Abstract

For a two-level system, it is believed that a far-off-resonant driving cannot help coherent population transfer between two states. In this work, we propose a scheme to implement the coherent transfer with far-off-resonant driving. The scheme works well with both constant driving and Gaussian driving. The total time to finish population transfer is also minimized by optimizing the detuning and coupling constants. We find that the scheme is sensitive to spontaneous emission much more than dephasing. It might find potential applications in X-ray quantum optics and population transfer in Rydberg atoms as well.

© 2016 Optical Society of America

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References

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2015 (5)

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]

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]

M. J. Hwang, R. Puebla, and M. B. Plenio, “Quantum phase transition and universal dynamics in the Rabi model,” Phys. Rev. Lett. 115, 180404 (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]

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]

2014 (1)

2013 (3)

B. W. Adams, C. Butha, S. M. Cavalettob, J. Eversb, Z. Harmanbc, C. H. Keitelb, A. Pálffyb, A. Picóna, R. Röhlsbergerd, Y. Rostovtseve, and K. Tamasakuf, “X-ray quantum optics,” J. Mod. Opt. 60, 2–21 (2013).
[Crossref]

G. Cao, H. O. Li, T. Tu, L. Wang, C. Zhou, M. Xiao, G. C. Guo, H. W. Jiang, and G. P. Guo, “Ultrafast universal quantum control of a quantum-dot charge qubit using Landau-Zener-Stückelberg interference,” Nat. Commun. 4, 1401 (2013).
[Crossref]

E. Dupont-Ferrier, B. Roche, B. Voisin, X. Jehl, R. Wacquez, M. Vinet, M. Sanquer, and S. De Franceschi, “Coherent coupling of two dopants in a Silicon nanowire probed by Landau-Zener-Stückelberg interferometry,” Phys. Rev. Lett. 110, 136802 (2013).
[Crossref]

2012 (4)

A. Ferrón, Daniel Domínguez, and M. Sánchez, “Tailoring population inversion in Landau-Zener-Stückelberg interferometry of flux qubits,” Phys. Rev. Lett. 109, 237005 (2012).
[Crossref]

Q. H. Chen, C. Wang, S. He, T. Liu, and K. L. Wang, “Exact solvability of the quantum Rabi model using Bogoliubov operators,” Phys. Rev. A 86, 023822 (2012).
[Crossref]

A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum Rabi model,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref] [PubMed]

V. V. Albert, “Quantum Rabi model for N-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
[Crossref] [PubMed]

2011 (4)

E. Solano, “Viewpoint: The dialogue between quantum light and matter,” Physics 4, 68 (2011).
[Crossref]

D. Braak, “Integrability of the Rabi model,” Phys. Rev. Lett. 107, 100401 (2011).
[Crossref] [PubMed]

W. T. Liao, A. Pálffy, and C. H. Keitel, “Nuclear coherent population transfer with X-ray laser pulses,” Phys. Lett. B 705, 134–138 (2011).
[Crossref]

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).

2010 (2)

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

M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82, 2313 (2010).
[Crossref]

2009 (2)

M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, and M. L. Roukes, “Nanomechanical measurements of a superconducting qubit,” Nature 459, 960–964 (2009).
[Crossref] [PubMed]

J. Johansson, M. H. S. Amin, A. J. Berkley, P. Bunyk, V. Choi, R. Harris, M. W. Johnson, T. M. Lanting, Seth Lloyd, and G. Rose, “Landau-Zener transitions in a superconducting flux qubit,” Phys. Rev. B 80, 012507 (2009).
[Crossref]

2008 (2)

A. Izmalkov, S. H. W. van der Ploeg, S. N. Shevchenko, M. Grajcar, E. Il’ichev, U. Hübner, A. N. Omelyanchouk, and H. G. Meyer, “Consistency of ground state and spectroscopic measurements on flux qubits,” Phys. Rev. Lett. 101, 017003 (2008).
[Crossref] [PubMed]

A. Pálffy, J. Evers, and C. H. Keitel, “Electric-dipole-forbidden nuclear transitions driven by super-intense laser fields,” Phys. Rev. C 77, 044602 (2008).
[Crossref]

2007 (1)

