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  • Featured in Physics
  • Open Access

Entangled Photon-Pair Emission in Waveguide Circuit QED from a Cooper Pair Splitter

Michele Governale1, Christian Schönenberger2, Pasquale Scarlino3,†, and Gianluca Rastelli4,5,*

  • *Contact author: gianluca.rastelli@ino.cnr.it
  • †Contact author: pasquale.scarlino@epfl.ch
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PRX Quantum 6, 020339 – Published 29 May, 2025

DOI: https://doi.org/10.1103/PRXQuantum.6.020339

Abstract

As a waveguide circuit QED architecture, we investigate theoretically the single-photon pair emission of a Cooper pair splitter composed of two double quantum dots, each coupled to a microwave transmission line. We find that this system can generate frequency-entangled photon pairs in the left and right transmission lines, specifically a superposition of two photon wave packets at different frequencies. The frequency entanglement of the two photons arises from the particle-hole coherent superposition (i.e., Andreev bound states) involving the delocalized entangled spin singlet. We also estimate a lower bound for the efficiency of entangled photon-pair generation, accounting for the presence of nonradiative processes such as phonon emissions. Our proposal is realistic and achievable with state-of-the-art techniques in quantum microwave engineering with electostatically defined semiconducting quantum dots.

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Physics Subject Headings (PhySH)

synopsis

Creating Entangled Pairs of Microwave Photons

Published 29 May, 2025

Researchers have demonstrated a method to make entangled microwave photons from the entangled electrons in superconductors.

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Popular Summary

Entanglement refers to the nonclassical correlations between distinct parts of a quantum system and is a fundamental resource for quantum technologies. The ability to transfer entanglement between different types of entities, such as electrons and photons, is crucial for further technological developments in this field. Quantum correlations can arise from interactions between the fundamental constituents of matter. For instance, a BCS superconductor represents a quantum condensate formed by spin-entangled Cooper pairs. Individual spin-entangled pairs can be extracted from the superconductor and separated using nanodevices called Cooper pair splitters. However, the direct observation and measurement of the entanglement encoded within a single pair remains an open challenge.

In our work, we propose a scheme to convert the entanglement of a single Cooper pair into frequency-entangled photon pairs using a circuit QED architecture operating in the microwave domain. We demonstrate the ability to generate pairs of entangled photons that propagate through two separate transmission lines. The entangled photons are emitted by a Cooper pair splitter made of quantum dots, which is the core of our setup. Our scheme combines cutting-edge components of nanoscale quantum hardware by requiring the integration of superconducting nanocontacts, semiconducting quantum dots, and microwave quantum devices.

Our scheme has the potential to become a powerful resource for quantum technologies since it enables the generation of entangled photons in the microwave domain using electronically driven devices integrated on a chip.

Article Text

References (174)

