Abstract

We propose a method to improve the stimulated Raman adiabatic passage (STIRAP) via dissipative quantum dynamics, taking into account the dephasing effects. Fast and robust population transfer can be obtained with the scheme by the designed pulses and detuning, even though the initial state of the system is imperfect. With a concrete three-level system as an example, the influences of the imperfect initial state, variations in the control parameters, and various dissipation effects are discussed in detail. The numerical simulation shows that the scheme is insensitive to moderate fluctuations of experimental parameters and the relatively large dissipation effects of the excited state. Furthermore, the dominant dissipative factors, namely, the dephasing effects of the ground states and the imperfect initial state are no longer undesirable, in fact, they are the important resources to the scheme. Therefore, the scheme could provide more choices for the realization of the complete population transfer in the strong dissipative fields where the standard stimulated Raman adiabatic passage or shortcut schemes are invalid.

© 2016 Optical Society of America

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References

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2016 (3)

K. S. Kumar, A. Vepsäläinen, S. Danilin, and G. S. Paraoanu, “Stimulated Raman adiabatic passage in a three-level superconducting circuit,” Nat. Commun. 7, 10628 (2016).
[Crossref] [PubMed]

Z. Chen, Y. H. Chen, Y. Xia, J. Song, and B. H. Huang, “Fast generation of three-atom singlet state by transitionless quantum driving,” Sci. Rep. 6, 22202 (2016).
[Crossref] [PubMed]

Y. H. Chen, Y. Xia, Q. C. Wu, B. H. Huang, and J. Song, “Method for constructing shortcuts to adiabaticity by a substitute of counterdiabatic driving terms,” Phys. Rev. A 93(5), 052109 (2016).
[Crossref]

2015 (4)

Y. H. Chen, Y. Xia, J. Song, and B. H. Huang, “Shortcuts to adiabatic passage for fast generation of Greenberger-Horne-Zeilinger states by transitionless quantum driving,” Sci. Rep. 5, 15616 (2015).
[Crossref] [PubMed]

Y. H. Chen, Y. Xia, Q. Q. Cheng, and J. Song, “Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states,” Phys. Rev. A 91(1), 012325 (2015).
[Crossref]

K. Kotru, D. L. Butts, J. M. Kinast, and R. E. Stoner, “Large-Area Atom Interferometry with Frequency-Swept Raman Adiabatic Passage,” Phys. Rev. Lett. 115(10), 103001 (2015).
[Crossref] [PubMed]

J. Randall, S. Weidt, E. D. Standing, K. Lake, S. C. Webster, D. F. Murgia, T. Navickas, K. Roth, and W. K. Hensinger, “Efficient preparation and detection of microwave dressed-state qubits and qutrits with trapped ions,” Phys. Rev. A 91(1), 012322 (2015).
[Crossref]

2014 (9)

B. T. Torosov, G. D. Valle, and S. Longhi, “Non-Hermitian shortcut to stimulated Raman adiabatic passage,” Phys. Rev. A 89(6), 063412 (2014).
[Crossref]

S. Ibáñez and J. G. Muga, “Adiabaticity condition for non-Hermitian Hamiltonians,” Phys. Rev. A 89(3), 033403 (2014).
[Crossref]

L. Giannelli and E. Arimondo, “Three-level superadiabatic quantum driving,” Phys. Rev. A 89(3), 033419 (2014).
[Crossref]

Y. Sun and H. Metcalf, “Nonadiabaticity in stimulated Raman adiabatic passage,” Phys. Rev. A 90(3), 033408 (2014).
[Crossref]

Y. H. Issoufa and A. Messikh, “Effect of dephasing on superadiabatic three-level quantum driving,” Phys. Rev. A 90(5), 055402 (2014).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Cheng, and J. Song, “Shortcuts to adiabatic passage for multiparticles in distant cavities: applications to fast and noise-resistant quantum population transfer, entangled states’ preparation and transition,” Laser. Phys. Lett. 11(11), 115201 (2014).
[Crossref]

Y. H. Chen, Y. Xia, Q. Q. Cheng, and J. Song, “Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems,” Phys. Rev. A 89(3), 033856 (2014).
[Crossref]

M. Lu, Y. Xia, L. T. Shen, J. Song, and N. B. An, “Shortcuts to adiabatic passage for population transfer and maximum entanglement creation between two atoms in a cavity,” Phys. Rev. A 89(1), 012326 (2014).
[Crossref]

S. Martínez-Garaot, E. Torrontegui, X. Chen, and J. G. Muga, “Shortcuts to adiabaticity in three-level systems using Lie transforms,” Phys. Rev. A 89(5), 053408 (2014).
[Crossref]

2013 (5)

S. Ibáñez, X. Chen, and J. G. Muga, “Improving shortcuts to adiabaticity by iterative interaction pictures,” Phys. Rev. A 87(4), 043402 (2013).
[Crossref]

X. J. Lu, X. Chen, A. Ruschhaupt, D. Alonso, S. Guerin, and J. G. Muga, “Fast and robust population transfer in two-level quantum systems with dephasing noise and/or systematic frequency errors,” Phys. Rev. A 88(3), 033406 (2013).
[Crossref]

