Canonical quantisation of telegrapher’s equations coupled by ideal nonreciprocal elements

Adrian Parra-Rodriguez and Iñigo L. Egusquiza

Department of Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

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We develop a systematic procedure to quantise canonically Hamiltonians of light-matter models of transmission lines coupled through lumped linear lossless ideal nonreciprocal elements, that break time-reversal symmetry, in a circuit QED set-up. This is achieved through a description of the distributed subsystems in terms of both flux and charge fields. We prove that this apparent redundancy is required for the general derivation of the Hamiltonian for a wider class of networks. By making use of the electromagnetic duality symmetry in transmission lines (waveguides), we provide unambiguous identification of the physical degrees of freedom, separating out the nondynamical parts. This doubled description can also treat the case of other extended lumped interactions in a regular manner that presents no spurious divergences, as we show explicitly in the example of a circulator connected to a Josephson junction through a transmission line. This theory enhances the quantum engineering toolbox to design complex networks with nonreciprocal elements.

Superconducting circuits are a crucial platform for scientific and technological advances in quantum information storing and processing. In this context, the introduction of scalable nonreciprocal elements that actually do work in the quantum regime is one of the most relevant objectives in the near term. These elements, when controllably realized, will be polyvalent, with applications such as routing coherent quantum information in complex paths, reducing the overhead of control lines on the chips, and isolating noise sources.

The classical description of these devices is well established. The fundamental nonreciprocal device, the two-port gyrator, introduces a π-phase shift to input signals in only one of the two directions of information flow. The more sophisticated N-port circulator, which can be described in terms of a network of gyrators, allows for one-way routing of information, e.g. energy flow entering from port N leaves from port N + 1. The existing nonreciprocal devices for microwave circuits are relatively bulky and operate in a classical regime. Recently, there have been several proposals for constructing these nonreciprocal devices to integrate them with the rest of the superconducting quantum technology working at mK, thus in a quantum regime. There is therefore a need to have quantum theoretical descriptions and tools that can serve the introduction of all these quantum nonreciprocal devices and the corresponding technology.

In this work, we include systematically ideal nonreciprocal elements (generalized gyrators and circulators) in canonical Hamiltonian models when they are coupled to transmission lines. To achieve that goal, we start from the telegrapher’s Lagrangian written in terms of flux and charge fields, i. e. a double configuration space, with a magnetic coupling between them. The apparent redundancy introduced in the description becomes the cornerstone for the correct expansion of fields when the lines are connected to nonreciprocal devices. The correct identification of independent modes is explicitly carried out in the Hamiltonian through a symplectic transformation. We illustrate this theory by applying it to a pedagogical example of a Josephson junction, from which superconducting qubits are constructed, capacitively coupled to a 3-port circulator through a transmission line. Due to the correct basis expansion, the quantized Hamiltonian inherits convergent multi-mode properties, such as Lamb shifts or multi-partite effective couplings.

A direct consequence of our study is that the canonical quantization of networks with ideal nonreciprocal N-port devices described in terms of scattering matrices can be easily carried out in terms of a combination of flux and charge variables. This is the outcome of the duality symmetry in Maxwell’s equations, which, in the present context of one-dimensional circuits, simplifies to the telegrapher’s equations.

To summarise, our work provides a common ground for the canonical quantization of superconducting distributed networks with nonreciprocal devices, sets up a general framework for their correct modeling in the quantum regime, and increases the toolbox of all researchers in quantum superconducting circuit technologies, as well as of those in quantum information processing and simulation.

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[1] M. Devoret and R. Schoelkopf, Science 339, 1169 (2013).

[2] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Nature 574, 505 (2019).

[3] S. E. Nigg, H. Paik, B. Vlastakis, G. Kirchmair, S. Shankar, L. Frunzio, M. H. Devoret, R. J. Schoelkopf, and S. M. Girvin, Physical Review Letters 108, 240502 (2012).

