Abstract

Motivated by the progress on shortcuts to adiabaticity, we propose three schemes for speeding up (fractional) stimulated Raman adiabatic passage, and achieving rapid and non-adiabatic creation and transfer of maximal coherence in a triple-quantum-dot system. These different but relevant protocols, designed from counter-diabatic driving, dress-state method, and resonant technique, require their own pumping fields, applied gate voltages and varying tunneling couplings between two spatially separated dots. Such fast and reliable shortcuts not only allow for feasibly experimental realization in solid-state architectures but also may have potential applications in quantum information processing and quantum control.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (5)

H. Edlbauer, S. Takada, G. Roussely, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, T. Meunier, and C Bäuerle, Non-universal transmission phase behaviour of a large quantum dot, Nat. Commun. 8, 1710 (2017).
[Crossref] [PubMed]

B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelarated quantum control using superadiabatic dynamics in solid-state lambda system,” Nat. Phys. 13, 330–334 (2017).
[Crossref]

A. Benseny, A. Kiely, Y.-P. Zhang, T. Busch, and A. Ruschhaupt, “Spatial non-adiabatic passage using geometric phases,” EPJ Quantum Technol. 4, 3 (2017).
[Crossref]

N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89, 015006 (2017).
[Crossref]

Bruce W. Shore, “Picturing stimulated Raman adiabatic passage: a STIRAP tutorial,” Adv. Opt. Photon. 9, 563–719 (2017).
[Crossref]

2016 (6)

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 074401 (2016).
[Crossref] [PubMed]

A. Baksic, H. Ribeiro, and A. A. Clerk, “Speeding up adiabatic quantum state transfer by using dressed states,” Phys. Rev. Lett. 116, 230503 (2016).
[Crossref] [PubMed]

Y.-C. Li and X. Chen, “Shortcut to adiabatic population transfer in quantum three-level systems: effective two-level problems and feasible counter-diabatic driving,” Phys. Rev. A 94, 063411 (2016).
[Crossref]

Y.-X. Du, Z.-T. Liang, Y.-C. Li, X.-X. Yue, Q.-X. Lv, W. Huang, X. Chen, H. Yan, and S.-L. Zhu, “Experimental realization of stimulated Raman shortcut-to-adiabatic passage with cold atoms,” Nat. Commun. 7, 12479 (2016).
[Crossref] [PubMed]

M. Kotzian, F. Gallego-Marcos, G. Platero, and R. J. Haug, “Channel blockade in a two-path triple-quantum-dot system,” Phys. Rev. B 94, 035442 (2016).
[Crossref]

S.-C. Tian, R.-G. Wan, C.-L. Wang, S.-L. Shu, L.-J. Wang, and C.-Z. Tong, “Creation and transfer of coherence via technique of stimulated Raman adiabatic passage in triple quantum dots,” Nanoscale Res. Lett. 11, 219–226 (2016).
[Crossref] [PubMed]

2015 (1)

E. Ferraro, M. De Michielis, M. Fanciulli, and E. Prati, “Coherent tunneling by adiabatic passage of an exchange-only spin qubit in a double quantum dot chain,” Phys. Rev. B 91, 075435 (2015).
[Crossref]

2014 (1)

Y. Ban and X. Chen, “Counter-diabatic driving for fast spin control in a two-electron double quantum dot,” Sci. Rep. 4, 6258 (2014).
[Crossref] [PubMed]

2013 (8)

J. Zhang, L. Greenman, X.-T. Deng, I. M. Hayes, and K. Birgitta Whaley, “Optimal control for electron shuttling,” Phys. Rev. B 87, 235324 (2013).
[Crossref]

S. Oh, Y-P. Shim, J. Fei, M. Friesen, and X. D. Hu, “Resonant adiabatic passage with three qubits,” Phys. Rev. A 87, 022332 (2013).
[Crossref]

J. Huneke, G. Platero, and S. Kohler, “Steady-state coherent transfer by adiabatic passage,” Phys. Rev. Lett. 110, 036802 (2013).
[Crossref] [PubMed]

