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Justification of canonical quantization of Josephson effect in various physical systems

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Quantum devices based on the Josephson effect in superconductors are usually described by a Hamiltonian obtained by commonly used canonical quantization.
However, this recipe has not yet been rigorously and systematically justified. We show that this approach is indeed correct in a certain range of parameters. We find the condition of validity of such quantization and the systematic corrections to the Josephson energy EJ: namely, the capacitance
energy Ec = e^2/2C must be much smaller than the superconducting gap Δ. Moreover, we find an experimentally testable modification of Josephson energy at large capacitance energy also with nonlinear capacitance.

Reference
[1]. K.Pomorski, A.Bednorz, "Justification of canonical quantization of Josephson effect and its modification due to large capacitance energy", J. Phys. A: Math. Theor. 49, 2016
(http://iopscience.iop.org/article/10.1088/1751-8113/49/12/125002/meta )
[2]. K.Pomorski, A.Bednorz, "Justification of canonical quantization of Josephson junction", 2015 (http://arxiv.org/abs/1502.00511)

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Justification of canonical quantization of Josephson effect in various physical systems

  1. 1. Justification of canonical quantization of Josephson effect (and its modifications due to large capacitance energy) Krzysztof Pomorski and Adam Bednorz Uniwersytet Warszawski Wydział Fizyki Instytut Fizyki Teoretycznej E-mail:pomorski@fuw.edu.pl 27 października 2015 K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 1 / 29
  2. 2. Content 1 Introduction Simplest model of tunneling JJ RCSJ model of JJ and concept of washboard potential with quasiparticles Phase sc qubit Charge sc qubit 2 Research results Commonly used canonical quantization Second quantization JJ Hamiltonians JJ non-linear capacitance (combination of I and II quantization) Literature confirmation Corrections to Cooper pair box in I quantization Conclusions Future persepctives Literature Technical issues K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 2 / 29
  3. 3. Tunneling Josephson junction:simplest model Rysunek : Tunneling Josephson junction (JJ)[from F.Nori, Nature 2005] : H = HL + HR + HT with HL = EL|L >< L|, HR = ER |R >< R|, HT = ET (|R >< L| + |L >< R|) with |ψ >= ψL|L > +ψR |R >, ER , EL >> ET . H|ψ >= EL ET ET ER ψL ψR = i d dt ψL ψR (1) We obtain DC and AC Josephson relations having ∆φ = φR − φL = 2φ with I = I0 sin(2φ) and 2e d dt 2φ(t) = V (t). K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 3 / 29
  4. 4. RCSJ model and concept of washboard potential. I(t) = I0 sin(2φ) + 2e 1 R 2dφ dt + 2e C d22φ dt2 (2) 0 = d 2dφ U(2φ) + 2e 1 R 2dφ dt + 2e C d22φ dt2 , where (3) U(2φ, t) = I0 cos(2φ) − 2φI(t) is washboard-potential. Obvious analogies with anharmonic mechanical oscillator! For small Josephson junction R → ∞. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 4 / 29
  5. 5. Concept of washboard potential+quasiparticles:phase qubit We set R → ∞ and ∆φ = 2e x2 x1 Ax (x)dx quite much equivalent to London relation j = const × A. H|2φ >= (( i d 2dφ )2 −I ×2φ+EJ cos(2φ))|2φ >= (H0+U(2φ))|2φ > (4) Operators 2φ, i d 2dφ play role x and p. The commutation relation [x, p] = i is analogical to [ i d 2dφ , 2φ] = i . K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 5 / 29
