Abstract

By periodical two-step modulation, we demonstrate that the dynamics of a multilevel system can evolve even in a multiple large detunings regime and provide the effective Hamiltonian (of interest) for this system. We then illustrate this periodical modulation in quantum state engineering, including achieving direct transition from the ground state to the Rydberg state or the desired superposition of two Rydberg states without satisfying the two-photon resonance condition, switching between the Rydberg blockade regime and the Rydberg antiblockade regime, stimulating distinct atomic transitions by the same laser field, and implementing selective transitions in the same multilevel system. Particularly, it is robust against perturbation of control parameters. Another advantage is that the waveform of the laser field has a simple square-wave form, which is readily implemented in experiments. Thus, it offers us a novel method of quantum state engineering in quantum information processing.

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

Full Article  |  PDF Article
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2017 (10)

Y. C. Yang, S. N. Coppersmith, and M. Friesen, “Achieving high-fidelity single-qubit gates in a strongly driven silicon-quantum-dot hybrid qubit,” Phys. Rev. A 95, 062321(2017).
[Crossref]

D. Pagel and H. Fehske, “Non-Markovian dynamics of few emitters in a laser-driven cavity,” Phys. Rev. A 96, 041802 (2017).
[Crossref]

R. Desbuquois, M. Messer, F. Görg, K. Sandholzer, G. Jotzu, and T. Esslinger, “Controlling the Floquet state population and observing micromotion in a periodically driven two-body quantum system,” Phys. Rev. A 96, 053602 (2017).
[Crossref]

S. A. Malinovskaya, “Design of many-body spin states of Rydberg atoms excited to highly tunable magnetic sublevels,” Opt. Lett. 42, 314–317 (2017).
[Crossref]

S. L. Su, Y. Gao, E. Liang, and S. Zhang, “Fast Rydberg antiblockade regime and its applications in quantum logic gates,” Phys. Rev. A 95, 022319 (2017).
[Crossref]

H. Ribeiro, A. Baksic, and A. A. Clerk, “Systematic Magnus-based approach for suppressing leakage and nonadiabatic errors in quantum dynamics,” Phys. Rev. X 7, 011021 (2017).

N. Thaicharoen, A. Schwarzkopf, and G. Raithel, “Control of spatial correlations between Rydberg excitations using rotary echo,” Phys. Rev. Lett. 118, 133401 (2017).
[Crossref]

Y. C. Zhang, X. F. Zhou, X. X. Zhou, G. C. Guo, and Z. W. Zhou, “Cavity-assisted single-mode and two-mode spin-squeezed states via phase-locked atom-photon coupling,” Phys. Rev. Lett. 118, 083604 (2017).
[Crossref] [PubMed]

X. F. Shi and T. A. B. Kennedy, “Annulled van der Waals interaction and fast Rydberg quantum gates,” Phys. Rev. A 95, 043429 (2017).
[Crossref]

S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, “Optical control of the resonant dipole-dipole interaction between Rydberg atoms,” Phys. Rev. Lett. 119, 053202 (2017).
[Crossref] [PubMed]

2016 (6)

T. Keating, C. H. Baldwin, Y. Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, “Arbitrary Dicke-state control of symmetric Rydberg ensembles,” Phys. Rev. Lett. 117, 213601 (2016).
[Crossref] [PubMed]

C. Tresp, C. Zimmer, I. Mirgorodskiy, H. Gorniaczyk, A. Paris-Mandoki, and S. Hofferberth, “Single-photon absorber based on strongly interacting Rydberg atoms,” Phys. Rev. Lett. 117, 223001 (2016).
[Crossref] [PubMed]

M. B. Kenmoe and L. C. Fai, “Periodically driven three-level systems,” Phys. Rev. B,  94, 125101 (2016).
[Crossref]

K. Iwahori and N. Kawakami, “Long-time asymptotic state of periodically driven open quantum systems,” Phys. Rev. B 94, 184304 (2016).
[Crossref]