M. N. Kiselev, K. Kikoin, and M. B. Kenmoe, “SU(3) Landau-Zener interferometry,” EPL 104, 57004 (2013).
[Crossref]

J.-F. Zhang, J. H. Shim, I. Niemeyer, T. Taniguchi, T. Teraji, H. Abe, S. Onoda, T. Yamamoto, T. Ohshima, J. Isoya, and D. Suter, “Experimental implementation of assisted quantum adiabatic passage in a single spin,” Phys. Rev. Lett. 110, 240501 (2013).
[Crossref] [PubMed]

E. Torrontegui, S. Ibanez, S. Martinez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. Atom. Mol. Opt. Phys. 62, 117–169 (2013).
[Crossref]

M. Seo, H. K. Choi, S.-Y. Lee, N. Kim, Y. Chung, H.-S. Sim, V. Umansky, and D. Mahalu, “Charge frustration in a triangular triple quantum dot”, Phys. Rev. Lett. 110, 046803 (2013).
[Crossref] [PubMed]

A. del Campo, “Shortcuts to adiabaticity by counterdiabatic driving,” Phys. Rev. Lett. 111, 100502 (2013).
[Crossref]

2012 (4)

M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nat. Phys. 8, 147–152 (2012).
[Crossref]

S. Ibánez, X. Chen, E. Torrontegui, J. G. Muga, and A. Ruschhaupt, “Multiple Schrödinger pictures and dynamics in shortcuts to adiabaticity,” Phys. Rev. Lett. 109, 100403 (2012).
[Crossref]

R. Rahman, R. P. Muller, J. E. Levy, M. S. Carroll, G. Klimeck, A. D. Greentree, and L. C. L. Hollenberg, “Coherent electron transport by adiabatic passage in an imperfect donor chain,” Phys. Rev. B 82, 155315 (2012).
[Crossref]

X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86, 033405 (2012).
[Crossref]

2011 (1)

B. Chen, W. Fan, and Y. Xu, “Adiabatic quantum state transfer in a nonuniform triple-quantum-dot system,” Phys. Rev. A 83, 014301 (2011).
[Crossref]

2010 (2)

L. M. Jong and A. D. Greentree, “Interferometry using spatial adiabatic passage in quantum dot networks,” Phys. Rev. B 81, 035311 (2010).
[Crossref]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett. 105, 123003 (2010).
[Crossref] [PubMed]

2009 (4)

E. Vernek, C. A. Büsser, G. B. Martins, E. V. Anda, N. Sandler, and S. E. Ulloa, “Kondo regime in triangular arrangements of quantum dots: Molecular orbitals, interference, and contact effects,” Phys. Rev. B 80, 035119 (2009).
[Crossref]

C. Pöltl, C. Emary, and T. Brandes, “Two-particle dark state in the transport through a triple quantum dot,” Phys. Rev. B 80, 115313 (2009).
[Crossref]

T. Kostyrko and B. R. Bułka, “Symmetry-controlled negative differential resistance effect in a triangular molecule,” Phys. Rev. B 79, 075310 (2009).
[Crossref]

M. V. Berry, “Transitionless quantum driving,” J. Phys. A 142, 365303 (2009).
[Crossref]

2008 (2)

J. H. Cole, A. D. Greentree, L. C. L. Hollenberg, and S. Das Sarma, “Spatial adiabatic passage in a realistic triple well structure,” Phys. Rev. B 77, 235418 (2008).
[Crossref]

M. C. Rogge and R. J. Haug, “Two-path transport measurements on a triple quantum dot”, Phys. Rev. B 77, 193306 (2008).
[Crossref]

2006 (1)

T. Kuzmenko, K. Kikoin, and Y. Avishai, “Magnetically tunable Kondo-Aharonov–Bohm effect in a triangular quantum dot,” Phys. Rev. Lett. 96, 046601 (2006).
[Crossref]

2005 (2)

J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
[Crossref] [PubMed]

M. Demirplak and S. A. Rice, “Assisted adiabatic passage revisited,” J. Phys. Chem. B 109, 6838–6844 (2005).
[Crossref]

2004 (1)

A. D. Greentree, J. H. Cole, A. R. Hamilton, and L. C. L. Hollenberg, “Coherent electronic transfer in quantum dot systems using adiabatic passage,” Phys. Rev. B 70, 235317 (2004).
[Crossref]

2003 (2)

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107, 9937–9945 (2003).
[Crossref]