  6. 6. Superconducting Cooper pair box qubit Rysunek : Schematics of single Cooper pair box, from Physica Scripta 77, V.Bouchiat, 1997 N|n >= n|n >, Hint = −EJ (|i >< i+1|+|i+1 >< i|), Hint = −EJcos(2φ), (5) [n, 2φ] = i , n = i d 2dφ (6) H = EC (N − n0)2 + Hint (7) K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 6 / 29
  7. 7. 3 basic superconducting qubit architectures Rysunek : From ’Physics Today’ F.Nori 2005 K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 7 / 29
  8. 8. Structures considered-Josephson junction biased by different circuits K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 8 / 29
  9. 9. Commonly used canonical quantization of Josephson effect, which was never fully justified! In first quantization we have: HC = Q2 /2C = CV 2 /2 (8) where Q is the charge and V is the voltage between 1 and 2. HL = LI2 /2 = ( φ/e)2 /2L (9) HJ = −e I1 cos(2φ)/2 (10) The canonical quantization means that we take the Hamiltonian H(φ, Q) = HC (Q) + HL/I (φ) + HJ(φ) (11) as a function of conjugate variables φ and Q and replace everything by their quantum counterparts , including conversion of Poisson bracket (φ, Q) = e/ into commutator between [φ, Q] = ie. It essentially means Q → e∂/i∂φ. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 9 / 29
  10. 10. Hamiltonian of the point-like tunneling JJ in second quantization (and 2 parameters Θ and φ) . Having Hamiltonian parts responsible for bulk superconductor hBCS and non-superconducting region heT (in second quantization using simplified Bogoliubov-de Gennes formalism) of the following form hBCS = −i∂x ∆ ∆ i∂x eh , heT = −τδ(x) 0 eiφ e−iφ 0 12 . (12) Since HT transfers between 1 and 2 the states θ are in different superconductors. In momentum space we have k = ∆ sinh θ and |θ± = (2 cosh θ)−1/2 e±θ±/2 ±e θ±/2 (13) in the eh basis, with ±(θ) = ±∆ cosh θ and Ee = +(θ+) − −(θ−). K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 10 / 29
  11. 11. We want to obtain/ compare it with Josephson junction Hamiltonian in first quantization H = HL + HJ + HC . (14) We define H = H − HL = +HJ + HC (15) K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 11 / 29
  12. 12. The full eigenproblem reads E|ψ = H|ψ (16) where the zero order state is |g with energy 0. Since HT is small, all states |gφ will also reduce to |g in zero order. Therefore we can write an ansatz for |ψ in the form |ψ = dφ(ψ(φ)|gφ + e ψe(φ)|eφ ), (17) where the latter sum is of higher order. Assuming explicit knowledge of EC = − e2 2C ∂2 φ with E = E − HL/I = EJ − e2 2C ∂2 φ and plugging this form into the eigenproblem we get H |ψ >= E ψdφ|gφ + e E ψe|eφ = dφ Egφ − e2 2C ∂2 φ ψ|gφ = + e Eeφ − e2 2C ∂2 φ ψe|eφ (18) . Sandwiching it with gφ| we get K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 12 / 29
  13. 13. Non-linear capacitance in tunneling JJ E ψ = Egφψ − e2 2C ∂2 φψ − (∂φψ)(e2 /C) gφ|∂φ|gφ −ψ(e2 /2C) gφ|∂2 φ|gφ − e2 2C e gφ|∂2 φψe|eφ . (19) The factor gφ|∂φ|gφ is an analogue to the differential Berry phase, which can be arbitrarily chosen. For our purpose, it is convenient to assume that it is zero. Only relative Berry phase for excited states would matter but only at high perturbation order. In the lowest order, we obtain E ψ Egφψ − e2 2C ∂2 φψ − ψ(e2 /2C) gφ|∂2 φ|gφ . (20) K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 13 / 29
  14. 14. Simple identities The term gφ|∂2 φ|gφ can be evaluated inserting identity between derivatives, gφ|∂2 φ|gφ = gφ|∂φ|gφ gφ∂φ|gφ + e gφ|∂φ|eφ eφ|∂φ|gφ (21) since we assume zero differential ground Berry phase, and from orthogonality between g and e we get gφ|∂2 φ|gφ = − e | gφ|∂φ|eφ |2 = − e | eφ|∂φ|gφ |2 (22) . K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 14 / 29
