Hamiltonian for coupled flux qubits

Physical Review B - Condensed Matter and Materials Physics

Volumn 71, Issue 6, 2005, Pages

  Van Den Brink, Alec Maassen ^a  

^a D WAVE SYSTEMS INC   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; CALCULATION; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; OSCILLATION; PHYSICS; QUANTUM MECHANICS; QUANTUM THEORY; SUPERCONDUCTOR;

EID: 16344373601     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.71.064503     Document Type: Article

References (38)

1. T. P. Orlando, J. E. Mooij, L. Tian, C. H. van der Wal, L. Levitov, S. Lloyd, and J. J. Mazo, Phys. Rev. B 60, 15 398 (1999).
2. i's differing by multiples of 2π correspond to physically distinguishable magnetic field configurations.
3. A nontrivial capacitance matrix will be essential in Eq. (1.3). Note also that, for this pure series circuit, H does not depend on how the inductance is distributed along the loop.
4. D. S. Crankshaw and T. P. Orlando, IEEE Trans. Appl. Supercond. 11, 1006 (2001).
5. E.g., Ya. S. Greenberg, A. Izmalkov, M. Grajcar, E. Il'ichev, W. Krech, H.-G. Meyer, M. H. S. Amin, and A. Maassen van den Brink, Phys. Rev. B 66, 214525 (2002), Eq. (13).
6. x=LI.
7. Since superconductivity is not a "classical" phenomenon, the term is a bit tenuous. Indeed, if H is written in terms of phases (fluxes), the inductive (Josephson) terms would involve ℏ in SI units. Of course, the dynamics (2.1) do emerge from the full quantum theory in the limit of large capacitances (cf. Sec. V).
8. m∝φ) in Eq. (13) of Ref. 4 satisfy the same criterion. They correspond to choosing two zero entries in Eq. (2.4) rather than Eq. (2.3), and are physically equivalent to χ, θ.
9. b in Eqs. (4.4) and (4.5).
10. The negative sign of the second term in (2.13) is most readily understood for degenerate bias, where any self-flux will effectively reduce the maximum flux frustration imposed externally.
11. J. Q. You, J. S. Tsai, and F. Nori, Phys. Rev. Lett. 89, 197902 (2002); J. Q. You, Y. Nakamura, and F. Nori, cond-mat/0309491 and references therein (unpublished).
12. J. Q. You, J. S. Tsai, and F. Nori, Phys. Rev. Lett. 89, 197902 (2002); J. Q. You, Y. Nakamura, and F. Nori, cond-mat/0309491 and references therein (unpublished).
13. 1=2+α, but this should have few physical consequences.
14. xeven though the saddle-points themselves will move along this line.
15. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
16. Only solutions bounded in φ are acceptable.
17. χ0,θ0= 〈φδ(χ-χ0)δ(θ-θ0)〉/ 〈δ(χ-χ0)δ(θ-θ0)〉 is a conditional quantum-mechanical expectation. The vector version below Eq. (4.10) is analogous.
18. 0 (a trivial change in normalization). However, our focus is on deriving the effective theory, not solving it.
19. 1 in second-order perturbation theory, not a new term in H̃.
20. 0〉, cf. Ref. 17. However, in practice direct numerical solution of H̃-though no more accurate-seems preferable over a perturbative approach.
21. zp (usually an irrelevant constant) is included in H̃.
22. Increasing m too fast, e.g., successive doubling,. wastes grid points on the wave function's large-|φ| tail.
23. The ground state tends to be found with the highest accuracy for a given Δ, but other states are readily found and compared as well.
24. The used matrix representation [W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd Edition (Cambridge University Press, Cambridge, England, 1992)] makes it convenient to allocate only half of a symmetric matrix in a way that is transparent to the rest of the program, as long as the latter never accesses matrix elements above the main diagonal.
25. M. H. S. Amin, Phys. Rev. B 71, 024504 (2005).
26. c is not given.
27. In general, not taking independent capacitances and Josephson couplings makes it more difficult to conceptually separate the effects of the two.
28. a, etc., which one would naively discard as a small part of the "free" Hamiltonian. That is, while total energy is conserved, the designation of an "interaction" part can be somewhat arbitrary, so it is best to be as consistent as possible and retain all terms in Eq. (4.3). The electrostatic counterpart may be more familiar in the field [cf. the remark below Eq. (4.4)],
29. E.g., J. F. Ralph, T. D. Clark, M. J. Everitt, and P. Stiffell, Phys. Rev. B 64, 180504(R) (2001).