A. R. P. Rau and W.-C. Zhao, “Decoherence in a driven three-level system,” Phys. Rev. A 68, 052102 (2003).
[Crossref]

1999 (1)

N. V. Vitanov, K.-A. Suominen, and B. W. Shore, “Creation of coherent atomic superpositions by fractional stimulated Raman adiabatic passage,” J. Phys. B 32, 4535–4546 (1999).
[Crossref]

1998 (1)

K. Bergmann, H. Theuer, and B. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

1990 (1)

C. E. Carroll and F. T. Hioe, “Analytic solutions for three-state systems with overlapping pulses,” Phys. Rev. A 42, 1522–1531 (1990).
[Crossref] [PubMed]

Abe, H.

J.-F. Zhang, J. H. Shim, I. Niemeyer, T. Taniguchi, T. Teraji, H. Abe, S. Onoda, T. Yamamoto, T. Ohshima, J. Isoya, and D. Suter, “Experimental implementation of assisted quantum adiabatic passage in a single spin,” Phys. Rev. Lett. 110, 240501 (2013).
[Crossref] [PubMed]

Ahufinger, V.

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 074401 (2016).
[Crossref] [PubMed]

Anda, E. V.

E. Vernek, C. A. Büsser, G. B. Martins, E. V. Anda, N. Sandler, and S. E. Ulloa, “Kondo regime in triangular arrangements of quantum dots: Molecular orbitals, interference, and contact effects,” Phys. Rev. B 80, 035119 (2009).
[Crossref]

Arimondo, E.

M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nat. Phys. 8, 147–152 (2012).
[Crossref]

Auer, A.

B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelarated quantum control using superadiabatic dynamics in solid-state lambda system,” Nat. Phys. 13, 330–334 (2017).
[Crossref]

Avishai, Y.

T. Kuzmenko, K. Kikoin, and Y. Avishai, “Magnetically tunable Kondo-Aharonov–Bohm effect in a triangular quantum dot,” Phys. Rev. Lett. 96, 046601 (2006).
[Crossref]

Awschalom, D. D.

B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelarated quantum control using superadiabatic dynamics in solid-state lambda system,” Nat. Phys. 13, 330–334 (2017).
[Crossref]

Baksic, A.

B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelarated quantum control using superadiabatic dynamics in solid-state lambda system,” Nat. Phys. 13, 330–334 (2017).
[Crossref]

A. Baksic, H. Ribeiro, and A. A. Clerk, “Speeding up adiabatic quantum state transfer by using dressed states,” Phys. Rev. Lett. 116, 230503 (2016).
[Crossref] [PubMed]

Ban, Y.

Y. Ban and X. Chen, “Counter-diabatic driving for fast spin control in a two-electron double quantum dot,” Sci. Rep. 4, 6258 (2014).
[Crossref] [PubMed]

Bason, M. G.

M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nat. Phys. 8, 147–152 (2012).
[Crossref]

Bäuerle, C

H. Edlbauer, S. Takada, G. Roussely, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, T. Meunier, and C Bäuerle, Non-universal transmission phase behaviour of a large quantum dot, Nat. Commun. 8, 1710 (2017).
[Crossref] [PubMed]

Benseny, A.

A. Benseny, A. Kiely, Y.-P. Zhang, T. Busch, and A. Ruschhaupt, “Spatial non-adiabatic passage using geometric phases,” EPJ Quantum Technol. 4, 3 (2017).
[Crossref]

R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 074401 (2016).
[Crossref] [PubMed]

Bergmann, K.

N. V. Vitanov, A. A. Rangelov, B. W. Shore, and K. Bergmann, “Stimulated Raman adiabatic passage in physics, chemistry, and beyond,” Rev. Mod. Phys. 89, 015006 (2017).
[Crossref]

K. Bergmann, H. Theuer, and B. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70, 1003–1025 (1998).
[Crossref]

Berry, M. V.

M. V. Berry, “Transitionless quantum driving,” J. Phys. A 142, 365303 (2009).
[Crossref]

Birgitta Whaley, K.

J. Zhang, L. Greenman, X.-T. Deng, I. M. Hayes, and K. Birgitta Whaley, “Optimal control for electron shuttling,” Phys. Rev. B 87, 235324 (2013).
[Crossref]

Brandes, T.