A. Pálffy, J. Evers, and C. H. Keitel, “Isomer triggering via nuclear excitation by electron capture,” Phys. Rev. Lett. 99, 172502 (2007).
[Crossref] [PubMed]

2006 (2)

T. J. Bürvenich, J. Evers, and C. H. Keitel, “Nuclear quantum optics with X-ray laser pulses,” Phys. Rev. Lett. 96, 142501 (2006).
[Crossref] [PubMed]

M. Sillanpää, T. Lehtinen, A. Paila, Y. Makhlin, and P. Hakonen, “Continuous-time monitoring of Landau-Zener interference in a Cooper-pair box,” Phys. Rev. Lett. 96, 187002 (2006).
[Crossref] [PubMed]

2005 (2)

W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, “Mach-Zehnder interferometry in a strongly driven superconducting qubit,” Science 310, 1653–1657 (2005).
[Crossref] [PubMed]

A. Aprahamian and Y. Sun, “Nuclear physics: Long live isomer research,” Nature Physics 1, 81–82 (2005).
[Crossref]

2004 (1)

A. Izmalkov, M. Grajcar, E. Il’ichev, N. Oukhanski, T. Wagner, H. G. Meyer, W. Krech, M. H. S. Amin, A. Maassen van den Brink, and A. M. Zagoskin, “Observation of macroscopic Landau-Zener transitions in a superconducting device,” Europhys. Lett. 65, 844 (2004).
[Crossref]

2003 (1)

K. W. D. Ledingham, P. McKenna, and R. P. Singhal, “Applications for nuclear phenomena generated by ultra-intense lasers,” Science 300, 1107–1111 (2003).
[Crossref] [PubMed]

1999 (1)

P. Walker and G. Dracoulis, “Energy traps in atomic nuclei,” Nature 399, 35–40 (1999).
[Crossref]

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]

1963 (1)

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[Crossref]

1937 (1)

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652 (1937).
[Crossref]

1936 (1)

I. I. Rabi, “On the process of space quantization,” Phys. Rev. 49, 324 (1936).
[Crossref]

Adams, B. W.

B. W. Adams, C. Butha, S. M. Cavalettob, J. Eversb, Z. Harmanbc, C. H. Keitelb, A. Pálffyb, A. Picóna, R. Röhlsbergerd, Y. Rostovtseve, and K. Tamasakuf, “X-ray quantum optics,” J. Mod. Opt. 60, 2–21 (2013).
[Crossref]

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]

Albert, V. V.

V. V. Albert, “Quantum Rabi model for N-state atoms,” Phys. Rev. Lett. 108, 180401 (2012).
[Crossref] [PubMed]

Altarelli, M.

M. Altarelli, R. Brinkmann, M. Chergui, W. Decking, B. Dobson, S. Düsterer, G. Grübel, W. Graeff, H. Graafsma, J. Hajdu, J. Marangos, J. Pflüger, H. Redlin, D. Riley, I. Robinson, J. Rossbach, A. Schwarz, K. Tiedtke, T. Tschentscher, I. Vartaniants, H. Wabnitz, H. Weise, R. Wichmann, K. Witte, A. Wolf, M. Wulff, and M. Yurkov, “The European X-Ray Free-Electron Laser Technical design report,” (DESY, 2007).

Amin, M. H. S.

J. Johansson, M. H. S. Amin, A. J. Berkley, P. Bunyk, V. Choi, R. Harris, M. W. Johnson, T. M. Lanting, Seth Lloyd, and G. Rose, “Landau-Zener transitions in a superconducting flux qubit,” Phys. Rev. B 80, 012507 (2009).
[Crossref]

A. Izmalkov, M. Grajcar, E. Il’ichev, N. Oukhanski, T. Wagner, H. G. Meyer, W. Krech, M. H. S. Amin, A. Maassen van den Brink, and A. M. Zagoskin, “Observation of macroscopic Landau-Zener transitions in a superconducting device,” Europhys. Lett. 65, 844 (2004).
[Crossref]

Aprahamian, A.

A. Aprahamian and Y. Sun, “Nuclear physics: Long live isomer research,” Nature Physics 1, 81–82 (2005).
[Crossref]

Ashhab, S.