Z. C. Shi, W. Wang, and X. X. Yi, “Population transfer driven by far-off-resonant fields,” Opt. Express,  24, 21971–21985 (2016).
[Crossref] [PubMed]

X. Luo, L. Wu, J. Chen, Q. Guan, K. Gao, Z. F. Xu, L. You, and R. Wang, “Tunable atomic spin-orbit coupling synthesized with a modulating gradient magnetic field,” Sci. Rep. 6, 18983 (2016).
[Crossref] [PubMed]

2015 (7)

K. Jiménez-García, L. J. LeBlanc, R. A. Williams, M. C. Beeler, C. Qu, M. Gong, C. Zhang, and I. B. Spielman, “Tunable spin-orbit coupling via strong driving in ultracold-atom systems,” Phys. Rev. Lett. 114, 125301 (2015).
[Crossref] [PubMed]

H. Schempp, G. Günter, S. Wüster, M. Weidemüller, and S. Whitlock, “Correlated exciton transport in Rydberg-dressed-atom spin chains,” Phys. Rev. Lett. 115, 093002 (2015).
[Crossref] [PubMed]

N. Thaicharoen, A. Schwarzkopf, and G. Raithel, “Measurement of the van der Waals interaction by atom trajectory imaging,” Phys. Rev. A 92, 040701(2015).
[Crossref]

C. Chen, J. H. An, H. G. Luo, C. P. Sun, and C. H. Oh, “Floquet control of quantum dissipation in spin chains,” Phys. Rev. A 91, 052122 (2015).
[Crossref]

P. Ponte, Z. Papić, F. Huveneers, and D. A. Abanin, “Many-body localization in periodically driven systems,” Phys. Rev. Lett. 114, 140401 (2015).
[Crossref] [PubMed]

M. P. Silveri, K. S. Kumar, J. Tuorila, J. Li, A. Vepsäläinen, E. V. Thuneberg, and G. S. Paraoanu, “Stückelberg interference in a superconducting qubit under periodic latching modulation,” New J. Phys. 17, 043058 (2015).
[Crossref]

D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, “Coherent excitation transfer in a spin chain of three Rydberg atoms,” Phys. Rev. Lett. 114, 113002 (2015).
[Crossref] [PubMed]

2014 (6)

N. Goldman and J. Dalibard, “Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields,” Phys. Rev. X 4, 031027 (2014).

E. Kuznetsova, G. Liu, and S. A Malinovskaya, “Adiabatic rapid passage two-photon excitation of a Rydberg atom,” Phys. Scr. T160, 014024 (2014).
[Crossref]

L. E. F. Foa Torres, P. M. Perez-Piskunow, C. A. Balseiro, and G. Usaj, “Multiterminal conductance of a floquet topological insulator,” Phys. Rev. Lett. 113, 266801 (2014).
[Crossref]

M. Benito, A. Gómez-León, V. M. Bastidas, T. Brandes, and G. Platero, “Floquet engineering of long-range p-wave superconductivity,” Phys. Rev. B 90, 205127 (2014).
[Crossref]

Q. J. Tong, J. H. An, L. C. Kwek, H. G. Luo, and C. H. Oh, “Simulating Zeno physics by a quantum quench with superconducting circuits,” Phys. Rev. A 89, 060101 (2014).
[Crossref]

P. M. Perez-Piskunow, G. Usaj, C. A. Balseiro, and L. E. F. Foa Torres, “Floquet chiral edge states in graphene,” Phys. Rev. B 89, 121401 (2014).
[Crossref]

2013 (3)

D. E. Liu, A. Levchenko, and H. U. Baranger, “Floquet Majorana fermions for topological qubits in superconducting devices and cold atom systems,” Phys. Rev. Lett. 111, 047002 (2013).
[Crossref]

Y. T. Katan and D. Podolsky, “Modulated Floquet topological insulators,” Phys. Rev. Lett. 110, 016802 (2013).
[Crossref]

M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3, 031005 (2013).