C. Pöltl, C. Emary, and T. Brandes, “Two-particle dark state in the transport through a triple quantum dot,” Phys. Rev. B 80, 115313 (2009).
[Crossref]

Bulka, B. R.

T. Kostyrko and B. R. Bułka, “Symmetry-controlled negative differential resistance effect in a triangular molecule,” Phys. Rev. B 79, 075310 (2009).
[Crossref]

Burkard, G.

B. B. Zhou, A. Baksic, H. Ribeiro, C. G. Yale, F. J. Heremans, P. Jerger, A. Auer, G. Burkard, A. A. Clerk, and D. D. Awschalom, “Accelarated quantum control using superadiabatic dynamics in solid-state lambda system,” Nat. Phys. 13, 330–334 (2017).
[Crossref]

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J. Huneke, G. Platero, and S. Kohler, “Steady-state coherent transfer by adiabatic passage,” Phys. Rev. Lett. 110, 036802 (2013).
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M. Seo, H. K. Choi, S.-Y. Lee, N. Kim, Y. Chung, H.-S. Sim, V. Umansky, and D. Mahalu, “Charge frustration in a triangular triple quantum dot”, Phys. Rev. Lett. 110, 046803 (2013).
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X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two- and three-level atoms,” Phys. Rev. Lett. 105, 123003 (2010).
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H. Edlbauer, S. Takada, G. Roussely, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, T. Meunier, and C Bäuerle, Non-universal transmission phase behaviour of a large quantum dot, Nat. Commun. 8, 1710 (2017).
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M. Seo, H. K. Choi, S.-Y. Lee, N. Kim, Y. Chung, H.-S. Sim, V. Umansky, and D. Mahalu, “Charge frustration in a triangular triple quantum dot”, Phys. Rev. Lett. 110, 046803 (2013).
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M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nat. Phys. 8, 147–152 (2012).
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M. G. Bason, M. Viteau, N. Malossi, P. Huillery, E. Arimondo, D. Ciampini, R. Fazio, V. Giovannetti, R. Mannella, and O. Morsch, “High-fidelity quantum driving,” Nat. Phys. 8, 147–152 (2012).
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J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots,” Science 309, 2180–2184 (2005).
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[Crossref]

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E. Vernek, C. A. Büsser, G. B. Martins, E. V. Anda, N. Sandler, and S. E. Ulloa, “Kondo regime in triangular arrangements of quantum dots: Molecular orbitals, interference, and contact effects,” Phys. Rev. B 80, 035119 (2009).
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R. Menchon-Enrich, A. Benseny, V. Ahufinger, A. D. Greentree, T. Busch, and J. Mompart, “Spatial adiabatic passage: a review of recent progress,” Rep. Prog. Phys. 79, 074401 (2016).
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H. Edlbauer, S. Takada, G. Roussely, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, T. Meunier, and C Bäuerle, Non-universal transmission phase behaviour of a large quantum dot, Nat. Commun. 8, 1710 (2017).
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E. Torrontegui, S. Ibanez, S. Martinez-Garaot, M. Modugno, A. del Campo, D. Guéry-Odelin, A. Ruschhaupt, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity,” Adv. Atom. Mol. Opt. Phys. 62, 117–169 (2013).
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Adv. Atom. Mol. Opt. Phys. (1)

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Adv. Opt. Photon. (1)

EPJ Quantum Technol. (1)

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Nat. Commun. (2)

H. Edlbauer, S. Takada, G. Roussely, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, T. Meunier, and C Bäuerle, Non-universal transmission phase behaviour of a large quantum dot, Nat. Commun. 8, 1710 (2017).
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Phys. Rev. B (11)

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Figures (7)