  15. 15. Definition of c modifying factor One can write down c(φ) in terms of first-quantized (oc-cupied and em-pty) single particles states c(φ) = oc,em | ψem|∂φ|ψoc |2 . (23) One can also use the adiabatic identity gφ|∂φ|eφ = gφ|(∂φHT )|eφ Eeφ − Egφ (24) because only HT depends on φ. Here time is replaced with phase difference and adiabatic approximation is taken! K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 15 / 29
  16. 16. Reference to Berry phase and scattering matrix For the nondegenerate eigenvalues the differential Berry phase is defined by ψ|∂φ|ψ = iγ (25) where γ is real because 0 = ∂φ ψ|ψ = i(γ − γ∗). For degenerate case we collect all the states of the same energy. We construct matrix γ with ψm|∂φ|ψn = iγmn (26) Note that 0 = ∂φ ψm|ψn = i(γmn − γ∗ nm) which shows that γ is Hermitian. Therefore for scattering matrices S iγ = S† ∂φS/2 (27) because the outgoing waves are halves of full waves. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 16 / 29
  17. 17. Numerical evaluation of c(φ) 0 0.2 0.4 0.6 0.8 1 t 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ 0 0.5 1 1.5 2 2.5 3 c(φ) Rysunek : The exact dependence of c(φ) on φ ∈ [0, π/2] and t referenced at φ = 0, c is even in φ and has the period π. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 17 / 29
  18. 18. Literature confirmation The next order term renormalizes capacitance e2 /2C → e2 /2C − t2 (3 − cos 2φ)e4 /32C2 (28) Note that higher order terms will also contain boundary deviations of ∆ near the tunneling point (also in Egφ). However, our correction remains dominant in the case of many independent channels. Anyway, one can calculate exact c(φ) analytically for all t (not only small) assuming constant (bulk) ∆, K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 18 / 29
  19. 19. Large capacitance energy and Cooper pair box qubit Rysunek : The relevant states of occupation for the Cooper pair box. H|n > EC (N − n0)2 − EJ i (|i >< +2| + |i + 2 >< i|). (29) By diagonalization, the difference of the energy levels of the qubit reads 2 4N2 g E2 C + E2 J . K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 19 / 29
  20. 20. Corrections to Cooper pair box energy We work in the base |N > explicitly assume that Josephson energy is of the form HJ = N EJ(|N + 2 N| + |N N + 2|) (30) In the vicinity of Ng = 0 the relevant states are |N = ±1 = |± and the total effective Hamiltonian can be written as H = (N2 g + 1)EC + 2Ng EC (|− −| − |+ +|) + EJ(|+ −| + |− −|) (31) The qubit operation range is achievable if Ng EC ∼ EJ even if EC is large (for sufficiently small Ng ). By diagonalization, the difference of the energy levels of the qubit reads 2 4N2 g E2 C + E2 J . However, if EC ∼ ∆, the above approach is incorrect because the Josephson term is different. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 20 / 29
  21. 21. The modified Josephson energy reads EJ = t 2t2 ∆2 dθ+dθ− (2π)2 (∆(cosh θ+ + cosh θ−) − EC )−1 (32) . 0.5 1.0 1.5 2.0 EC 1.5 2.0 2.5 E’J EJ Rysunek : The dependence of EJ /EJ on EC /∆ in the Cooper pair qubit. At EC → 2∆ there is logarithmic divergence K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 21 / 29
  22. 22. Conclusions We show that this approach is indeed correct in certain range of parameters. We find the condition of the validity of such quantization and the lowest corrections to the Josephson energy. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 22 / 29
