [|Q |/Qb ∝ (t /t) × (∆0 /t) explains this ΦΦ small value.] mΦΦ ≃ v¯− [q 2 + ǫJ ǫC + (F0′ Qzz v1− )¯ q a], q,11 in §4 is approximated as mq,11 z,z+1 /¯ q 1 0 − ¯ b Qzz /(2|QLR |). This is roughly and the coefficient of q¯a is F0′ Qzz v1 ≃ ǫJ ǫC Q z,z+1 z,z+1 /¯. (100 eV)2 for the above values. Therefore, a response to a nonuniform field occurs at higher energies and is unrelated to microwave absorption. Our calculation does not take account of anharmonicity, and this can be included by performing a higher-order perturbative expansion of Φ. The results expected by doing. this expansion would be a shift of the energy level at which the absorption occurs and a finite width of the absorption spectrum. Therefore, a microscopic investigation of a small Josephson junction will possibly be useful for studying the decoherence effect, which is not yet well understood. Acknowledgement The numerical computation in this work was carried out at the Yukawa Institute Computer Facility. Appendix A: Derivation of eq. (1) Firstly, we perform the Hubbard-Stratonovich transformation for the interaction terms. h Z i e2 1 X X ¯ ˆ ′ δn−q,z ′ exp − dτ δn V q,z q;z,z 2 N2 q z,z ′ Z Z io n ie X ¯ 1 XhX −1 (δnq,z ϕ−q,z + ϕ¯q,z δn−q,z ) . ϕ¯q,z Vˆq;z,z = D[ϕL ϕR ]exp − dτ ′ ϕ−q,z ′ + 2 q N z ′ z,z. 12/17.

(13) J. Phys. Soc. Jpn.. Here, Vˆq;z,z ′ =. DRAFT ′ Vq;z,z ′ Vq;z,z ′ ′ Vq;z,z ′. Vq;z,z ′. . , δnq,z =. δnL q,z (τ ) , δnR (τ ) q,z. ¯ q,z = δn. R δnL q,z (τ ) δnq,z (τ ). −1 written in the same way as δnq,z . The inverse of Vˆq is written as Vˆq;z,z. , and ϕq,z is ′ 2 = a2πq¯ vvz′ vvzz , z. −qd ¯. −2¯ qa. −2¯ q(d+a). . 1+e 1 −e ′ ′ vz=1 = (1−e1−e qa , vz=Nz = 1−e−2¯ qa , vz=1 = 1−e−2¯ −2¯ qd )(1−e−2¯ qa ) , v1<z<Nz = 1−e−2¯ qd , v1<z≤Nz = 2 ¯ −1 −1 −e−qa 1 0 for 1 < z ≤ N , and otherwise V ˆ −1 ′ = ˆ0. 0; Vˆq;z,z−1 = Vˆq;z−1,z = a2πq¯ 1−e z −2¯ qa 0 1 q;z,z. After a similar transformation for the superconducting interaction and shift-. ing the phase of the wave functions; ψiσ (τ ) → exp[iΦi (τ )/2]ψiσ (τ ) and ∆i (τ ) → exp[iΦi (τ )]∆i (τ ), (hereafter we neglect the amplitude fluctuation, and then we re-. place ∆i (τ ) in the right-hand side by ∆0 ; the shift of the phases is