2011 (3)

P. Huang, J. Zhou, F. Fang, X. Kong, X. Xu, C. Ju, and J. Du, “Landau-Zener-Stückelberg interferometry of a single electronic spin in a noisy environment,” Phys. Rev. X 1, 011003 (2011).

N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductor quantum wells,” Nat. Phys. 7, 490 (2011).
[Crossref]

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402(2011).
[Crossref] [PubMed]

2010 (5)

T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82, 235114 (2010).
[Crossref]

L. Zhou and L. M. Kuang, “Zeno-anti-Zeno crossover via external fields in a one-dimensional coupled-cavity waveguide,” Phys. Rev. A 82, 042113 (2010).
[Crossref]

S. N. Shevchenko, S. Ashhab, and F. Nori, “Landau-Zener-Stückelberg interferometry,” Phys. Rep. 492, 1–30 (2010).
[Crossref]

T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of Antiblockade in an Ultracold Rydberg Gas,” Phys. Rev. Lett. 104, 013001 (2010).
[Crossref]

Z. X. Chen, Z. W. Zhou, X. Zhou, X. F. Zhou, and G. C. Guo, “Quantum simulation of Heisenberg spin chains with next-nearest-neighbor interactions in coupled cavities,” Phys. Rev. A 81, 022303 (2010).
[Crossref]

2009 (2)

A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115 (2009).
[Crossref]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110 (2009).
[Crossref]

2008 (1)

J. Cho, D. G. Angelakis, and S. Bose, “Fractional quantum Hall state in coupled cavities,” Phys. Rev. Lett. 101, 246809 (2008).
[Crossref] [PubMed]

2007 (3)

C. Ates, T. Pohl, T. Pattard, and J. M. Rost, “Antiblockade in Rydberg Excitation of an Ultracold Lattice Gas,” Phys. Rev. Lett. 98, 023002 (2007).
[Crossref] [PubMed]

S. A. Malinovskaya and V. S. Malinovsky, “Chirped-pulse adiabatic control in coherent anti-Stokes Raman scattering for imaging of biological structure and dynamics,” Opt. Lett. 32, 707–709 (2007).
[Crossref]

S. Malinovskaya, “Chirped pulse control methods for imaging of biological structure and dynamics,” Int. J. Quant. Chem. 107, 3151–3158 (2007).
[Crossref]

2001 (1)

M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, “Dipole blockade and quantum information processing in mesoscopic atomic ensembles,” Phys. Rev. Lett. 87, 037901 (2001).
[Crossref]

2000 (1)

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85, 2208 (2000).
[Crossref] [PubMed]

1998 (1)

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

1973 (1)

H. Sambe, “Steady states and quasienergies of a quantum-mechanical system in an oscillating feld,” Phys. Rev. A 7, 2203 (1973).
[Crossref]

1965 (1)

J. H. Shirley, “Solution of the Schrödinger equation with a Hamiltonian periodic in time,” Phys. Rev. 138, B979 (1965).
[Crossref]

Abanin, D. A.

P. Ponte, Z. Papić, F. Huveneers, and D. A. Abanin, “Many-body localization in periodically driven systems,” Phys. Rev. Lett. 114, 140401 (2015).
[Crossref] [PubMed]

Adams, C. S.

D. Barredo, H. Labuhn, S. Ravets, T. Lahaye, A. Browaeys, and C. S. Adams, “Coherent excitation transfer in a spin chain of three Rydberg atoms,” Phys. Rev. Lett. 114, 113002 (2015).
[Crossref] [PubMed]

Akhmerov, A. R.

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402(2011).
[Crossref] [PubMed]

Alicea, J.

L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, “Majorana fermions in equilibrium and in driven cold-atom quantum wires,” Phys. Rev. Lett. 106, 220402(2011).
[Crossref] [PubMed]

Amthor, T.

T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of Antiblockade in an Ultracold Rydberg Gas,” Phys. Rev. Lett. 104, 013001 (2010).
[Crossref]

An, J. H.