  1. M. Wallquist, K. Hammerer, O. Rabl, M. Lukin, and P. Zoller, Hybrid quantum devices and quantum engineering, Phys. Scr. 2009, 014001 (2009).
  2. Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems, Rev. Mod. Phys. 85, 623 (2013).
  3. A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, Hybrid quantum systems with circuit quantum electrodynamics, Nat. Phys. 16, 257 (2020).
  4. Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary ed. (Cambridge University Press, Cambridge, UK, 2010).
  5. Z.-S. Yuana, X.-H. Bao, C.-Y. Lua, J. Zhanga, C.-Z. Penga, and J.-W. Pana, Entangled photons and quantum communication, Phys. Rep. 497, 1 (2010).
  6. C. K. Hong and L. Mandel, Theory of parametric frequency down conversion of light, Phys. Rev. A 31, 2409 (1985).
  7. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Theory of two-photon entanglement in type-II optical parametric down-conversion, Phys. Rev. A 50, 5122 (1994).
  8. O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, Regulated and entangled photons from a single quantum dot, Phys. Rev. Lett. 84, 2513 (2000).
  9. A. Orieux, M. Versteegh, K. Jöns, and S. Ducci, Semiconductor devices for entangled photon pair generation: A review, Rep. Prog. Phys. 80, 076001 (2017).
  10. N. Akopian, N. H. Lindner, E. Poem, Y. Berlatzky, J. Avron, D. Gershoni, B. D. Gerardot, and P. M. Petroff, Entangled photon pairs from semiconductor quantum dots, Phys. Rev. Lett. 96, 130501 (2006).
  11. C. Salter, R. Stevenson, I. Farrer, C. Nicoll, D. Ritchie, and A. Shields, An entangled-light-emitting diode, Nature 465, 594 (2010).
  12. J. Budich and B. Trauzettel, Entanglement transfer from electrons to photons in quantum dots: an open quantum system approach, Nanotechnology 21, 274001 (2010).
  13. C. Emary, B. Trauzettel, and C. W. J. Beenakker, Emission of polarization-entangled microwave photons from a pair of quantum dots, Phys. Rev. Lett. 95, 127401 (2005).
  14. M. Müller, S. Bounouar, K. D. Jöns, M. Glässl, and P. Michler, On-demand generation of indistinguishable polarization-entangled photon pairs, Nat. Photon. 8, 224 (2014).
  15. J. Torrès and T. Martin, Positive and negative Hanbury-brown and Twiss correlations in normal metal-superconducting devices, Eur. Phys. J. B 12, 319 (1999).
  16. M.-S. Choi, C. Bruder, and D. Loss, Spin-dependent Josephson current through double quantum dots and measurement of entangled electron states, Phys. Rev. B 62, 13569 (2000).
  17. P. Recher, E. V. Sukhorukov, and D. Loss, Andreev tunneling, Coulomb blockade, and resonant transport of nonlocal spin-entangled electrons, Phys. Rev. B 63, 165314 (2001).
  18. G. Lesovik, T. Martin, and G. Blatter, Electronic entanglement in the vicinity of a superconductor, Eur. Phys. J. B 24, 287 (2001).
  19. N. M. Chtchelkatchev, G. Blatter, G. B. Lesovik, and T. Martin, Bell inequalities and entanglement in solid-state devices, Phys. Rev. B 66, 161320(R) (2002).
  20. V. Bouchiat, N. Chtchelkatchev, D. Feinberg, G. B. Lesovik, T. Martin, and J. Torrés, Single-walled carbon nanotube-superconductor entangler: Noise correlations and Einstein-Podolsky-Rosen states, Nanotechnology 14, 77 (2002).
  21. D. Beckmann, H. B. Weber, and H. Löhneysen, Evidence for crossed Andreev reflection in superconductor-ferromagnet hybrid structures, Phys. Rev. Lett. 93, 197003 (2004).
  22. S. Russo, M. Kroug, T. M. Klapwijk, and A. F. Morpurgo, Experimental observation of bias-dependent nonlocal Andreev reflection, Phys. Rev. Lett. 95, 027002 (2005).
  23. P. Cadden-Zimansky, J. Wei, and V. Chandrasekhar, Cooper-pair-mediated coherence between two normal metals, Nat. Phys. 5, 393 (2009).
  24. A. Kleine, A. Baumgartner, J. Trbovic, and C. Schönenberger, Contact resistance dependence of crossed Andreev reflection, Europhys. Lett. 87, 27011 (2009).
  25. V. Wei and J. nad Chandrasekhar, Positive noise cross-correlation in hybrid superconducting and normal-metal three-terminal devices, Nat. Phys. 6, 494 (2010).
  26. L. Hofstetter, S. Csonka, J. Nygård, and C. Schönenberger, Cooper pair splitter realized in a two-quantum-dot y-junction, Nature 461, 960 (2009).
  27. L. G. Herrmann, F. Portier, P. Roche, A. L. Yeyati, T. Kontos, and C. Strunk, Carbon nanotubes as Cooper-pair beam splitters, Phys. Rev. Lett. 104, 026801 (2010).
  28. J. Schindele, A. Baumgartner, and C. Schönenberger, Near-unity Cooper pair splitting efficiency, Phys. Rev. Lett. 109, 157002 (2012).
  29. J. Schindele, A. Baumgartner, R. Maurand, M. Weiss, and C. Schönenberger, Nonlocal spectroscopy of Andreev bound states, Phys. Rev. B 89, 045422 (2014).