performed on the fluctuating part and the static phase remains in ∆0 ; see, for example, ref. 18) R R R ¯ L ∆L ] D[∆ ¯ R ∆R ] D[ϕLϕR ]e−S ′ ) is written as S ′ = the effective action (e−S = D[∆ h i h i o R R P n L ∂ΦL ∂ΦR ∂ ∂ L R R i (τ ) i (τ ) dτ iσ ψ¯iσ (τ ) ∂τ + 2i ∂τ ψiσ (τ ) + ψ¯iσ (τ ) ∂τ + 2i ∂τ ψiσ (τ ) + dτ [HL′ (τ ) + ′ HR (τ ) + HC′ (τ ) + HT′ (τ )]. Here, X X L L L L L L HL′ (τ ) = − t {eaij ψ¯iσ (τ )ψjσ (τ ) + eaji ψ¯jσ (τ )ψiσ (τ )} − µ ψ¯iσ (τ )ψiσ (τ ) <i,j>σ. iσ. X1 L L L L (τ )ψ¯i↓ (τ ) − ∆L∗ + [ |∆L0 |2 − ∆L0 ψ¯i↑ 0 ψi↓ (τ )ψi↑ (τ )] g i R r with eaij = exp[ie rji dr · A(r) − 2i ΦLi (τ ) + 2i ΦLj (τ )], X HT′ (τ ) = t′ {eaLR ψ¯iL′ σ (τ )ψiR′ σ (τ ) + eaRL ψ¯iR′ σ (τ )ψiL′ σ (τ )}i′ =(x,y,z=1) , i′ σ. and. HC′ (τ ) =. 1 XX ie X X ¯ −1 ϕ¯q,z Vˆq;z,z δnq,z ϕ−q,z + ϕ¯q,z δn−q,z . ′ ϕ−q,z ′ + 2 q 2N q z ′ z,z. As a preparation for integrating out the electrons, we rewrite S ′ as follows. 2 ϕL (−ωl ) |∆R 1 XX L |∆L0 |2 0| −q,z ′ −1 2 ′ 2 ( ϕq,z (ωl ) ϕRq,z (ωl ) )Vˆq;z,z +βN Nz + S = βN Nz +S L+S R +S LR . ′ R ϕ (−ω ) l ′ g g 2 l,q z,z ′ −q,z. Here, ∆L0 = |∆L0 |eiφL , β = 1/T , and T is the temperature. ωl = 2lπT and ǫn = (2n+1)πT. are Matsubara frequencies. √ X T 2 XX X ¯L L L L −1 L ˆ L ′ ′ (ǫn −ǫ′ )ΨL′ ′ (ǫ′ ), ¯ ˆ Ψkζ (ǫn )U S = Ψkζ (ǫn )[−Gkζ (ǫn ) ]Ψkζ (ǫn )+ n n kζ k−k ,ζζ N N + 1 z ′ ′ ′ n,k,ζ n,n k,k. √. . ζ,ζ. L ′ N ′ ˆ k−k′ ,ζζ ′ (ǫn − ǫ ) ΨkR′ ζ′ (ǫn′ ) . √ δk,k′ δn,n′ Tˆζζ ′ + W S n Ψk′ ζ ′ (ǫn ) T ψkζ↑ ¯ = The definitions of these quantities are as follows: Ψk,ζ = ψ¯−kζ↓ , Ψk,ζ P i x ˆ k,ζ (ǫn )−1 = iǫn−ξ∗ kζ ∆0 , ξkζ = ψ¯kζ↑ ψ−kζ↓ , G ∆0 iǫn +ξkζ i=x,y,ζ ξk − µ [ξk = −2tcos(kx a)], LR. T 2 XXX = N Nz + 1 n,n′ k,k′ ζ,ζ ′. ¯ R (ǫn ) ¯ L (ǫn ) Ψ Ψ kζ kζ. . 13/17.

(14) J. Phys. Soc. Jpn.. DRAFT. √ P 1 ′ ′ ˆ ˆ k−k′,ζζ ′ (ωl ) = wk−k′ (ωl ) ˆ0 −ˆ1 t′ sinkζ sink ′ + T W ζ l′ ,q 2 wq (ωl )wk−k ′ −q (ωl − ωl )Tζζ ′ , ˆ N 1 ˆ 0 R d/2 ˆ Tˆζζ ′ = t′ sinkζ sinkζ′ τˆ0 τˆˆ03 , wk−k′ (ωl ) = −ie −d/2 Azk−k′,z ′ (ωl )dz ′ + 2i [ΦLk−k′ ,z=1(ωl ) − 3. ΦR k−k ′ ,z=1 (ωl )],. L Uˆk−k ′ ,ζζ ′ (ωl ) =. Nz X. ˆ L(1)′ (ωl )+Uˆ L(2)′ (ωl )]+ szζζ ′ [U k−k ,z k−k ,z. z=1. z=1. szζζ ′. =. sin(kζ z)sin(kζ′ z),. N z −1 X. sz+ ζζ ′. =. sin[kζ (z + 1)]sin(kζ′ z),. (a = 0, x, y and q0 = ωl ),. z+ ˆ L(z1) z+ z+ ˆ L(z2) [(sz+ ζζ ′ −sζ ′ ζ )Uk−k ′ ,z (ωl )+(sζζ ′ +sζ ′ ζ )Uk−k ′ ,z (ωl )], aL L AãL q,z (ωl ) = Aq,z (ωl ) − iqa Φq,z (ωl )/2e. L(1) Uˆk−k′ ,z (ωl ) = ie[ϕLk−k′ ,z (ωl ) + A˜0k−k′,z (ωl )]τˆ3 − e. L(2) Uˆk−k′ ,z (ωl ). =. √. X (ξ i − ξ i ′ ) k k ˆ A˜iL k−k ′ ,z (ωl )1, ′ k − k i i i=x,y. i − ξki ′+q iL ′ iL T X e2 X ξki + ξki ′ − ξk−q A˜q,z (ωl )A˜k−k′ −q,z (ωl − ωl′ )τˆ3 , ′ N l′ ,q 2 i=x,y qi (ki − ki − qi ). √ P L(z1) T −t zL ′ zL ′ ˆ ˆ L(z2) Uˆk−k′,z (ωl ) = tuzL k−k ′ ,z (ωl )1, Uk−k ′ ,z (ωl ) = N l′ ,q 2 uq,z (ωl )uk−k ′ −q,z (ωl − ωl )τˆ3 , and R z+1 z uzL Ak−k′ ,z ′ (ωl )dz ′ + 2i [ΦLk−k′ ,z+1(ωl ) − ΦLk−k′,z (ωl )]. k−k ′ ,z (ωl ) = −ie z. By performing the integration for electrons, the ′ ( D[ψ¯L ψ L ]D[ψ¯R ψ R ]e−S = e−Seff ) is written as 2 |∆R 1 XX L |∆L0 |2 0| −1 2 2 ( ϕq,z (ωl ) ϕRq,z (ωl ) )Vˆq;z,z +βN Nz + Seff = βN Nz ′ g g 2 l,q ′ R. z,z. −1 ˆ GˆL 0 −1 ˆ 0 GˆR. ˆ −1 = Here, G. effective. action. ϕL (−ωl ) −q,z ′ ϕR (−ωl ) −q,z ′. ˆ −1 +Uˆ +W ˆ +Tˆ ). −Trln(−G. ˆ , and Tˆ are given above. We perform the perturbaand Uˆ , W ˆ −1 + U ˆ +W ˆ + Tˆ ) ≃ −Trln(−G ˆ −1 ) + Tr(G ˆ Uˆ ) + tive expansion of ϕ, Φ, and t′ ; −Trln(−G 1 ˆ Uˆ G ˆ Uˆ Tr(G 2. . ˆW ˆG ˆW ˆ +G ˆ Tˆ G ˆW ˆ +G ˆW ˆG ˆ Tˆ ). Our perturbative expansion up to the +G. second order of Φ corresponds to EJ /EC ≫ 1 and neglects the anharmonicity, which is. close to the case of the transmon.6 With this approximation, Seff results in eq. (1). Appendix B: Calculation of Q. P explicit expressions for Q introduced in §2. ( ′ := P P P P ′′ 2 T T := e2 N 2 Nz +1 n,k ζ .) Here, we omit the sufe2 N 2 Nz +1 n,k ζ,ζ ′ and ˆ have the suffix L or R in fices L and R for simplicity except for QLR ; Q and G We . present 2 P 2. common. Q00 q,zz ′ (ωl ) = −. Qij q,zz ′ (ωl ) P′′. szζζ. ′ ˆ k+qζ (ǫn + ωl )ˆ ˆ kζ ′ (ǫn )ˆ τ3 ], τ3 G szζζ ′ szζζ ′ Tr[G j i i P′ z z ′ (ξk+q −ξk )(ξk+q −ξkj ) ˆ k+qζ (ǫn = sζζ ′ sζζ ′ Tr[G qi qj. P′. i i −2ξki +ξk+q ξk−q ˆ kζ (ǫn )ˆ Tr[G τ3 ]δij , qi2. i0 Q0i q,zz ′ (ωl ) = Qq,zz ′ (ωl ) = −i. P′. ′. szζζ ′ szζζ ′. i −ξki ξk+q ˆ k+qζ (ǫn Tr[G qi. 14/17. ˆ kζ ′ (ǫn )ˆ1] + + ωl )ˆ1G. ˆ kζ ′ (ǫn )ˆ1], + ωl )ˆ τ3 G.