C. Chen, J. H. An, H. G. Luo, C. P. Sun, and C. H. Oh, “Floquet control of quantum dissipation in spin chains,” Phys. Rev. A 91, 052122 (2015).
[Crossref]

Q. J. Tong, J. H. An, L. C. Kwek, H. G. Luo, and C. H. Oh, “Simulating Zeno physics by a quantum quench with superconducting circuits,” Phys. Rev. A 89, 060101 (2014).
[Crossref]

Angelakis, D. G.

J. Cho, D. G. Angelakis, and S. Bose, “Fractional quantum Hall state in coupled cavities,” Phys. Rev. Lett. 101, 246809 (2008).
[Crossref] [PubMed]

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Figures (12)

Fig. 1
Fig. 1 (a) The maximum population P k m a x of level | k ( k = 2 , 3 ) during the whole evolution versus E 2 / E 3 by periodically modulating the coupling strength { Ω 1 , Ω 1   } , where the initial state is | 1 and Δ 1 / Ω 1 = 60 , Ω 2 ( ) / Ω 1 = 2 , Ω 1 = Ω 1 , Δ k   = Δ k ( k = 1 , 2 ), τ 1 ( ) = π E 3 ( ) E 1 ( ) . (b-d) The time evolution of populations Pk ( k = 1 , 2 , 3 ) of each levels with different E 2 / E 3 . Except for pink-dash line, all system dynamics are simulated by using Hamiltonian (11).
Fig. 2
Fig. 2 (a) The population P3 versus evolution time and coupling strength Ω 1   under two-step modulation of the coupling strength { Ω 1 , Ω 1   } , where Ω 2 / Ω 1 = 2 , Δ 1 / Ω 1 = 60 , Δ 2 / Ω 1 = 30 , Ω 2   = Ω 2 , Δ k   = Δ k ( k = 1 , 2 ), τ 1 ( ) = π E 3 ( ) E 1 ( ) . (b) The population P3 versus evolution time and coupling strength Ω 2   under two-step modulation of the coupling strength { Ω 2 , a 2   } , where Ω 2 / Ω 1 = 2 , Δ 1 / Ω 1 = 60 , Δ 2 / Ω 1 = 20 , Ω 1   = Ω 1 , Δ k   = Δ k ( k = 1 , 2 ), τ 1 ( ) = π E 3 ( ) E 1 ( ) . (c) The population P3 versus evolution time and detuning Δ 1   under two-step modulation of the detuning { Δ 1 , Δ 1   } , where Ω 2 / Ω 1 = 2 , Δ 1 / Ω 1 = 60 , Δ 2 / Ω 1 = 30 , Ω k   = Ω k ( k = 1 , 2 ), Δ 2   = Δ 2 , τ 1 ( ) = π E 3 ( ) E 1 ( ) . (d) The population P3 versus evolution time and detuning Δ 2   under two-step modulation of the detuning { Δ 2 , Δ 2   } , where Ω 2 / Ω 1 = 2 , Δ 1 / Ω 1 = 60 , Δ 2 / Ω 1 = 20 , Ω k   = Ω k ( k = 1 , 2 ), Δ 1   = Δ 1 , τ 1 ( ) = π E 3 ( ) E 1 ( ) .
Fig. 3
Fig. 3 (a) The structure of Rydberg atom coupled by two laser fields and a microwave field. (b) The structure of two identical atoms coupled by laser fields and Rydberg-Rydberg interaction.
Fig. 4
Fig. 4 (a) The population P3 of level | 3 versus Δ2 in STIRAP, where Ω 1 ( t ) = Ω 1 e ( t τ ) 2 τ 2 , Ω 2 ( t ) = Ω 1 e t 2 τ 2 , τ = 200, and Δ 1 / Ω 1 = 30 . (b) The time evolution of populations Pk ( k = 1 , 2 , 3 ) of each levels by periodically modulating coupling strength { Ω 1 , Ω 1 } in three-level system, where Ω 1 / Ω 1 = 1 , Δ 1 ( ) / Ω 1 = 60 , Δ 2 ( ) / Ω 1 = 30 , Ω 2 ( ) / Ω 1 = 2 , τ 1 ( ) = π E 3 ( ) E 1 ( ) . (c-d) The time evolution of populations Pk ( k = 1 , 2 , 3 , 4 ) of each levels by periodically modulating coupling strength { Ω 1 , Ω 1 } in four-level system, where Ω 1 / Ω 1 = 1 , Δ 1 ( ) / Ω 1 = 60 , Δ 2 ( ) / Ω 1 = 30 , Ω 2 ( ) / Ω 1 = 2 , Δ 3 ( ) / Ω 1 = 28.8 , Ω 3 ( ) / Ω 1 = 2 . The time interval satisfies τ 1 ( ) = π E 3 ( ) E 1 ( ) in panel (c) while the time interval satisfies τ 1 ( ) = π E 4 ( ) E 1 ( ) in panel (d). All system dynamics are simulated by Hamiltonian (11).