[4] F. Solgun and D. DiVincenzo, Annals of Physics 361, 605 (2015).

[5] K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, Physical Review X 5, 041020 (2015).

[6] B. J. Chapman, E. I. Rosenthal, J. Kerckhoff, B. A. Moores, L. R. Vale, J. A. B. Mates, G. C. Hilton, K. Lalumière, A. Blais, and K. W. Lehnert, Physical Review X 7, 041043 (2017).

[7] C. Müller, S. Guan, N. Vogt, J. H. Cole, and T. M. Stace, Physical Review Letters 120, 213602 (2018).

[8] J. Kerckhoff, K. Lalumière, B. J. Chapman, A. Blais, and K. W. Lehnert, Physical Review Applied 4, 034002 (2015).

[9] G. Viola and D. P. DiVincenzo, Physical Review X 4, 021019 (2014).

[10] A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, Physical Review X 7, 011007 (2017).

[11] S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. Dieterle, O. Painter, and J. Fink, Nature Communications 8, 953 (2017).

[12] B. D. H. Tellegen, Philips Research Reports 3, 81 (1948).

[13] B. Yurke and J. Denker, Physical Review A 29, 1419 (1984).

[14] S. Chakravarty and A. Schmid, Physical Review B 33, 2000 (1986).

[15] B. Yurke, Journal of the Optical Society of America B 4, 1551 (1987).

[16] M. J. Werner and P. D. Drummond, Physical Review A 43, 6414 (1991).

[17] M. H. Devoret, in Proceedings of the Les Houches Summer School, Session LXIII (Elsevier, edited by S. Reynaud, E. Giacobino, and J. Zinn-Justin, 1995).

[18] E. Paladino, F. Taddei, G. Giaquinta, and G. Falci, Physica E: Low-Dimensional Systems and Nanostructures 18, 39 (2003).

[19] G. Burkard, R. H. Koch, and D. P. DiVincenzo, Physical Review B 69, 064503 (2004).

[20] G. Burkard, Physical Review B 71, 144511 (2005).

[21] C. Bergenfeldt and P. Samuelsson, Physical Review B 85, 045446 (2012).

[22] J. Bourassa, F. Beaudoin, J. M. Gambetta, and A. Blais, Physical Review A 86, 013814 (2012).

[23] M. Bamba and T. Ogawa, Physical Review A 89, 023817 (2014).

[24] F. Solgun, D. W. Abraham, and D. P. DiVincenzo, Physical Review B 90, 134504 (2014).

[25] H. L. Mortensen, K. Mølmer, and C. K. Andersen, Physical Review A 94, 053817 (2016).

[26] M. Malekakhlagh and H. E. Türeci, Physical Review A 93, 012120 (2016).

[27] M. Malekakhlagh, A. Petrescu, and H. E. Türeci, Physical Review Letters 119, 073601 (2017).

[28] M. F. Gely, A. Parra-Rodriguez, D. Bothner, Y. M. Blanter, S. J. Bosman, E. Solano, and G. A. Steele, Physical Review B 95, 245115 (2017).

[29] A. Parra-Rodriguez, E. Rico, E. Solano, and I. L. Egusquiza, Quantum Science and Technology 3, 024012 (2018).

[30] A. Parra-Rodriguez, I. L. Egusquiza, D. P. DiVincenzo, and E. Solano, Physical Review B 99, 014514 (2019).

[31] Z. K. Minev, Z. Leghtas, S. O. Mundhada, L. Christakis, I. M. Pop, and M. H. Devoret, ``Energy-participation quantization of josephson circuits,'' (2020), arXiv:2010.00620 [quant-ph].

[32] M. Mariantoni, ``The energy of an arbitrary electrical circuit, classical and quantum,'' (2020), arXiv:2007.08519 [class-ph].