(15) J. Phys. Soc. Jpn.. DRAFT. P ′ ˆ k+qζ (ǫn + ωl )ˆ1G ˆ kζ ′ (ǫn )ˆ1] + ′′ (2ξζ sz δz,z ′ + szζζ ′ szζζ ′ [−(ξζ − ξζ ′ )2 ]Tr[G ζζ z+ z′+ ˆ 2tsζζ δz ′ ,z+1 + 2tsζζ δz,z ′ +1 )Tr[Gkζ (ǫn )ˆ τ3 ], i i P ′ z ′ ξk+q −ξk z zi ˆ k+qζ (ǫn + ωl )ˆ1G ˆ kζ ′ (ǫn )ˆ1], s (ω ) = − s (ω ) = −Q Qiz (ξζ − ξζ ′ )Tr[G ′ ′ ′ ′ l l ζζ ζζ q,zz q,zz qi P′ z z ′ z0 ˆ k+qζ (ǫn + ωl )ˆ ˆ kζ ′ (ǫn )ˆ1], and τ3 G sζζ ′ sζζ ′ (ξζ − ξζ ′ )Tr[G Q0z q,zz ′ (ωl ) = −Qq,zz ′ (ωl ) = i P′ ′ z=1 2 ˆ L (ǫn + ωl )ˆ1G ˆ R ′ (ǫn )ˆ1 − ˆ L (ǫn )ˆ ˆ R ′ (ǫn )ˆ τ3 − G τ3 G QLR = (t sζζ ′ ) Tr[2G q (ωl ) k+qζ kζ kζ kζ R L ˆ ˆ ˆ ˆ Gk+qζ (ǫn + ωl )1Gkζ ′ (ǫn )1]. P P µν ¯ µν := The relation z szζζ ′ = δζζ ′ is used to calculate the quantity Q q,z z ′ Qq,zz ′ , and ¯ µν by using the following relations: Qzz Qµν is calculated with Qµν and Q = Qzz q,zz ′ (ωl ) =. P′. q;z,z+1. −Qzz q;z,z /2,. q,zz. Q00 q;z,z+1. q,z. q;z,z+1. ¯ 00 − Q00 )/2, and Q00 = Q ¯ 00 − Q00 . We set the tempera= (Q q,z q;z,z q;1,2 q,z=1 q;1,1. ture at absolute zero (T → 0) after the summation over the Matsubara frequency. The. results for q0 , qi = 0 are calculated as follows. 2 P 2 2 2 2|∆0 | cos(φL −φR ) ′ LR 2 1 Qq=0 = −2e N 2 Nz +1 k,ζ,ζ ′ (t sinkζ sinkζ ′ ) Ekζ Ekζ ′ (Ekζ +Ekζ ′ ) , ¯ 00 = e2 12 2 P [sin(kζ z)]2 |∆30 |2 , Q q=0,z k,ζ N Nz +1 Ekζ P 2 j ij 2 1 i 2 |∆0 | 2 ¯ Q 3 , q=0,z = e N 2 Nz +1 k,ζ vk vk [sin(kζ z)] Ekζ 2 P Ekζ Ekζ ′ −ξkζ ξkζ ′ +|∆0 |2 2 2 ′ 2 00 2 1 Qq=0;z,z = e N 2 Nz +1 , and sin (k z)sin (k z) ′ ζ ζ k,ζ,ζ Ekζ Ekζ ′ (Ekζ +Ekζ ′ ) 2 P 2 2 2 ′ 2 2 Ekζ Ekζ ′ −ξkζ ξkζ ′ −|∆0 | 2 1 ′) − sin (k z)sin (k z)(ξ − ξ Qzz = e ′ 2 ζ ζ ζ q=0;z,z ζ k,ζ,ζ N Nz +1 Ekζ Ekζ ′ (Ekζ +Ekζ ′ ) P 2 1 0| e2 N12 Nz2+1 k,ζ (2tsinkζ )2 |∆ 3 (1 − 2 δz=1 ). Ekζ q 2 + |∆0 |2 and further calculations are performed by numerical ξkζ Here, Ekζ = computation.. Appendix C: Determinant at the interface P 2 ˆ˜ φφ = s2ζ m Here, we show the derivation of eq. (6). By using m ˆ φφ q,12 and q,1ζ ζ s2ζ /cζ = P φφ φφ φφ φφ φφ −1 ˆ ˆ˜ (m ˆ ˆ φφ −1 for large Nz (cζ := −2 + 2coskζ ), m ˆ q,11 − ζ m ˜ q,1ζ = m ˆ q,11 + m q,12 + q,1ζ ˜ q,ζ ) m i h P s22ζ φφ φφ φφ φφ ˆ˜ )−1 m ˆ q,12 − cζ m ˆ q,12 (m ˆ q,12 . Each term is written as follows. q,ζ ζ cζ m   ± i ¯ 0b 00 ¯ Q q v¯ + Qq,1 2e q,1 b .  q,1 m ˆ φφ ˆ φφ q,11 + m q,12 = ±zz −i 1 a0 ab ¯ ¯ ¯ qa Q qb − Q ) 2 (qa Q 2e. q,1. q,1. 