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

Baranov, D. G.

Barredo, D.

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]

Ber, R.

R. Ber and M. Schwartz, “Unusual transitions made possible by superoscillations,” arXiv:1502.01406 (2015).

Berggren, K. K.

W. D. Oliver, Y. Yu, J. C. Lee, K. K. Berggren, L. S. Levitov, and T. P. Orlando, “Mach-Zehnder interferometry in a strongly driven superconducting qubit,” Science 310, 1653–1657 (2005).
[Crossref] [PubMed]

Bergmann, K.

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]

Berkley, A. J.

J. Johansson, M. H. S. Amin, A. J. Berkley, P. Bunyk, V. Choi, R. Harris, M. W. Johnson, T. M. Lanting, Seth Lloyd, and G. Rose, “Landau-Zener transitions in a superconducting flux qubit,” Phys. Rev. B 80, 012507 (2009).
[Crossref]

Braak, D.

D. Braak, “Integrability of the Rabi model,” Phys. Rev. Lett. 107, 100401 (2011).
[Crossref] [PubMed]

Breuer, H. P.

H. P. Breuer and F. Petruccione, The Theory of Open Quantum System (Oxford University, 2002).

Brinkmann, R.

M. Altarelli, R. Brinkmann, M. Chergui, W. Decking, B. Dobson, S. Düsterer, G. Grübel, W. Graeff, H. Graafsma, J. Hajdu, J. Marangos, J. Pflüger, H. Redlin, D. Riley, I. Robinson, J. Rossbach, A. Schwarz, K. Tiedtke, T. Tschentscher, I. Vartaniants, H. Wabnitz, H. Weise, R. Wichmann, K. Witte, A. Wolf, M. Wulff, and M. Yurkov, “The European X-Ray Free-Electron Laser Technical design report,” (DESY, 2007).

Browaeys, A.

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]

Bunyk, P.

J. Johansson, M. H. S. Amin, A. J. Berkley, P. Bunyk, V. Choi, R. Harris, M. W. Johnson, T. M. Lanting, Seth Lloyd, and G. Rose, “Landau-Zener transitions in a superconducting flux qubit,” Phys. Rev. B 80, 012507 (2009).
[Crossref]

Bürvenich, T. J.

T. J. Bürvenich, J. Evers, and C. H. Keitel, “Nuclear quantum optics with X-ray laser pulses,” Phys. Rev. Lett. 96, 142501 (2006).
[Crossref] [PubMed]

Butha, C.

B. W. Adams, C. Butha, S. M. Cavalettob, J. Eversb, Z. Harmanbc, C. H. Keitelb, A. Pálffyb, A. Picóna, R. Röhlsbergerd, Y. Rostovtseve, and K. Tamasakuf, “X-ray quantum optics,” J. Mod. Opt. 60, 2–21 (2013).
[Crossref]

Cao, G.

G. Cao, H. O. Li, T. Tu, L. Wang, C. Zhou, M. Xiao, G. C. Guo, H. W. Jiang, and G. P. Guo, “Ultrafast universal quantum control of a quantum-dot charge qubit using Landau-Zener-Stückelberg interference,” Nat. Commun. 4, 1401 (2013).
[Crossref]

Cavalettob, S. M.

B. W. Adams, C. Butha, S. M. Cavalettob, J. Eversb, Z. Harmanbc, C. H. Keitelb, A. Pálffyb, A. Picóna, R. Röhlsbergerd, Y. Rostovtseve, and K. Tamasakuf, “X-ray quantum optics,” J. Mod. Opt. 60, 2–21 (2013).
[Crossref]

Chen, Q. H.

Q. H. Chen, C. Wang, S. He, T. Liu, and K. L. Wang, “Exact solvability of the quantum Rabi model using Bogoliubov operators,” Phys. Rev. A 86, 023822 (2012).
[Crossref]

Chergui, M.