Fig. 5
Fig. 5 The time evolution of populations Pm ( m = g g , T , r r ) of each levels by periodically modulating coupling strength { Ω e f f , Ω e f f } , where Ω e f f   = 0.5 Ω e f f , Δ 1 / Ω e f f = 23 , V / Ω e f f = 39 . (a) τ 1 ( ) = π E T ( ) E g g ( ) , (b) τ 1 ( ) = π E r r ( ) E g g ( ) . (c) The time evolution of populations Pm ( m = g g , T , r r ) of each levels without two-step modulation, Δ 1 / Ω e f f = 23 , V / Ω e f f = 39 .
Fig. 6
Fig. 6 (a) The structure of three-level system coupled by a laser field, where the detuning Δ1 exactly matches with the transition frequency ω23. (b) The structure of Rubidium atom driven by single laser field with large detunings. (c-d) The structure of Ne* atom coupled by laser fields ε 1 and ε 2 , where the degeneracy of sublevels are removed by magnetic field B .
Fig. 7
Fig. 7 The time evolution of populations Pm ( m = 1 , 2 , 3 ) of each levels (a) without, (b) with, periodically modulating coupling strength { Ω 1 , Ω 1   } , where Ω 1   = Ω 1 , Δ 1 / Ω 1 = 48 , τ 1 = π E 2 E 1 , u 1   = π E 3   E 1   .
Fig. 8
Fig. 8 The time evolution of populations Pm ( m = 1 , 2 , 3 ) of each levels by periodically modulating coupling strength { Ω 1 , Ω 1 } with different time interval τ 1 ( ) , where Ω 1 = Ω 1 , Δ 1 / Ω 1 = 30 , τ 2 / Ω 1 = 53 , Δ 3 / Ω 1 = 100 . (a) τ 1 ( ) = π E 2 ( ) E 1 ( ) , (b) τ 1 ( ) = π E 3 ( ) E 1 ( ) , (c) τ 1 ( ) = π E 4 ( ) E 1 ( ) .
Fig. 9
Fig. 9 The time evolution of populations Pm ( m = 1 , 2 , 3 , 2 , 3 ) of each levels by periodically modulating coupling strength { Ω 1 , Ω 1 } in five-level system, where Ω 1 = Ω 1 , Ω 2 / Ω 1 = 2 , Δ 1 / Ω 1 = 33 , l t a 2 / Ω 1 = 9 , Δ 3 / Ω 1 = 36 , Δ 4 / Ω 1 = 6 . (c) τ 1 ( ) = π E 3 ( ) E 1 ( ) , (d) τ 1 ( ) = π E 3 ( ) E 1 ( ) .
Fig. 10
Fig. 10 The time evolution of populations Pk ( k = 1 , 2 , 3 , 3 ) of each levels by periodically modulating coupling strength { Ω 1 , Ω 1 } , where Ω 2 / Ω 1 = 2 , Δ 1 / Ω 1 = 60 , Δ 2 / Ω 1 = 30 , Δ 3 / Ω 1 = 28 , Ω 1 = Ω 1 , Ω 2 = Ω 2 , Δ k   = Δ k , ( k = 1 , 2 , 3 ) . (a) τ 1 ( ) = ( 2 n + 1 ) π E 3 ( ) E 1 ( ) . (b) τ 1 ( ) = ( 2 n + 1 ) π E 3 ( ) E 1 ( ) .
Fig. 11
Fig. 11 The population P3 of Rydberg state versus the perturbations δ Ω 1 and δ Ω 2 in two-step modulation of coupling strength { Ω 1 , Ω 1 } , where Ω 1 / Ω 1 = 1 , Δ 1 ( ) / Ω 1 = 60 , Δ 2 ( ) / Ω 1 = 30 , Ω 2 ( ) / Ω 1 = 2 , τ 1 ( ) = π E 3 ( ) E 1 ( ) .
Fig. 12
Fig. 12 (a) The population P3 of Rydberg state (the left-blue vertical axis) and the evolution time t s achieving the maximum population P3 (the right-orange vertical axis) as a function of γ in two-step modulation of coupling strength { Ω 1 , Ω 1 } , where Ω 1 / Ω 1 = 1 , Δ 1 ( ) / Ω 1 = 60 , Δ 2 ( ) / Ω 1 = 30 , Ω 2 ( ) / Ω 1 = 2 , τ 1 ( ) = π E 3 ( ) E 1 ( ) . (b-d) The shapes of coupling strength Ω 1 ( t ) with different γ. (b) γ = 50. (c) γ = 100. (d) γ = 1000.