Fig. 1
Fig. 1 (a) Schematic diagram of a TQD. The pumping laser is shed on QD1 to excite one exciton, while the tunneling rates Ω2 between QD1 and QD2, Ω3 between QD1 and QD3 are controlled by gate electrodes. (b) The four-level system is comprised of: the ground state |0〉 without any excitations in any dots; the first excited state |1〉 where the exciton, including both the electron and the hole, is in the first dot; the states |2〉 and |3〉, where the electron is in the second and the third dot, respectively.
Fig. 2
Fig. 2 F-STIRAP (step I) and STIRAP (step II) for creation and transfer of coherence in a TQD. (a) pumping pulse Ω1 (blue, solid) and tunneling pulses Ω2 (red, dashed), Ω3 (black, dotted). (b) Time evolution of quantum states, where populations P0 (blue, solid), P1 (red, dashed), P2 (black, dotted), P3 (purple, dash-dotted), with t f I = t f II = 100. Noting that the Rabi frequency is scaled by 5 × 2π MHz and the corresponding operation times are in units of 2π0(= 0.01μs).
Fig. 3
Fig. 3 Accelerated F-STIRAP (step I) and STIRAP (step II) by using counter-diabatic driving along unitary transformation in protocol A: (a) modified pumping pulse Ω̃1 (blue, solid) and tunneling pulses Ω̃2 (red, dashed), Ω̃3 (black, dash-dotted). (b) Time evolution of quantum states, where P0 (blue, solid), P1 (red, dashed), P2 (black, dotted), P3 (purple, dot-dashed), with t f I = t f II = 1. Noting that the Rabi frequency is scaled by 5 × 2π MHz and the operation time is scaled by 0.01μs.
Fig. 4
Fig. 4 Accelerated F-STIRAP (step I) and STIRAP (step II) by using dressed-state method in protocol B: (a) modified pumping pulse Ω̃1 (blue, solid) and tunneling pulses Ω̃2 (red, dashed), Ω̃3 (black, dash-dotted). (b) Time evolution of quantum states, where P0 (blue, solid), P1 (red, dashed), P2 (black, dotted), P3 (purple, dot-dashed), with t f I = t f II = 1. Noting that the Rabi frequency is scaled by 5 × 2π MHz and the operation time is scaled by 0.01μs.
Fig. 5
Fig. 5 With the suppression of the population excitations in the intermediate state, accelerated F-STIRAP (step I) and STIRAP (step II) by using dressed-state method in protocol B: (a) modified pumping pulse Ω̃1 (blue, solid) and tunneling pulses Ω̃2 (red, dashed), Ω̃3 (black, dash-dotted). (b) Time evolution of quantum states, where P0 (blue, solid), P1 (red, dashed), P2 (black, dotted), P3 (purple, dot-dashed), with t f I = t f II = 1. Noting that the Rabi frequency is scaled by 5 × 2π MHz and the operation time is scaled by 0.01μs.
Fig. 6
Fig. 6 Fast non-adiabatic creation (step I) and transfer (step II) of coherence in a TQD by using resonant pulses in protocol C: (a) designed pumping pulse Ω̃1 (blue, solid) and tunneling pulses Ω̃2 (red, dashed), Ω̃3 (black, dash-dotted). (b) Time evolution of quantum states, where population P0 (blue, solid), P1 (red, dashed), P2 (black, dotted), P3 (purple, dot-dashed), with t f I = 1.24 and t f II = 0.61. Noting that the Rabi frequency is scaled by 5 × 2π MHz and the operation time is scaled by 0.01μs.
Fig. 7
Fig. 7 Infidelity log(1 − Fi) (i =I, II) versus operation time for two steps, where pumping and tunneling pulses are Ω1 = Ω sin(ωt), Ω2 = Ω cos(ωt), and Ω3 = Ω cos(ωt). Parameters: ω = π / ( 4 t f I ) and Ω = 5 for step I (red, dashed) and ω = π / ( 2 t f II ) and Ω = 10 for step II (blue, solid). Here the operation times t f I = 1.24 and t f II = 0.61 are highlighted by two vertical (black, dashed) lines.