M. Altarelli, R. Brinkmann, M. Chergui, W. Decking, B. Dobson, S. Düsterer, G. Grübel, W. Graeff, H. Graafsma, J. Hajdu, J. Marangos, J. Pflüger, H. Redlin, D. Riley, I. Robinson, J. Rossbach, A. Schwarz, K. Tiedtke, T. Tschentscher, I. Vartaniants, H. Wabnitz, H. Weise, R. Wichmann, K. Witte, A. Wolf, M. Wulff, and M. Yurkov, “The European X-Ray Free-Electron Laser Technical design report,” (DESY, 2007).

Choi, V.

J. Johansson, M. H. S. Amin, A. J. Berkley, P. Bunyk, V. Choi, R. Harris, M. W. Johnson, T. M. Lanting, Seth Lloyd, and G. Rose, “Landau-Zener transitions in a superconducting flux qubit,” Phys. Rev. B 80, 012507 (2009).
[Crossref]

Chuang, I.

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010).
[Crossref]

Crespi, A.

A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum Rabi model,” Phys. Rev. Lett. 108, 163601 (2012).
[Crossref] [PubMed]

Cummings, F. W.

E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE 51, 89–109 (1963).
[Crossref]

De Franceschi, S.

E. Dupont-Ferrier, B. Roche, B. Voisin, X. Jehl, R. Wacquez, M. Vinet, M. Sanquer, and S. De Franceschi, “Coherent coupling of two dopants in a Silicon nanowire probed by Landau-Zener-Stückelberg interferometry,” Phys. Rev. Lett. 110, 136802 (2013).
[Crossref]

Decking, W.

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Europhys. Lett. (1)

A. Izmalkov, M. Grajcar, E. Il’ichev, N. Oukhanski, T. Wagner, H. G. Meyer, W. Krech, M. H. S. Amin, A. Maassen van den Brink, and A. M. Zagoskin, “Observation of macroscopic Landau-Zener transitions in a superconducting device,” Europhys. Lett. 65, 844 (2004).
[Crossref]

J. Mod. Opt. (1)

B. W. Adams, C. Butha, S. M. Cavalettob, J. Eversb, Z. Harmanbc, C. H. Keitelb, A. Pálffyb, A. Picóna, R. Röhlsbergerd, Y. Rostovtseve, and K. Tamasakuf, “X-ray quantum optics,” J. Mod. Opt. 60, 2–21 (2013).
[Crossref]

Nat. Commun. (1)

G. Cao, H. O. Li, T. Tu, L. Wang, C. Zhou, M. Xiao, G. C. Guo, H. W. Jiang, and G. P. Guo, “Ultrafast universal quantum control of a quantum-dot charge qubit using Landau-Zener-Stückelberg interference,” Nat. Commun. 4, 1401 (2013).
[Crossref]

Nature (2)

M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, and M. L. Roukes, “Nanomechanical measurements of a superconducting qubit,” Nature 459, 960–964 (2009).
[Crossref] [PubMed]

P. Walker and G. Dracoulis, “Energy traps in atomic nuclei,” Nature 399, 35–40 (1999).
[Crossref]

Nature Physics (1)

A. Aprahamian and Y. Sun, “Nuclear physics: Long live isomer research,” Nature Physics 1, 81–82 (2005).
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Figures (14)