  30. L. Hofstetter, S. Csonka, A. Baumgartner, G. Fülöp, S. d’Hollosy, J. Nygård, and C. Schönenberger, Finite-bias Cooper pair splitting, Phys. Rev. Lett. 107, 136801 (2011).
  31. A. Das, Y. Ronen, M. Heiblum, D. Mahalu, A. Kretinin, and H. Shtrikman, High-efficiency Cooper pair splitting demonstrated by two-particle conductance resonance and positive noise cross-correlation, Nat. Commun. 3, 1165 (2012).
  32. G. Fülöp, S. d’Hollosy, A. Baumgartner, P. Makk, V. A. Guzenko, M. H. Madsen, J. Nygård, C. Schönenberger, and S. Csonka, Local electrical tuning of the nonlocal signals in a cooper pair splitter, Phys. Rev. B 90, 235412 (2014).
  33. G. Fülöp, F. Domínguez, S. d’Hollosy, A. Baumgartner, P. Makk, M. H. Madsen, V. A. Guzenko, J. Nygård, C. Schönenberger, A. Levy Yeyati, and S. Csonka, Magnetic field tuning and quantum interference in a Cooper pair splitter, Phys. Rev. Lett. 115, 227003 (2015).
  34. Z. B. Tan, D. Cox, T. Nieminen, P. Lähteenmäki, D. Golubev, G. B. Lesovik, and P. J. Hakonen, Cooper pair splitting by means of graphene quantum dots, Phys. Rev. Lett. 114, 096602 (2015).
  35. I. Borzenets, Y. Shimazaki, G. Jones, M. Craciun, S. Russo, M. Yamamoto, and S. Tarucha, High efficiency CVD graphene-lead (pb) cooper pair splitter, Sci. Rep. 6, 23051 (2016).
  36. Z. Tan, N. S. Laitinen, A. Kirsanov, A. Galda, V. M. Vinokur, M. Haque, A. Savin, D. S. Golubev, G. B. Lesovik, and P. J. Hakonen, Thermoelectric current in a graphene Cooper pair splitter, Nat. Commun. 12, 138 (2016).
  37. P. Pandey, R. Danneau, and D. Beckmann, Ballistic graphene Cooper pair splitter, Phys. Rev. Lett. 126, 147701 (2021).
  38. O. Kürtössy, Z. Scherübl, G. Fülöp, I. Lukács, T. Kanne, J. Nygård, P. Makk, and S. Csonka, Parallel InAs nanowires for Cooper pair splitters with Coulomb repulsion, Npj Quantum Mater. 7, 88 (2022).
  39. O. Sauret, D. Feinberg, and T. Martin, Quantum master equations for the superconductor–quantum dot entangler, Phys. Rev. B 70, 245313 (2004).
  40. J. P. Morten, A. Brataas, and W. Belzig, Circuit theory of crossed Andreev reflection, Phys. Rev. B 74, 214510 (2006).
  41. D. Futterer, M. Governale, M. G. Pala, and J. König, Nonlocal Andreev transport through an interacting quantum dot, Phys. Rev. B 79, 054505 (2009).
  42. J. Eldridge, M. G. Pala, M. Governale, and J. König, Superconducting proximity effect in interacting double-dot systems, Phys. Rev. B 82, 184507 (2010).
  43. D. Chevallier, J. Rech, T. Jonckheere, and T. Martin, Current and noise correlations in a double-dot Cooper-pair beam splitter, Phys. Rev. B 83, 125421 (2011).
  44. J. Rech, D. Chevallier, T. Jonckheere, and T. Martin, Current correlations in an interacting Cooper-pair beam splitter, Phys. Rev. B 85, 035419 (2012).
  45. P. Trocha and I. Weymann, Spin-resolved Andreev transport through double-quantum-dot cooper pair splitters, Phys. Rev. B 91, 235424 (2015).
  46. F. Dominguez and A. Levy Yeyati, Quantum interference in a cooper pair splitter: The three sites model, Phys. E 75, 322 (2016).
  47. E. Amitai, R. P. Tiwari, S. Walter, T. L. Schmidt, and S. E. Nigg, Nonlocal quantum state engineering with the Cooper pair splitter beyond the Coulomb blockade regime, Phys. Rev. B 93, 075421 (2016).
  48. R. Hussein, A. Braggio, and M. Governale, Entanglement-symmetry control in a quantum-dot Cooper-pair splitter, Phys. Status Solidi B 254, 1600603 (2017).
  49. N. Walldorf, F. Brange, C. Padurariu, and C. Flindt, Noise and full counting statistics of a Cooper pair splitter, Phys. Rev. B 101, 205422 (2020).
  50. B. R. Bułka, Cooper pair splitter in a photonic cavity: Detection of Andreev scatterings, Phys. Rev. B 106, 085424 (2022).
  51. F. Brange, K. Prech, and C. Flindt, Dynamic Cooper pair splitter, Phys. Rev. Lett. 127, 237701 (2021).
  52. M. Tam, C. Flindt, and F. Brange, Optimal entanglement witness for Cooper pair splitters, Phys. Rev. B 104, 245425 (2021).
  53. A. Ranni, F. Brange, E. Mannila, C. Flindt, and V. Maisi, Real-time observation of Cooper pair splitting showing strong non-local correlations, Nat. Commun. 12, 6358 (2021).
  54. M. C. Hels, B. Braunecker, K. Grove-Rasmussen, and J. Nygård, Noncollinear spin-orbit magnetic fields in a carbon nanotube double quantum dot, Phys. Rev. Lett. 117, 276802 (2016).
  55. A. Bordoloi, V. Zannier, L. Sorba, C. Schönenberger, and A. Baumgartner, Spin cross-correlation experiments in an electron entangler, Nature 612, 454 (2022).
  56. G. Wang, G. P. Dvir, T. Mazur, C.-X. Liu, N. van Loo, S. ten Haaf, A. Bordin, S. Gazibegovic, G. Badawy, E. Bakkers, M. Wimmer, and L. Kouwenhoven, Singlet and triplet Cooper pair splitting in hybrid superconducting nanowires, Nature 612, 448 (2022).