4e. ± ± + Here, v¯q,1 = vq,11 + vq,12 ; v¯q,1 |q→0 ≃. ¯ µν Q q,1. =. Qµν q,11. +. q,1. a2 q¯ (2 2π. Qµν q,12 .. . − + d/2a + a/2d), v¯q,1 |q→0 ≃. . . a2 , 2πd. . i i ′ a a b b φφ φφ −1 φφ 2e  2e 0  D0  D′  0 ˆ + c . m ˆ φφ − c m ˆ ( m ˜ ) m ˆ = ζ Dζ ζ q,12 q,12 q,12 q,ζ Dζ −i −i ′ 1 1 ′ b d b d 0 0 2 2 2e 4e 2e 4e 2 2 2 ′ ′ ′ ′2 2 2 ¯ 00 , Here, 4e D0 = a0 d0 − b0 , 4e D = a d − b , 4e Dζ = aζ dζ − bζ ; a0 = v¯q,z + Q q,z. ¯ 0b qb , Q q,z. d0 =. 2qa Qaz q;z,z+1. ¯ ab qb , qa Q q,z. −. ′. a =. Qzz q;z,z+1 ,. ′. vq;z,z+1 +Q00 q;z,z+1,. aζ = v˜q,ζ +. ˜ 00 Q q,ζ. ′. b =. 0z Q0b q;z,z+1 qb −Qq;z,z+1 , ′. = a0 + cζ a , bζ =. 15/17. ˜ 0b qb Q q,ζ. and. ′. b0 =. qa Qab q;z,z+1 qb −. d = ˜ 0z = b0 + cζ b′ , −Q q,ζ.

(16) J. Phys. Soc. Jpn.. DRAFT. ˜ ab qb − 2qa Q ˜ az − Q ˜ zz = d0 + cζ d′ . v¯q,z = vq;z,z + 2vq;z,z+1 = dζ = qa Q q,ζ q,ζ q,ζ µν µν ¯ for q → 0, and Qq,z = Qq;z,z + 2Qµν q;z,z+1 .. ¯ ) a2 q¯(1−e−qa ¯ ) 2π(1+e−qq. ≃. a3 q¯2 4π. We use the approximation Qq;1,2 ≃ Qq:z,z+1 (z ≥ 2) to perform this matrix calcula-. tion. This approximation is good for Qzz and Q0z but poor for Q00 . However, this does not change F0 (defined below, which induces a peculiar q-dependence) but modifies other coefficients. Then the collective mode at the interface is determined by X φφ φφ ˆ˜ (m ˆ φφ −1 ˆ˜ φφ m 4e2 Det m ˆ q,11 − q,1ζ ˜ q,ζ ) m q,1ζ ζ. ± ¯ 00 + a′ F0 + a0 F ′)(qa Q ¯ ab qb − Q ¯ ±zz + d′F0 + d0 F ′ ) − (Q ¯ 0b qb + b′ F0 + b0 F ′ )2 = 0. = (¯ vq,1 +Q q,1 q,1 q,1 q,1. (C·1) s22ζ D0 /(cζ Dζ ) and F ′ :=. P s22ζ D ′ /Dζ . The summation ζ is approxRπ P imated by integration for large Nz ( Nz2+1 ζ ≃ π2 0 dkζ ), and the calculated results Here, F0 :=. P. ζ. P. ζ. are. and. p F0 = − D0′. q. D0′ + 2 + 2. p. q p 1 − 4D0′ D ′′ − D0′ + 2 − 2 1 − 4D0′ D ′′ p , 2 1 − 4D0′ D ′′. q q p p ′ ′ ′′ D0 + 2 + 2 1 − 4D0 D − D0′ + 2 − 2 1 − 4D0′ D ′′ 1 p p p F ′ = ( D0′ + 4 + 16D ′′ + D0′ ) − . 2 8 1 − 4D0′ D ′′. [D0′ = (a0 d0 − b20 )/(−a′ d0 − a0 d′ + 2b0 b′ ) and D ′′ = (a′ d′ − b′2 )/(−a′ d0 − a0 d′ + 2b0 b′ )]. ¯ ab qb + qi (Q ¯ ij Q ¯ 00 ¯ i0 ¯ 0j j ∝ q¯2 for small q, and then Here, a0 d0 − b0 = v¯q,z qa Q q,z q,z q,z − Qq,z Qq,z )q r ¯ ij q 2 v¯q,z +qi Q q,z qj ′ F0 ≃ 0 in this limit. On the other hand, F 6= 0. F0 ≃ − −q2 v0q;z,z+1 for small q zz +Q 0 q;z,z+1 r vq;z,z+1 +Q00 q;z,z+1 ′ and F |q0,q=0 = 14 − − 12 . ¯ 00 Q q,z. By