Equations (22)

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H 0 = k = 1 3 ω k | k k | + k = 1 2 Ω k e i ω k l t | k k + 1 | + H . c . ,
H 0 = k = 1 2 Δ k | k + 1 k + 1 | + Ω k | k k + 1 | + H . c . ,
H 0 = ( 0 Ω 1 0 Ω 1 Δ 1 Ω 2 0 Ω 2 Δ 2 ) .
0 = S + H 0 S = ( 0 Ω 1 sin  α Ω 1 cos  α Ω 1 sin  α ξ 1 0 Ω 1 cos  α 0 ξ 2 ) ,
E 1 ξ 1 x 1 2 ξ 2 x 2 2 ,    | E 1 | 1 x 1 | ξ 1 x 2 | ξ 2 , E 2 ξ 1 + ξ 1 x 1 2 ,       | E 2 | ξ 1 + x 1 | 1 , E 3 ξ 2 + ξ 2 x 2 2 ,       | E 3 | ξ 2 + x 2 | 1 .
U ( t ) = e i 0 t = ( 1 x 1 ( e i Θ 1 1 ) x 2 ( e i Θ 2 1 ) x 1 ( e i Θ 1 1 ) e i Θ 1 x 1 x 2 x 2 ( e i Θ 2 1 ) x 1 x 2 e i Θ 2 ) ,
  U ( t ) = S U ( t ) S +   ( 1 ( e i Θ 2 1 ) x 1 sin α + ( e i Θ 1 1 ) x 2 cos α ( e i Θ 2 1 ) x 1 cos α + ( 1 e i Θ 1 ) x 2 sin α ( e i Θ 2 1 ) x 1 sin α + ( e i Θ 1 1 ) x 2 cos α e i Θ 2 sin 2 α + e i Θ 1 cos 2 α ( e i Θ 2 e i Θ 1 ) cos α sin α ( e i Θ 2 1 ) x 1 cos α + ( 1 e i Θ 1 ) x 2 sin α ( e i Θ 2 e i Θ 1 ) cos α sin α e i Θ 2 cos 2 α + e i Θ 1 sin 2 α ) .
H ( t ) = { H 0 = k = 1 2 Δ k | k + 1 k + 1 | + Ω k | k k + 1 | + H . c . ,   t [ m T , m T + τ 1 ) , H 0   ' = k = 1 2 Δ k   ' | k + 1 k + 1 | + Ω k   ' | k k + 1 | + H . c . ,   t [ m T + τ 1 , ( m + 1 ) T ) ,
U ( T ) = U ( τ 1   ' ) U ( τ 1 ) = e i H 0   ' τ 1   ' e i H 0 τ 1 ( 1 z 1 z 2 z 4 1 z 3 z 5 z 6 e i ( ϕ + ϕ ) ) ,
z 1 = y 1 y 1   + y 2   y 3 ,   z 2 = y 2 + e i ϕ y 2   + y 1   y 3 ,   z 3 = y 1   y 2 y 3 + e i ϕ y 3   , z 4 = y 1   y 1 + y 2 y 3   ,   z 5 = y 2   + e i ϕ y 2 + y 1 y 3   ,   z 6 = y 1 y 2   y 3   + e i ϕ y 3 , y 1 ( ) = ( e i ϕ ( ) 1 ) x 1 ( ) sin  α ( ) 2 x 2 ( ) cos  α ( ) ,   y 2 ( ) = ( e i ϕ ( ) 1 ) x 1 ( ) cos  α ( ) 2 x 2 ( ) sin  α ( ) , y 3 ( ) = ( e i ϕ ( ) + 1 ) cos  α ( ) sin  α ( ) .