  57. S. Kawabata, Test of Bell’s inequality using the spin filter effect in ferromagnetic semiconductor microstructures, J. Phys. Soc. Jpn. 70, 1210 (2001).
  58. P. Samuelsson, E. V. Sukhorukov, and M. Büttiker, Orbital entanglement and violation of Bell inequalities in mesoscopic conductors, Phys. Rev. Lett. 91, 157002 (2003).
  59. B. Braunecker, P. Burset, and A. Levy Yeyati, Entanglement detection from conductance measurements in carbon nanotube Cooper pair splitters, Phys. Rev. Lett. 111, 136806 (2013).
  60. H. Soller, L. Hofstetter, and D. Reeb, Entanglement witnessing in superconducting beamsplitters, Euro Phys. Lett. 102, 50009 (2013).
  61. W. Kłobus, A. Grudka, A. Baumgartner, D. Tomaszewski, C. Schönenberger, and J. Martinek, Entanglement witnessing and quantum cryptography with nonideal ferromagnetic detectors, Phys. Rev. B 89, 125404 (2014).
  62. P. Busz, D. Tomaszewski, and J. Martinek, Spin correlation and entanglement detection in Cooper pair splitters by current measurements using magnetic detectors, Phys. Rev. B 96, 064520 (2017).
  63. A. Bednorz and W. Belzig, Proposal for a cumulant-based bell test for mesoscopic junctions, Phys. Rev. B 83, 125304 (2011).
  64. Z. Scherübl, A. Pályi, and S. Csonka, Probing individual split cooper pairs using the spin qubit toolkit, Phys. Rev. B 89, 205439 (2014).
  65. V. Cerletti, O. Gywat, and D. Loss, Entanglement transfer from electron spins to photons in spin light-emitting diodes containing quantum dots, Phys. Rev. B 72, 115316 (2005).
  66. S. E. Nigg, R. P. Tiwari, S. Walter, and T. L. Schmidt, Detecting nonlocal Cooper pair entanglement by optical Bell inequality violation, Phys. Rev. B 91, 094516 (2015).
  67. A. Schroer and P. Recher, Detection of nonlocal spin entanglement by light emission from a superconducting p−n junction, Phys. Rev. B 92, 054514 (2015).
  68. A. Chatterjee, P. Stevenson, S. De Franceschi, A. Morello, N. P. de Leon, and F. Kuemmeth, Semiconductor qubits in practice, Nat. Rev. Phys. 3, 157 (2021).
  69. X. Zhang, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, and G.-P. Guo, Semiconductor quantum computation, Natl. Sci. Rev. 6, 32 (2019).
  70. G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Semiconductor spin qubits, Rev. Mod. Phys. 95, 025003 (2023).
  71. A. Cottet, M. Dartiailh, M. Desjardins, T. Cubaynes, L. Contamin, M. Delbecq, J. Viennot, L. Bruhat, B. Douçot, and T. Kontos, Cavity QED with hybrid nanocircuits: From atomic-like physics to condensed matter phenomena, J. Phys.: Condens. Matter 29, 433002 (2017).
  72. G. Burkard, M. J. Gullans, X. Mi, and J. R. Petta, Superconductor-semiconductor hybrid-circuit quantum electrodynamics, Nat. Rev. Phys. 2, 129 (2020).
  73. T. Frey, P. J. Leek, M. Beck, A. Blais, T. Ihn, K. Ensslin, and A. Wallraff, Dipole coupling of a double quantum dot to a microwave resonator, Phys. Rev. Lett. 108, 046807 (2012).
  74. M. R. Delbecq, V. Schmitt, F. D. Parmentier, N. Roch, J. J. Viennot, G. Fève, B. Huard, C. Mora, A. Cottet, and T. Kontos, Coupling a quantum dot, fermionic leads, and a microwave cavity on a chip, Phys. Rev. Lett. 107, 256804 (2011).
  75. K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, J. M. Taylor, A. A. Houck, and J. R. Petta, Circuit quantum electrodynamics with a spin qubit, Nature 490, 380 (2012).
  76. J. Basset, D.-D. Jarausch, A. Stockklauser, T. Frey, C. Reichl, W. Wegscheider, T. M. Ihn, K. Ensslin, and A. Wallraff, Single-electron double quantum dot dipole-coupled to a single photonic mode, Phys. Rev. B 88, 125312 (2013).
  77. F. R. Braakman, J. Danon, L. R. Schreiber, W. Wegscheider, and L. Vandersypen, Dynamics of spin-flip photon-assisted tunneling, Phys. Rev. B 89, 075417 (2014).
  78. C. Rössler, D. Oehri, O. Zilberberg, G. Blatter, M. Karalic, J. Pijnenburg, A. Hofmann, T. Ihn, K. Ensslin, C. Reichl, and W. Wegscheider, Transport spectroscopy of a spin-coherent dot-cavity system, Phys. Rev. Lett. 115, 166603 (2015).
  79. J. J. Viennot, M. C. Dartiailh, A. Cottet, and T. Kontos, Coherent coupling of a single spin to microwave cavity photons, Science 349, 408 (2015).
  80. V. Ranjan, G. Puebla-Hellmann, M. Jung, T. Hasler, A. Nunnenkamp, M. Muoth, C. Hierold, A. Wallraff, and C. Schönenberger, Clean carbon nanotubes coupled to superconducting impedance-matching circuits, Nat. Commun. 6, 7165 (2015).
  81. J. Viennot, M. R. Delbecq, L. E. Bruhat, M. C. Dartiailh, M. M. Desjardins, M. Baillergeau, A. Cottet, and T. Kontos, Towards hybrid circuit quantum electrodynamics with quantum dots, C. R. Phys. 17, 705 (2016).