[33] Z. K. Minev, T. G. McConkey, M. Takita, A. D. Corcoles, and J. M. Gambetta, ``Circuit quantum electrodynamics (cqed) with modular quasi-lumped models,'' (2021), arXiv:2103.10344 [quant-ph].

[34] M. Rymarz, Master Thesis: The Quantum Electrodynamics of Singular and Nonreciprocal Superconducting Circuits (RWTH Aachen, 2018).

[35] M. Rymarz, S. Bosco, A. Ciani, and D. P. DiVincenzo, Physical Review X 11, 011032 (2021).

[36] I. L. Egusquiza and A. Parra-Rodriguez, ``Algebraic canonical quantization of lumped superconducting networks,'' (2022), arXiv:2203.06167 [quant-ph].

[37] J. Ulrich and F. Hassler, Physical Review B 94, 094505 (2016).

[38] D. Jeltsema and A. J. Van Der Schaft, Reports on Mathematical Physics 63, 55 (2009).

[39] D. M. Pozar, Microwave Engineering, 4th ed. (John Wiley & Sons, Hoboken, New York, 2009).

[40] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Physical ReviewA 69, 062320 (2004).

[41] A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Physical Review Letters 101, 080502 (2008).

[42] J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, Physical Review A 80, 032109 (2009).

[43] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Reviews of Modern Physics 82, 1155 (2010).

[44] J. Koch, A. A. Houck, K. L. Hur, and S. M. Girvin, Physical Review A 82, 043811 (2010).

[45] S. Filipp, M. Göppl, J. Fink, M. Baur, R. Bianchetti, L. Steffen, and A. Wallraff, Physical Review A 83, 063827 (2011).

[46] B. Peropadre, J. Lindkvist, I.-C. Hoi, C. M. Wilson, J. J. Garcia-Ripoll, P. Delsing, and G. Johansson, New Journal of Physics 15, 035009 (2013).

[47] N. M. Sundaresan, Y. Liu, D. Sadri, L. J. Szőcs, D. L. Underwood, M. Malekakhlagh, H. E. Türeci, and A. A. Houck, Physical Review X 5, 021035 (2015).

[48] A. Roy and M. Devoret, Comptes Rendus Physique 17, 740 (2016).

[49] U. Vool and M. Devoret, International Journal of Circuit Theory and Applications 45, 897 (2017).

[50] A. Roy and M. Devoret, Physical Review B 98, 045405 (2018).

[51] P. A. M. Dirac, Canadian Journal of Mathematics 2, 129 (1950).

[52] P. A. M. Dirac, Physical Review 114, 924 (1959).

[53] W. Lamb and R. Retherford, Physical Review 72, 241 (1947).

[54] V. Weisskopf and E. Wigner, Zeitschrift für Physik 63, 54 (1930).

[55] L. Silberstein, Annalen der Physik 327, 579 (1907).

[56] M. G. Calkin, American Journal of Physics 33, 958 (1965).

[57] J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).

[58] S. Duinker, Philips Research Reports 14, 29 (1959).

[59] H. J. Carlin and A. B. Giordano, Network theory: An introduction to reciprocal and non reciprocal circuits, 1st ed. (Prentice Hall, Englewood Cliffs, New Jersey, 1964).

[60] R. W. Newcomb, Linear Multiport Synthesis (McGraw-Hill, New York, 1966).

[61] A. Parra-Rodriguez and I. L. Egusquiza, ``Quantum fluctuations in electrical multiport linear systems,'' (2021), arXiv:2110.14604 [quant-ph].

[62] A. Parra-Rodriguez, PhD Thesis: Canonical Quantization of Superconducting Circuits (Universidad del Pais Vasco, 2021).

[63] J. Walter, Mathematische Zeitschrift 133, 301 (1973).

[64] C. T. Fulton, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 77, 293 (1977).

Cited by

[1] A. Parra-Rodriguez and I. L. Egusquiza, "Quantum Fluctuations in Electrical Multiport Linear Systems", arXiv:2110.14604.

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