U ( T ) ( cos  ( ϕ 1 ) sin  ( ϕ 1 ) 0 sin  ( ϕ 1 ) cos  ( ϕ 1 ) 0 0 0 e i φ 1 ) ,
H e f f = Ω e f f | 1 2 | + Ω e f f * | 2 1 | ,
U ( T ) = U ( τ 1   ' ) U ( τ 1 ) = e i H 0   ' τ 1   ' e i H 0 τ 1 ( 1 z 1 z 2 z 1 e i 2 ϕ cos 2 α + sin 2 α z 3 z 2 z 3 e i 2 ϕ sin 2 α + cos 2 α ) ,
H e f f   ' = Ω e f f   ' | 1 3 | + Ω e f f '* | 3 1 | ,
H 0 = Ω e f f e i Δ 1 t ( | g 11 r | I 2 + I 1 | g 22 r | + H . c . ) + V | r r r r | ,
H 0 = Δ 1 | T T | + ( V 2 Δ 1 ) | r r r r | + 2 Ω e f f ( | T g g | + | T r r | + H . c . ) ,
H 0 = Δ 1 | 2 2 | + Ω 1 | 1 2 | + Ω 1 | 1 3 | + H . c .
H 0 = k = 2 4 Δ k 1 | k k | + Ω 1 | 1 k | + Ω 1 | k 1 | .
H 0 = Δ 1 | 2 2 | + Δ 3 | 2 2 | + ( Δ 1 + Δ 2 ) | 3 3 | + ( Δ 3 + Δ 4 ) | 3 3 | + Ω 1 | 1 2 | + Ω 1 | 1 2 | + Ω 2 | 2 3 | + Ω 2 | 2 3 | + H . c .
H 0 = Δ 1 | 2 2 | + Δ 2 | 3 3 | + Δ 3 | 3 3 | + Ω 1 | 1 2 | + Ω 2 | 2 3 | + Ω 2 | 2 3 | + H . c .
H ( t ) = { Δ 1 | 2 2 | + Δ 2 | 3 3 | + ( Ω 1 + δ Ω 1 ) | 1 2 | + ( Ω 2 + δ Ω 2 ) | 2 3 | + H . c . , t [ m T , m T + τ ) , Δ 1 | 2 2 | + Δ 2 | 3 3 | + ( Ω 1   ' + δ Ω 1 ) | 1 2 | + ( Ω 2 + δ Ω 2 ) | 2 3 | + H . c . , t [ m T + τ , ( m + 1 ) T ) .
Ω 1 ( t ) = { Ω 1 + Ω 1 Ω 1 1 + e γ mod ( t / T ) , mod ( t / T ) < τ 2 , Ω 1 + Ω 1 Ω 1 1 + e γ [ mod ( t / T ) τ ] , τ 2 mod ( t / T ) T τ 2 , Ω 1 + Ω 1 Ω 1 1 + e γ [ mod ( t / T ) T ] , mod ( t / T ) > T τ 2 .

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