  82. F. Beaudoin, D. Lachance-Quirion, W. Coish, and M. Pioro-Ladriére, Coupling a single electron spin to a microwave resonator: Controlling transverse and longitudinal couplings, Nanotechnology 27, 464003 (2016).
  83. L. E. Bruhat, J. J. Viennot, M. C. Dartiailh, M. M. Desjardins, T. Kontos, and A. Cottet, Cavity photons as a probe for charge relaxation resistance and photon emission in a quantum dot coupled to normal and superconducting continua, Phys. Rev. X 6, 021014 (2016).
  84. T. Brandes and N. Lambert, Steering of a bosonic mode with a double quantum dot, Phys. Rev. B 67, 125323 (2003).
  85. L. Childress, A. S. Sørensen, and M. D. Lukin, Mesoscopic cavity quantum electrodynamics with quantum dots, Phys. Rev. A 69, 042302 (2004).
  86. M. Trif, V. N. Golovach, and D. Loss, Spin dynamics in InAs nanowire quantum dots coupled to a transmission line, Phys. Rev. B 77, 045434 (2008).
  87. A. Cottet and T. Kontos, Spin quantum bit with ferromagnetic contacts for circuit QED, Phys. Rev. Lett. 105, 160502 (2010).
  88. P.-Q. Jin, M. Marthaler, A. Shnirman, and G. Schön, Strong coupling of spin qubits to a transmission line resonator, Phys. Rev. Lett. 108, 190506 (2012).
  89. X. Hu, Y. Liu, and F. Nori, Strong coupling of a spin qubit to a superconducting stripline cavity, Phys. Rev. B 86, 035314 (2012).
  90. C. Bergenfeldt and P. Samuelsson, Nonlocal transport properties of nanoscale conductor-microwave cavity systems, Phys. Rev. B 87, 195427 (2013).
  91. L. D. Contreras-Pulido, C. Emary, T. Brandes, and R. Aguado, Non-equilibrium correlations and entanglement in a semiconductor hybrid circuit-QED system, New J. Phys. 15, 095008 (2013).
  92. C. Kloeffel, M. Trif, P. Stano, and D. Loss, Circuit QED with hole-spin qubits in Ge/Si nanowire quantum dots, Phys. Rev. B 88, 241405(R) (2013).
  93. C. Bergenfeldt, P. Samuelsson, B. Sothmann, C. Flindt, and M. Büttiker, Hybrid microwave-cavity heat engine, Phys. Rev. Lett. 112, 076803 (2014).
  94. A. Cottet, T. Kontos, and B. Douçot, Electron-photon coupling in mesoscopic quantum electrodynamics, Phys. Rev. B 91, 205417 (2015).
  95. O. Dmytruk, M. Trif, C. Mora, and P. Simon, Out-of-equilibrium quantum dot coupled to a microwave cavity, Phys. Rev. B 93, 075425 (2016).
  96. A. Cottet, Z. Leghtas, and T. Kontos, Theory of interactions between cavity photons induced by a mesoscopic circuit, Phys. Rev. B 102, 155105 (2020).
  97. X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman, and J. R. Petta, Strong coupling of a single electron in silicon to a microwave photon, Science 355, 156 (2016).
  98. A. Stockklauser, P. Scarlino, J. V. Koski, S. Gasparinetti, C. K. Andersen, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, Strong coupling cavity QED with gate-defined double quantum dots enabled by a high impedance resonator, Phys. Rev. X 7, 011030 (2017).
  99. X. Mi, J. V. Cady, D. M. Zajac, J. Stehlik, L. F. Edge, and J. R. Petta, Circuit quantum electrodynamics architecture for gate-defined quantum dots in silicon, Appl. Phys. Lett. 110, 043502 (2017).
  100. P. Scarlino, D. J. van Woerkom, A. Stockklauser, J. V. Koski, M. C. Collodo, S. Gasparinetti, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, All-microwave control and dispersive readout of gate-defined quantum dot qubits in circuit quantum electrodynamics, Phys. Rev. Lett. 122, 206802 (2019).
  101. Y. Li, S.-X. Li, F. Gao, H.-O. Li, G. Xu, K. Wang, D. Liu, G. Cao, M. Xiao, T. Wang, J.-J. Zhang, G.-C. Guo, and G.-P. Guo, Coupling a germanium hut wire hole quantum dot to a superconducting microwave resonator, Nano Lett. 18, 2091 (2018).
  102. N. Samkharadze, G. Zheng, N. Kalhor, D. Brousse, A. Sammak, U. C. Mendes, A. Blais, G. Scappucci, and L. Vandersypen, Strong spin-photon coupling in silicon, Science 359, 1123 (2018).
  103. X. Mi, M. Benito, S. Putz, D. M. Zajac, J. M. Taylor, G. Burkard, and J. R. Petta, A coherent spin-photon interface in silicon, Nature 555, 599 (2018).
  104. A. J. Landig, J. V. Koski, P. Scarlino, U. C. Mendes, A. Blais, C. Reichl, W. Wegscheider, A. Wallraff, K. Ensslin, and T. Ihn, Coherent spin-photon coupling using a resonant exchange qubit, Nature 560, 179 (2018).
  105. C. Yu, S. Zihlmann, J. Abadillo-Uriel, V. Michal, N. Rambal, H. Niebojewski, T. Bedecarrats, M. Vinet, E. Dumur, M. Filippone, B. Bertrand, S. De Franceschi, Y.-M. Niquet, and R. Maurand, Strong coupling between a photon and a hole spin in silicon, Nat. Nanotechnol. 18, 741 (2023).