Fig. 1
Fig. 1 The population as a function of time where P0 (P1) is the population of state |0〉 (|1〉). (a) The intensity modulation, Δ/Ω1 = 30, Ω21 = 2. (b) The frequency modulation, Δ1/Ω = 20, Δ2/Ω = 30. (c) The constant Rabi frequency and detuning, Δ1/Ω = 20. The dash lines represent the function |sinΛt|2.
Fig. 2
Fig. 2 The minimal time ��f as a function of (a) the coupling constant Ω1 in the intensity modulation, Δ/Ω2 = 30; (b) the detuning Δ1 in the frequency modulation, Δ2/Ω = 30. The solid and dash lines are the exact and approximate results, respectively. The horizontal line (dot-dash line) is for �� f = π 2 4 Ω. (c) The population with different detunings Δk in the intensity modulation, where the square-well pulse is described by Ω12 = 3 and Δ12 = 100 for the transition |0〉 ↔ |1〉.
Fig. 3
Fig. 3 The population P1 as a function of time with different γ in the intensity modulation. (a) γ = 10000. (b) γ = 300. (c) γ = 100. (d) γ = 50. The bottom panels (e)–(h) are the square-well field corresponding to the top panels (a)–(d), respectively. The parameters of perfect square-well field are Ω21 = 3 and Δ/Ω1 = 50.
Fig. 4
Fig. 4 The population P1 as a function of time with noisy square-well field. (a) The noise ε(t) is generated randomly in the interval [−0.05, 0.05]. (b) The noise ε(t) is generated randomly in the interval [−0.3, 0.3]. The bottom panels (c) and (d) are the noisy square-well fields corresponding to the top panels (a) and (b), respectively. For perfect square-well fields, we employ Ω21 = 3, and Δ/Ω1 = 50 in the intensity modulation.
Fig. 5
Fig. 5 The population versus the dissipation rate γ01 and the dephasing rate γ11. The initial state is |0〉, Δ1/Ω = 30, Δ2/Ω = 300.
Fig. 6
Fig. 6 (a) The population as a function of time and detuning, Ω12 = 10. (b) The population as a function of time when Δ/Ω2 = 30 (the dash line), where the parameters approximately satisfy Eq. (10).
Fig. 7
Fig. 7 The population versus {��, ξ} of the Gaussian pulse, Δ/Ω = 2.
Fig. 8
Fig. 8 (a) The population of |e〉 and |g〉 as a function of time with resonant atom-field couplings (i.e., Δ = ω0ωb = 0). (b) The probability that there are n photons in the field at different time. The field is initially in a coherent state given by Eq. (14) with the average photon number 〈n〉 = 20. The photon number is truncated at 51 in numerical calculations.
Fig. 9
Fig. 9 (a) The population of |e〉 and |g〉 as a function of time in the intensity modulation, Δ/Ω1 = 300, Ω21 = 3. (b) The probability that there are n photons in the field at different time. The field is initially in a coherent state described by Eq. (14) with 〈n〉 = 20. (c) The occupation of No. m basis as a function of time, where the basis are ordered as {|g, 0〉, |e, 0〉, |g, 1〉, |e, 1〉, ..., |g, 50〉, |e, 50〉, |g, 51〉}. (d) The probability of n photons in the field as a function of time.
Fig. 10
Fig. 10 (a) The population at |e〉 and |g〉 as a function of time in the intensity modulation. (b) The probability that there are n photons in the field at different time. The field is in random state initially.
Fig. 11
Fig. 11 The level configuration of a three-level system.
Fig. 12
Fig. 12 The population as a function of time where the initial state is |2〉. (a) Δ/Ω1 = 50, Ω21 = 2. (b) Δ/Ω1 = 100, Ω21 = 2. (c) Δa1 = 50, Δb1 = 100, Ω21 = 2.
Fig. 13
Fig. 13 The population as a function of time in frequency modulation. The initial state is |0〉. (a) Δα1 = 40, Δβ1 = 20, δ1 = 10, Ω2 = Ω1. (b) δα = 0, δβ1 = 30, Δ/Ω1 = 10, Ω21 = Ω1.
Fig. 14
Fig. 14 The population as a function of time in intensity modulation. The initial state is |2〉, Δ/Ω1 = 40, δ = 0, Ω2 = Ω1, Ω′21 = −1.

Equations (22)