taking account of these properties of F0 and F ′ for small q, the determinant ¯ ab qb (¯ ¯ ±zz +d′ F0 )(¯ ¯ 00 + (×4e2 ) in eq. (C·1) is rewritten as qa Q v ± +a′ F0 +¯ vq,z F ′ )+(−Q v ± +Q F. q,1. q,1. q,1. F. ′ ¯ ab ¯ 00 ¯ i0 ¯ 0j ¯ ab ¯ ab ¯ 0b qb b′ F0 − (b′ F0 )2 + qi (Q ¯ ij Q a F0 + v¯q,z F ) − 2Q F F F − QF QF )qj (here, QF := Qq,1 + F Qq,z ). ′. ′. This leads to eq. (6).. 16/17.

(17) J. Phys. Soc. Jpn.. DRAFT. References 1) R. J. Schoelkopf and S. M. Girvin: Nature 451 (2008) 664. 2) J. Clarke and F. K. Wilhelm: Nature 453 (2008) 1031. 3) A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf: Phys. Rev. A 69 (2004) 062320. 4) A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf: Nature 431 (2004) 162. 5) M. Tinkham: Introduction to Superconductivity (McGraw-Hill, New York, 1996), Chap. 7. 6) J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf: Phys. Rev. A 76 (2007) 042319. 7) D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf: Nature 445 (2007) 515. 8) J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf: Nature 449 (2007) 443. 9) U. Eckern, G. Schön, and V. Ambegaokar: Phys. Rev. B 30 (1984) 6419. 10) P. A. Lee and M. O. Scully: Phys. Rev. B 3 (1971) 769. 11) E. Prada and F. Sols: Eur. Phys. J. B 40 (2004) 379. ′. ′. 12) The factor szζζ ′ szζζ ′ in Q (in Appendix B) is rewritten as 4szζζ ′ szζζ ′ = cos[kζ (z − z ′ )]cos[kζ ′ (z − z ′ )] − cos[kζ (z + z ′ )]cos[kζ ′ (z − z ′ )] − cos[kζ (z − z ′ )]cos[kζ ′ (z + z ′ )] +. cos[kζ (z + z ′ )]cos[kζ ′ (z + z ′ )], where 1 ≤ z, z ′ ≤ Nz . This indicates that these terms P oscillate rapidly with varying ζ for large |z ± z ′ |. Though the summation ζ,ζ ′ in ˆ kζ , Q gives finite values even in the case of z 6= z ′ because of the factor ξζ in G. its contribution is negligible for large |z ± z ′ |. Therefore, our approximation that Qz,z ′ 6= 0 only for |z − z ′ | ≤ 1 is considered to be valid.. 13) P. W. Anderson: Phys. Rev. 112 (1958) 1900. 14) R. Côté and A. Griffin: Phys. Rev. B 48 (1993) 10404. 15) S. E. Shafranjuk and J. B. Ketterson: Phys. Rev. B 74 (2006) 172501. 16) W.-C. Wu and A. Griffin: Phys. Rev. Lett. 74 (1995) 158. 17) J. Swihart: J. Appl. Phys. 32 (1961) 461. 18) R. M. Lutchyn, P. Nagornykh, and V. M. Yakovenko: Phys. Rev. B 77 (2008) 144516.. 17/17.

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