  106. L. E. Bruhat, T. Cubaynes, J. J. Viennot, M. C. Dartiailh, M. M. Desjardins, A. Cottet, and T. Kontos, Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs, Phys. Rev. B 98, 155313 (2018).
  107. D. de Jong, C. G. Prosko, L. Han, F. K. Malinowski, Y. Liu, L. P. Kouwenhoven, and W. Pfaff, Controllable single Cooper pair splitting in hybrid quantum dot systems, Phys. Rev. Lett. 131, 157001 (2023).
  108. S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, Discrete tunable color entanglement, Phys. Rev. Lett. 103, 253601 (2009).
  109. X. Guo, Y. Mei, and S. Du, Testing the bell inequality on frequency-bin entangled photon pairs using time-resolved detection, Optica 4, 388 (2017).
  110. N. Lingaraju, H.-H. Lu, F. Leaird, S. Estrella, J. Lukens, and A. M. Weiner, Bell state analyzer for spectrally distinct photons, Optica 9, 280 (2022).
  111. H.-H. Lu, M. Liscidini, A. L. Gaeta, A. M. Weiner, J. M. Lukens, and J. M. Lukens, Frequency-bin photonic quantum information, Optica 10, 1655 (2023).
  112. P. Imany, J. A. Jaramillo-Villegas, O. D. Odele, K. Han, D. E. Leaird, J. M. Lukens, P. Lougovski, M. Qi, and A. M. Weiner, 50-GHz-spaced comb of high-dimensional frequency-bin entangled photons from an on-chip silicon nitride microresonator, Opt. Express 26, 1825 (2018).
  113. F. Brange, R. Baruah, and C. Flindt, Adiabatic Cooper pair splitter, Phys. Rev. B 109, L081402 (2024).
  114. W. van der Wiel, S. De Franceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L. P. Kouwenhoven, Electron transport through double quantum dots, Rev. Mod. Phys. 75, 1 (2002).
  115. W. Legrand, S. Lopes, Q. Schaeverbeke, F. Montaigne, and M. Desjardins, Optimal design of nanomagnets for on-chip field gradients, Phys. Rev. Appl. 20, 044062 (2023).
  116. S. G. J. Philips, M. T. Mądzik, S. V. Amitonov, S. L. de Snoo, M. Russ, N. Kalhor, C. Volk, W. I. L. Lawrie, D. Brousse, L. Tryputen, B. P. Wuetz, A. Sammak, M. Veldhorst, G. Scappucci, and L. Vandersypen, Universal control of a six-qubit quantum processor in silicon, Nature 609, 919 (2022).
  117. C.-H. Chang, Y.-C. Li, Y.-J. Wu, C.-Y. Liao, M.-J. Lin, H.-Y. Tsao, and J.-Y. Li, A hyperbolic micromagnet for multiple spin qubits with fast Rabi oscillations and high addressability, IEEE Electron Device Lett. 45, 526 (2024).
  118. F. Bersan, M. Aldeghi, N. Martinolli, V. Boureau, T. Aboud, M. Ghini, P. Scarlino, G. Salis, and A. M. Ionescu, Integration of cobalt ferromagnetic control gates for electrical and magnetic manipulation of semiconductor quantum dots, Advanced Functional Materials 35, 2419940 (2025).
  119. N. Hendrickx, L. Massai, M. Mergenthaler, F. J. Schupp, S. Paredes, S. W. Bedell, G. Salis, and A. Fuhrer, Sweet-spot operation of a germanium hole spin qubit with highly anisotropic noise sensitivity, Nat. Mater. 23, 920 (2024).
  120. A. V. Rozhkov and D. P. Arovas, Interacting-impurity Josephson junction: Variational wave functions and slave-boson mean-field theory, Phys. Rev. B 62, 6687 (2000).
  121. T. Meng, S. Florens, and P. Simon, Self-consistent description of Andreev bound states in Josephson quantum dot devices, Phys. Rev. B 79, 224521 (2009).
  122. B. Yurke and J. S. Denker, Quantum network theory, Phys. Rev. A 29, 1419 (1984).
  123. M. Devoret, in Quantum Fluctuations: Les Houches Session LXIII, edited by S. Reynaud, E. Giacobino, and J. Zinn-Justin (Elsevier Science, Amsterdam, The Netherlands, 1997), p. 351.
  124. U. Vool and M. Devoret, Introduction to quantum electromagnetic circuits, Int. J. Circuit Theory Appl. 45, 897 (2017).
  125. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  126. A. Cottet, Microwave spectroscopy of a Cooper pair beam splitter, Phys. Rev. B 86, 075107 (2012).
  127. A. Cottet, T. Kontos, and A. L. Yeyati, Subradiant split Cooper pairs, Phys. Rev. Lett. 108, 166803 (2012).
  128. K. Wang, C. Payette, Y. Dovzhenko, P. W. Deelman, and J. R. Petta, Charge relaxation in a single-electron Si/SiGe double quantum dot, Phys. Rev. Lett. 111, 046801 (2013).
  129. A. Hofmann, C. Karlewski, A. Heimes, C. Reichl, W. Wegscheider, G. Schön, K. Ensslin, T. Ihn, and V. F. Maisi, Phonon spectral density in a GaAs/AlGaAs double quantum dot, Phys. Rev. Res. 2, 033230 (2020).
  130. E. Paladino, Y. M. Galperin, G. Falci, and B. L. Altshuler, 1/f noise: Implications for solid-state quantum information, Rev. Mod. Phys. 86, 361 (2014).
  131. Z. Peng, S. de Graaf, J. Tsai, and O. Astafiev, Tuneable on-demand single-photon source in the microwave range, Nat. Commun. 7, 12588 (2016).