30. For k ↑ 1, Eq. (4.4) seems to diverge. However, physically this limit can only be achieved with two loops right on top of each other. The resulting ferromagnetic interaction is described by M < 0 in Eq. (4.2), so that the last term in Eq. (4.4) tends to cancel the other two. Indeed, we have merely rewritten the finite Eq. (4.3).
31. Not all of our assumptions (cf. Ref. 3) may hold in a figure-8 geometry [J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, cond-mat/0308192 (unpublished)] if part of the inductance is distributed along the shared leg. Therefore, this case deserves special attention. Furthermore, incorporating a (large) Josephson 'junction into a shared leg is known to cause an antiferromagnetic coupling in close analogy to the inductive one discussed here. See L. S. Levitov, T. P. Orlando, J. B. Majer, and J. E. Mooij, cond-mat/0108266; J. R. Butcher, graduation thesis, Delft University of Technology, Delft, 2002; M. Grajcar, A. Izmalkov, S. H. W. van der Ploeg, S. Linzen, E. Il'ichev, Th. Wagner, U. Hübner, H.-G. Meyer, A. Maassen van den Brink, S. Uchaikin, and A. M. Zagoskin, cond-mat/0501085 (unpublished).
32. Not all of our assumptions (cf. Ref. 3) may hold in a figure-8 geometry [J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, cond-mat/0308192 (unpublished)] if part of the inductance is distributed along the shared leg. Therefore, this case deserves special attention. Furthermore, incorporating a (large) Josephson 'junction into a shared leg is known to cause an antiferromagnetic coupling in close analogy to the inductive one discussed here. See L. S. Levitov, T. P. Orlando, J. B. Majer, and J. E. Mooij, cond-mat/0108266; J. R. Butcher, graduation thesis, Delft University of Technology, Delft, 2002; M. Grajcar, A. Izmalkov, S. H. W. van der Ploeg, S. Linzen, E. Il'ichev, Th. Wagner, U. Hübner, H.-G. Meyer, A. Maassen van den Brink, S. Uchaikin, and A. M. Zagoskin, cond-mat/0501085 (unpublished).
33. Not all of our assumptions (cf. Ref. 3) may hold in a figure-8 geometry [J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, cond-mat/0308192 (unpublished)] if part of the inductance is distributed along the shared leg. Therefore, this case deserves special attention. Furthermore, incorporating a (large) Josephson 'junction into a shared leg is known to cause an antiferromagnetic coupling in close analogy to the inductive one discussed here. See L. S. Levitov, T. P. Orlando, J. B. Majer, and J. E. Mooij, cond-mat/0108266; J. R. Butcher, graduation thesis, Delft University of Technology, Delft, 2002; M. Grajcar, A. Izmalkov, S. H. W. van der Ploeg, S. Linzen, E. Il'ichev, Th. Wagner, U. Hübner, H.-G. Meyer, A. Maassen van den Brink, S. Uchaikin, and A. M. Zagoskin, cond-mat/0501085 (unpublished).
34. Not all of our assumptions (cf. Ref. 3) may hold in a figure-8 geometry [J. B. Majer, F. G. Paauw, A. C. J. ter Haar, C. J. P. M. Harmans, and J. E. Mooij, cond-mat/0308192 (unpublished)] if part of the inductance is distributed along the shared leg. Therefore, this case deserves special attention. Furthermore, incorporating a (large) Josephson 'junction into a shared leg is known to cause an antiferromagnetic coupling in close analogy to the inductive one discussed here. See L. S. Levitov, T. P. Orlando, J. B. Majer, and J. E. Mooij, cond-mat/0108266; J. R. Butcher, graduation thesis, Delft University of Technology, Delft, 2002; M. Grajcar, A. Izmalkov, S. H. W. van der Ploeg, S. Linzen, E. Il'ichev, Th. Wagner, U. Hübner, H.-G. Meyer, A. Maassen van den Brink, S. Uchaikin, and A. M. Zagoskin, cond-mat/0501085 (unpublished).
35. A. Izmalkov, M. Grajcar, E. Il'ichev, Th. Wagner, H.-G. Meyer, A. Yu. Smirnov, M. H. S. Amin, A. Maassen van den Brink, and A. M. Zagoskin, Phys. Rev. Lett. 93, 037003 (2004).
36. C. H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, S. Lloyd, and J. E. Mooij, Science 290, 773 (2000).
37. E. Il'ichev, N. Oukhanski, A. Izmalkov, Th. Wagner, M. Grajcar, H.-G. Meyer, A. Yu. Smirnov, A. Maassen van den Brink, M. H. S. Amin, and A. M. Zagoskin, Phys. Rev. Lett. 91, 097906 (2003).
38. M. Grajcar, A. Izmalkov, E. Il'ichev, Th. Wagner, N. Oukhanski, U. Hübner, T. May, I. Zhilyaev, H. E. Hoenig, Ya. S. Greenberg, V. I. Shnyrkov, D. Born, W. Krech, H.-G. Meyer, A. Maassen van den Brink, and M. H. S. Amin, Phys. Rev. B 69, 060501(R) (2004).