  23. 23. Future perspectives: 1 Extension results to triplet superconductor. 2 Extension results to multiband superconductor. 3 Extension results to triple superconductor Josephson junction. 4 Accounting for effects in flux-phase Josephson junction. 5 Josephson effect in superconducting quark-gluon plasma? 6 Considerations of structure Sc-qdot-Sc. 7 Considerations of Sc-Fe-Sc tunneling Josephson junction. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 23 / 29
  24. 24. References 1. B.D.Josephson, Phys. Lett. 1, 251 (1962) 2. M.Thinkham-Introduction to superconductivity (2004) 3. K.Pomorski, A.Bednorz, http://arxiv.org/abs/1502.00511 (2015) 4. U.Eckern, G.Schon, V. Ambegaokar, Phys. Rev. B 30 6419 (1984) 5. G.Schon, and A.D.Zaikin, Phys. Rep. 198, 237 (1990) 6. A.Barone- ’Physics and applications of Josephson effect’ 7. F.Nori- ’Physics today’ 2005 8. P.de-Gennes-’Superconductivity of Metals and Alloys’ 9. M.V.Berry, 1984, Proc. R. Soc. London, Ser. A 392, 45 10. ’Superconducting Qubits: A Short Review’, M. H. Devoret, A. Wallraff, J. M. Martinis 11. G.E.Blonder, M.Tinkham, T.M.Klapwijk, BTK paper, Physical Review B, vol. 25,no. 7, pp. 45154532, 1982. 12. F.Romeo, R.Citro , Physical Review B 91, 035427 (2015) Minimal model of point contact Andreev reĆection spectroscopy of multiband superconductors K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 24 / 29
  25. 25. Computed scattering matrix Normal scattering matrix is then obtained from solutions of the eigenproblem for (12) Se = r iteiφ ite−iφ r (33) with t = sin(τ/ ) and r = √ 1 − t2,Sh = ST e . One can add the overall phase eiα/ to the scattering matrix by modifying hT → hT + αδ(x) (accounting interface properties) but it will not change any of our results and hence we can safely disregard it. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 25 / 29
  26. 26. Scattering matrix S Therefore we finally get detailed version of scattering equation multiplied by M as 4MS = r sinh2 θ t sinh θ(i sinh θ cos φ − cosh θ sin φ) t sinh θ(i sinh θ cos φ + cosh θ sin φ) r sinh2 θ t2 sin φ(− cosh θ sin φ − i sinh θ cos φ) −rt sinh θ sin φ rt sinh θ sin φ t2 sin φ(− cosh θ sin φ + i sinh θ cos φ) t2 sin φ(− cosh θ sin φ + i sinh θ cos φ) −rt sinh θ sin φ rt sinh θ sin φ t2 sin φ(− cosh θ sin φ − i sinh θ cos φ) r sinh2 θ t sinh θ(−i sinh θ cos φ − cosh θ sin φ) t sinh θ(−i sinh θ cos φ + cosh θ sin φ) r sinh2 θ , where M = sinh2 θ + t2 sin2 φ. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 26 / 29
  27. 27. The scattering matrix connects the modes amplitudes Ao = ARo ALo =      A1Ro A2Ro A1Lo A2Lo      = S      A1Li A2Li A1Ri A2Ri      = S ALi ARi = SAi (34) S1 = Se 0 0 Sh K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 27 / 29
  28. 28. Rysunek : Energy spectrum of JJ, with the continuum parts above +∆ and below −∆ and two phase-dependent ABSs in the gap. Possible excitations of the ground state: (A) between ABSs, (B) between an ABS and continuum, (C) between lower and upper continuum. K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 28 / 29
  29. 29. Analytical evaluation of c(φ) In this approximation c(φ) 2t2 dθ−dθ+ (4π)2 |e(θ+−θ−)/2+iφ − e(θ−−θ+)/2−iφ|2 (cosh θ+ + cosh θ−)2 . (35) By introducing variables 2s = θ+ + θ− and 2w = θ+ − θ− we get c(φ) t2 dsdw (2π)2 sinh2 w + sin2 φ cosh2 s cosh2 w . (36) K.P and Adam Bednorz (UW) Canonical quantization of Josephson effect 27 października 2015 29 / 29

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