  132. S. Gasparinetti, M. Pechal, J.-C. Besse, M. Mondal, C. Eichler, and A. Wallraff, Correlations and entanglement of microwave photons emitted in a cascade decay, Phys. Rev. Lett. 119, 140504 (2017).
  133. P. Forn-Díaz, C. W. Warren, C. W. S. Chang, A. M. Vadiraj, and C. M. Wilson, On-demand microwave generator of shaped single photons, Phys. Rev. Appl. 8, 054015 (2017).
  134. Y. Lu, A. Bengtsson, J. Burnett, B. Suri, S. Sathyamoorthy, H. Nilsson, M. Scigliuzzo, J. Bylander, G. Johansson, and P. Delsing, Quantum efficiency, purity and stability of a tunable, narrowband microwave single-photon source, Npj Quantum Inf. 7, 4182 (2021).
  135. Y. Zhou, Z. Peng, Y. Horiuchi, O. Astafiev, and J. Tsai, Tunable microwave single-photon source based on transmon qubit with high efficiency, Phys. Rev. Appl. 13, 034007 (2020).
  136. F. De Palma, F. Oppliger, W. Jang, S. Bosco, M. Janík, S. Calcaterra, G. Katsaros, G. Isella, D. Loss, and P. Scarlino, Strong hole-photon coupling in planar Ge: probing the charge degree and Wigner molecule states, Nat. Commun. 15, 10177 (2024).
  137. P. Scarlino, J. H. Ungerer, D. J. van Woerkom, M. Mancini, P. Stano, C. Müller, A. J. Landig, J. V. Koski, C. Reichl, W. Wegscheider, T. Ihn, K. Ensslin, and A. Wallraff, In situ tuning of the electric-dipole strength of a double-dot charge qubit: Charge-noise protection and ultrastrong coupling, Phys. Rev. X 12, 031004 (2022).
  138. A. D’Arrigo, R. Lo Franco, G. Benenti, E. Paladino, and G. Falci, Recovering entanglement by local operations, Ann. Phys. (N. Y) 350, 211 (2014).
  139. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. Vandersypen, Spins in few-electron quantum dots, Rev. Mod. Phys. 79, 1217 (2007).
  140. P. Scarlino, E. Kawakami, P. Stano, M. Shafiei, C. Reichl, W. Wegscheider, and L. Vandersypen, Spin-relaxation anisotropy in a GaAs quantum dot, Phys. Rev. Lett. 113, 256802 (2014).
  141. L. C. Camenzind, L. Yu, P. Stano, J. D. Zimmerman, A. C. Gossard, D. Loss, and D. M. Zumbühl, Hyperfine-phonon spin relaxation in a single-electron GaAs quantum dot, Nat. Commun. 9, 3454 (2018).
  142. F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Silicon quantum electronics, Rev. Mod. Phys. 85, 961 (2013).
  143. W. I. L. Lawrie, N. W. Hendrickx, F. van Riggelen, M. Russ, L. Petit, A. Sammak, G. Scappucci, and M. Veldhorst, Spin relaxation benchmarks and individual qubit addressability for holes in quantum dots, Nano Lett. 20, 7237 (2020).
  144. N. W. Hendrickx, W. I. L. Lawrie, M. Russ, F. van Riggelen, S. L. de Snoo, R. N. Schouten, A. Sammak, G. Scappucci, and M. Veldhorst, A four-qubit germanium quantum processor, Nature 591, 580 (2021).
  145. M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leenstra, B. de Ronde, J. P. Dehollain, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak, An addressable quantum dot qubit with fault-tolerant control-fidelity, Nat. Nanotechnol. 9, 981 (2014).
  146. P. Stano and D. Loss, Review of performance metrics of spin qubits in gated semiconducting nanostructures, Nat. Rev. Phys. 4, 672 (2022).
  147. J. Shabani, M. Kjaergaard, H. J. Suominen, Y. Kim, F. Nichele, K. Pakrouski, T. Stankevic, R. M. Lutchyn, P. Krogstrup, R. Feidenhans’l, S. Kraemer, C. Nayak, M. Troyer, C. M. Marcus, and C. J. Palmstrøm, Two-dimensional epitaxial superconductor-semiconductor heterostructures: A platform for topological superconducting networks, Phys. Rev. B 93, 155402 (2016).
  148. H. J. Suominen, M. Kjaergaard, A. R. Hamilton, J. Shabani, C. J. Palmstrøm, C. M. Marcus, and F. Nichele, Zero-energy modes from coalescing Andreev states in a two-dimensional semiconductor-superconductor hybrid platform, Phys. Rev. Lett. 119, 176805 (2017).
  149. Q. Wang, S. L. D. ten Haaf, I. Kulesh, D. Xiao, C. Thomas, M. J. Manfra, and S. Goswami, Triplet correlations in Cooper pair splitters realized in a two-dimensional electron gas, Nat. Commun. 14, 4876 (2023).
  150. N. W. Hendrickx, D. P. Franke, A. Sammak, M. Kouwenhoven, D. Sabbagh, L. Yeoh, R. Li, M. L. V. Tagliaferri, M. Virgilio, G. Capellini, G. Scappucci, and M. Veldhorst, Gate-controlled quantum dots and superconductivity in planar germanium, Nat. Commun. 9, 2835 (2018).
  151. G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Katsaros, and M. Veldhorst, The germanium quantum information route, Nat. Rev. Mat. 6, 926 (2021).
  152. J. Krause, C. Dickel, E. Vaal, M. Vielmetter, J. Feng, R. Bounds, G. Catelani, J. M. Fink, and Y. Ando, Magnetic field resilience of three-dimensional transmons with thin-film al/alox/Al Josephson junctions approaching 1t, Phys. Rev. Appl. 17, 034032 (2022).