Equations (42)

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H 0 = ( 0 Ω 1 ( t ) 0 0 Ω 1 ( t ) δ p Ω 2 ( t ) Ω 3 ( t ) 0 Ω 2 ( t ) ( δ p ω 12 ) 0 0 Ω 3 ( t ) 0 ( δ p ω 13 ) ) ,
| Ψ ( t ) = Σ j = 0 3 c j ( t ) | j ,
H 0 I = ( 0 Ω 1 ( t ) 0 Ω 1 ( t ) 0 Ω 2 ( t ) 0 Ω 2 ( t ) 0 ) ,
H 0 II = ( 0 Ω 2 ( t ) Ω 3 ( t ) Ω 2 ( t ) 0 0 Ω 3 ( t ) 0 0 ) .
Ω 1 ( t ) = Ω 0 I { tanh [ ( t t 1 ) / τ 1 ] + 1 } ,
Ω 2 ( t ) = Ω 0 I { tanh [ ( t t 1 ) / τ 1 ] 3 } ,
Ω 2 ( t ) = Ω 0 II exp [ ( t t 2 ) 2 / ( 2 τ 2 2 ) ] ,
Ω 3 ( t ) = Ω 0 II exp [ ( t t 3 ) 2 / ( 2 τ 2 2 ) ] ,
| D = cos θ | 0 sin θ | 2 ,
| B ± = 1 2 ( sin θ | 0 ± 1 | 1 + cos θ | 2 ) ,
A ( t ) = ( sin θ / 2 cos θ sin θ / 2 1 / 2 0 1 / 2 cos θ / 2 sin θ cos θ / 2 ) .
ad I = ( Ω ( t ) 0 i θ ˙ 0 0 0 i θ ˙ 0 Ω ( t ) ) ,
cd I = ( 0 0 i θ ˙ 0 0 0 i θ ˙ 0 0 ) ,
H I = ( Ω 1 ( t ) λ 1 + Ω 2 ( t ) λ 6 θ ˙ λ 5 ) ,
λ 1 = ( 0 1 0 1 0 0 0 0 0 ) , λ 6 = ( 0 0 0 0 0 1 0 1 0 ) , λ 5 = ( 0 0 i 0 0 0 i 0 0 ) .
U ( t ) = e i ϕ λ 6 = ( 1 0 0 0 cos ϕ sin ϕ 0 i sin ϕ cos ϕ ) .
Ω ˜ 1 = Ω 1 cos ϕ + θ ˙ sin ϕ ,
Ω ˜ 2 = Ω 2 ϕ ˙ ,
Ω ˜ a = θ ˙ cos ϕ + Ω 1 sin ϕ .
Ω ˜ 1 = Ω 1 2 + θ ˙ 2
Ω ˜ 2 = Ω 2 ϕ ˙ .
Ω ˜ 2 = Ω 2 2 + θ ˙ 2 2 ,
Ω ˜ 3 = Ω 3 ϕ ˙ 2 ,
Ω 1 = Ω sin θ , Ω 2 = Ω cos θ .
ad I = Ω J z + θ ˙ J y ,
J x = 1 2 ( 0 1 0 1 0 1 0 1 0 ) , J y = 1 2 ( 0 i 0 i 0 i 0 i 0 ) , J z = ( 1 0 0 0 0 0 0 0 1 ) .
c I = g x ( t ) J x + g z ( t ) J z ,
V = exp [ i η ( t ) J z ] exp [ i μ ( t ) J y ] exp [ i ξ ( t ) J z ] ,
I = V ( ad I + c I ) V + i d d t V V ,
θ ˜ = θ arctan [ g x ( t ) Ω + g z ( t ) ] ,
Ω ˜ = [ Ω + g z ( t ) ] 2 + g x 2 ( t ) ,
g x ( t ) = μ ˙ cos ξ θ ˙ tan ξ .
g z ( t ) = Ω + ξ ˙ + μ ˙ sin ξ θ ˙ tan μ cos ξ .
μ = arctan ( θ ˙ Ω ) , g x ( t ) = μ ˙ , g z ( t ) = 0 .
Ω 2 = Ω sin θ 2 , Ω 3 = Ω cos θ 2 .
μ = arctan ( θ ˙ f ( t ) Ω ) , f ( t ) = 1 + A exp ( t 2 τ ) ,
g x ( t ) = μ ˙ , g z ( t ) = Ω ( t ) θ ˙ tan μ .
ad I = Ω J z + θ ˙ J y ,
0 t f Ω 2 + θ ˙ 2 d t = 2 k π ,
Ω 1 = Ω sin ( ω t ) , Ω 2 = Ω cos ( ω t ) ,
t f I ( k ) = 1 Ω 4 π 2 k 2 π 2 16 .
t f II ( k ) = 1 Ω 4 π 2 k 2 π 2 4 .

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