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H = d σ + ε 𝟙 ,
U ( t , 0 ) = e i H t = ( P ( t ) i Q ( t ) R ( t ) e i ( θ π 2 ) R ( t ) e i ( θ π 2 ) P ( t ) + i Q ( t ) ) ,
U ( T , 0 ) = e i H 2 t 2 e i H 1 t 1 = ( cos ϕ sin ϕ e i θ sin ϕ e i θ cos ϕ ) ,
H eff = 1 T ( 0 ϕ e i ( θ + π 2 ) ϕ e i ( θ + π 2 ) 0 ) .
U ( t , 0 ) = { e i H 1 t U ( n T , 0 ) , t [ 0 , t 1 ] e i H 2 ( t t 1 ) e i H 1 t 1 U ( n T , 0 ) , t [ t 1 , T ] ,
H 0 = ω 0 | 1 1 | + Ω e i ω l t | 0 1 | + H . c ,
𝒯 = ( 4 m + 1 ) π 2 ( | d 1 | + | d 2 | ) 4 ϕ | d 1 | | d 2 | , m = 0 , 1 , 2 ,
Ω ( t ) = { Ω 2 + Ω 1 Ω 2 1 + e γ ( t t 1 2 ) , t < 0.5 t 1 , Ω 2 + Ω 1 Ω 2 1 + e γ ( t T + t 1 2 ) , t 0.5 t 1 ,
ρ ˙ ( t ) = i [ H ( t ) , ρ ( t ) ] + 01 ( ρ ) + 11 ( ρ ) ,
d 1 x sin ( m ϕ ) + d 1 z cos ( m ϕ ) = | d 1 | , m = 1 , 2 , 3 , ,
U ( n T , 0 ) = ( cos ( n ϑ ) i Q sin ( n ϑ ) Q 2 + R 2 R Q 2 + R 2 sin ( n ϑ ) e i θ R Q 2 + R 2 sin ( n ϑ ) e i θ cos ( n ϑ ) + i Q sin ( n ϑ ) Q 2 + R 2 ) .
H = ω b a a + ω 0 2 σ z + Ω σ x ( a + a ) ,
H = ( ω 0 2 0 0 Ω 0 0 0 0 ω 0 2 Ω 0 0 0 0 0 Ω ω 0 2 + ω b 0 0 Ω 0 Ω 0 0 ω 0 2 + ω b Ω 0 0 0 0 0 Ω ω 0 2 + 2 ω b 0 0 0 0 Ω 0 0 ω 0 2 + 2 ω b Ω 0 0 0 0 0 Ω ω 0 2 + 3 ω b ) .
ρ ˙ ( t ) = i [ H , ρ ( t ) ] ,
ρ n n b ( 0 ) = n n e n n ! .
H = ( 0 0 Ω 1 0 0 Ω 2 Ω 1 Ω 2 Δ ) .
U ( t , 0 ) = e i H t = e i Δ t 2 ( B 1 B 2 B 4 B 2 B 3 B 5 B 4 B 5 B 6 ) ,
U ( T , 0 ) = e i H b t 2 e i H a t 1 = e i 2 ( Δ a t 1 + Δ b t 2 ) y a y b ( B 1 B 2 B 4 B 2 B 3 B 5 B 4 B 5 B 6 ) ,
U ( t , 0 ) = U ( n T , 0 ) = e i n 2 ( Δ a t 1 + Δ b t 2 ) ( y a y b ) n ( Ω 1 2 2 ( Ω 1 2 + Ω 2 2 ) ( s 1 n + s 2 n ) Ω 1 Ω 2 2 ( Ω 1 2 + Ω 2 2 ) ( s 1 n + s 2 n ) i Ω 1 2 Ω 1 2 + Ω 2 2 ( s 1 n s 2 n ) Ω 1 Ω 2 2 ( Ω 1 2 + Ω 2 2 ) ( s 1 n + s 2 n ) Ω 2 2 2 ( Ω 1 2 + Ω 2 2 ) ( s 1 n + s 2 n ) i Ω 2 2 Ω 1 2 + Ω 2 2 ( s 1 n s 2 n ) i Ω 1 2 Ω 1 2 + Ω 2 2 ( s 1 n s 2 n ) i Ω 2 2 Ω 1 2 + Ω 2 2 ( s 1 n s 2 n ) 1 2 ( s 1 n + s 2 n ) ) ,
| Ψ ( t ) = U ( n T , 0 ) | Ψ , 0 = 𝒩 ( Ω 1 sin ( n φ ) Ω 2 sin ( n φ ) i Ω 1 2 + Ω 2 2 cos ( n φ ) ) ,
𝒯 = π 2 tan 1 | 2 ( Δ a Δ b ) Ω 1 2 + Ω 2 2 4 Ω 1 2 + 4 Ω 2 2 + Δ a Δ b | ( π 4 Ω 1 2 + 4 Ω 2 2 + Δ a 2 + π 4 Ω 1 2 + 4 Ω 2 2 + Δ b 2 ) .
| Ψ T = 𝒩 ( Ω 1 | 0 + Ω 2 | 1 + + Ω N | N 1 ) ,

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