  153. S. Telkamp, T. Antonelli, C. Todt, M. Hinderling, M. Coraiola, D. Haxell, S. ten Kate, D. Sabonis, P. Zeng, R. Schott, E. Cheah, C. Reichl, F. Nichele, F. Krizek, and W. Wegscheider, Development of a Nb-based semiconductor-superconductor hybrid 2DEG platform, Adv. Electron. Mater. 2024, 2400687 (2025).
  154. D. M. Pozar, Microwave Engineering; 3rd ed. (Wiley, Hoboken, NJ, 2005).
  155. N. A. Masluk, I. M. Pop, A. Kamal, Z. K. Minev, and M. H. Devoret, Microwave characterization of Josephson junction arrays: Implementing a low loss superinductance, Phys. Rev. Lett. 109, 137002 (2012).
  156. N. Samkharadze, A. Bruno, P. Scarlino, G. Zheng, D. P. DiVincenzo, L. DiCarlo, and L. Vandersypen, High-kinetic-inductance superconducting nanowire resonators for circuit QED in a magnetic field, Phys. Rev. Appl. 5, 044004 (2016).
  157. L. Grünhaupt, M. Spiecker, D. Gusenkova, N. Maleeva, S. T. Skacel, I. Takmakov, F. Valenti, P. Winkel, H. Rotzinger, W. Wernsdorfer, A. V. Ustinov, and I. M. Pop, Granular aluminium as a superconducting material for high-impedance quantum circuits, Nat. Mater. 18, 816 (2019).
  158. R. Kuzmin, N. Mehta, N. Grabon, R. Mencia, and V. E. Manucharyan, Superstrong coupling in circuit quantum electrodynamics, Npj Quantum Inf. 5, 20 (2019).
  159. J. Puertas Martínez, S. Léger, N. Gheeraert, R. Dassonneville, L. Planat, F. Foroughi, Y. Krupko, O. Buisson, C. Naud, W. Hasch-Guichard, S. Florens, I. Snyman, and N. Roch, A tunable Josephson platform to explore many-body quantum optics in circuit-QED, Npj Quantum Inf. 5, 19 (2019).
  160. N. Mehta, R. Kuzmin, C. Ciuti, and V. E. Manucharyan, Down-conversion of a single photon as a probe of many-body localization, Nature 613, 650 (2023).
  161. D. Niepce, J. Burnett, and J. Bylander, High kinetic inductance NbN nanowire superinductors, Phys. Rev. Appl. 11, 044014 (2019).
  162. A. Shearrow, G. Koolstra, S. J. Whiteley, N. Earnest, P. S. Barry, F. J. Heremans, D. D. Awschalom, E. Shirokoff, and D. I. Schuster, Atomic layer deposition of titanium nitride for quantum circuits, Appl. Phys. Lett. 113, 212601 (2018).
  163. J.-C. Besse, S. Gasparinetti, M. C. Collodo, T. Walter, P. Kurpiers, M. Pechal, C. Eichler, and A. Wallraff, Single-shot quantum nondemolition detection of individual itinerant microwave photons, Phys. Rev. X 8, 021003 (2018).
  164. Z. Wang, L. Balembois, M. Rančić, E. Billaud, M. Le Dantec, A. Ferrier, P. Goldner, S. Bertaina, T. Chanelière, D. Esteve, D. Vion, P. Bertet, and E. Flurin, Single-electron spin resonance detection by microwave photon counting, Nature 619, 276 (2023).
  165. L. Balembois, J. Travesedo, L. Pallegoix, A. May, E. Billaud, M. Villiers, D. Estève, D. Vion, P. Bertet, and E. Flurin, Cyclically operated microwave single-photon counter with sensitivity of 10−22W/Hz sensitivity, Phys. Rev. Appl. 21, 014043 (2024).
  166. W. Khan, P. P. Potts, S. Lehmann, C. Thelander, K. A. Dick, P. Samuelsson, and V. F. Maisi, Efficient and continuous microwave photoconversion in hybrid cavity-semiconductor nanowire double quantum dot diodes, Nat. Commun. 12, 5130 (2021).
  167. O. Stanisavljević, J.-C. Philippe, J. Gabelli, M. Aprili, J. Estève, and J. Basset, Efficient microwave photon-to-electron conversion in a high-impedance quantum circuit, Phys. Rev. Lett. 133, 076302 (2024).
  168. E. Zakka-Bajjani, F. Nguyen, M. Lee, L. Vale, R. W. Simmonds, and J. Aumentado, Quantum superposition of a single microwave photon in two different colour states, Nat. Phys. 7, 599 (2011).
  169. E. V. Stolyarov, Generation of time-frequency entangled photon pairs propagating in separate waveguides in circuit qed setup, Phys. Rev. A 106, 052420 (2022).
  170. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  171. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  172. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  173. R. C. C. Leon, C. H. Yang, J. C. C. Hwang, J. Camirand Lemyre, T. Tanttu, W. Huang, K. W. Chan, K. Y. Tan, F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, M. Pioro-Ladrière, A. Saraiva, and A. S. Dzurak, Coherent spin control of s-, p-, d- and f-electrons in a silicon quantum dot, Nat. Commun. 11, 797 (2020).
  174. T. Dvir, G. Wang, N. van Loo, C.-X. Liu, G. P. Mazur, A. Bordin, S. L. Ten Haaf, J.-Y. Wang, D. van Driel, and F. Zatelli et al., Realization of a minimal Kitaev chain in coupled quantum dots, Nature 614, 445 (2023).

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PRX Quantum 6, 020339– Published 29 May, 2025
  • Received 7 October 2024
  • Revised 11 February 2025
  • Accepted 3 April 2025
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