S. Barz, J. F. Fitzsimons, E. Kashefi, and P. Walther, “Experimental verification of quantum computation,” Nat. Phys. 9(11), 727–731 (2013).
[Crossref]

J. Zhang, J. H. Shim, I. Niemeyer, T. Taniguchi, T. Teraji, H. Abe, S. Onoda, T. Yamamoto, T. Ohshima, J. Isoya, and et al., “Experimental implementation of assisted quantum adiabatic passage in a single spin,” Phys. Rev. Lett. 110(24), 240501 (2013).
[Crossref] [PubMed]

Q. C. Wu and X. Ji, “Generation of steady three- and four-dimensional entangled states via quantum-jump-based feedback,” Quantum Inf. Process. 12(10), 3167–3178 (2013).
[Crossref]

2012 (4)

X. Q. Shao, T. Y. Zheng, and S. Zhang, “Engineering steady three-atom singlet states via quantum-jump-based feedback,” Phys. Rev. A 85(4), 042308 (2012).
[Crossref]

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,” Nature Phys. 8(2), 147–152 (2012).
[Crossref]

A. Leclerc, D. Viennot, and G. Jolicard, “The role of the geometric phases in adiabatic population tracking for non-Hermitian Hamiltonians,” J. Phys. A: Math. Theor. 45(41), 415201 (2012).
[Crossref]

A. del Campo, M. M. Rams, and W. H. Zurek, “Assisted finite-rate adiabatic passage across a quantum critical point: Exact solution for the quantum Ising model,” Phys. Rev. Lett. 109(11), 115703 (2012);;H. Saberi, T. Opatrný, K. Mølmer, and A. del Campo, “Adiabatic tracking of quantum many-body dynamics,” Phys. Rev. A 90(6), 060301(R) (2014).
[Crossref] [PubMed]

2011 (6)

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83(6), 062116 (2011);;X. Chen and J. G. Muga, “Engineering of fast population transfer in three-level systems,” Phys. Rev. A 86(3), 033405 (2012).
[Crossref]

B. T. Torosov, S. Guérin, and N. V. Vitanov, “High-fidelity adiabatic passage by composite sequences of chirped pulses,” Phys. Rev. Lett. 106(23), 233001 (2011).
[Crossref] [PubMed]

A. del Campo, “Frictionless quantum quenches in ultracold gases: A quantum-dynamical microscope,” Phys. Rev. A 84(3), 031606 (2011);
[Crossref]

M. J. Kastoryano, F. Reiter, and A. S. Sørensen, “Dissipative preparation of entanglement in optical cavities,” Phys. Rev. Lett. 106(9), 090502 (2011).
[Crossref] [PubMed]

M. V. Berry and R. Uzdin, “Slow non-Hermitian cycling: exact solutions and the Stokes phenomenon,” J. Phys. A: Math. Theor. 44(43), 435303 (2011).
[Crossref]

L. T. Shen, X. Y. Chen, Z. B. Yang, H. Z. Wu, and S. B. Zheng, “Steady-state entanglement for distant atoms by dissipation in coupled cavities,” Phys. Rev. A 84(6), 064302 (2011).
[Crossref]

2010 (2)

J. Jing and T. Yu, “Non-Markovian relaxation of a three-level system: Quantum trajectory approach,” Phys. Rev. Lett. 105(24), 240403 (2010);;J. Jing, L. A. Wu, M. S. Sarandy, and J. G. Muga, “Inverse engineering control in open quantum systems,” Phys. Rev. A 88(5), 053422 (2013).
[Crossref]

X. Chen, I. Lizuain, A. Ruschhaupt, D. Guery-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two-and three-level atoms,” Phys. Rev. Lett. 105(12), 123003 (2010).
[Crossref] [PubMed]

2009 (4)

F. Verstraete, M. M. Wolf, and J. I. Cirac, “Quantum computation and quantum-state engineering driven by dissipation,” Nature Phys. 5(9), 633–636 (2009).
[Crossref]

G. Vacanti and A. Beige, “Cooling atoms into entangled states,” New J. Phys. 11(8), 083008 (2009).
[Crossref]

G. Dridi, S. Guérin, V. Hakobyan, H. R. Jauslin, and H. Eleuch, “Ultrafast stimulated Raman parallel adiabatic passage by shaped pulses,” Phys. Rev. A 80(4), 043408 (2009).
[Crossref]

M. V. Berry, “Transitionless quantum driving,” J. Phys. A 42(36), 365303 (2009).
[Crossref]

2008 (3)

X. Lacour, S. Guérin, and H. R. Jauslin, “Optimized adiabatic passage with dephasing,” Phys. Rev. A 78(3), 033417 (2008).
[Crossref]

J. G. Muga, J. Echanobe, A. del Campo, and I. Lizuain, “Generalized relation between pulsed and continuous measurements in the quantum Zeno effect,” J. Phys. B 41, 175501 (2008).
[Crossref]

D. Møller, L. B. Madsen, and K. Mølmer, “Geometric phases in open tripod systems,” Phys. Rev. A 77(2), 022306 (2008).
[Crossref]

2006 (1)

S. Maniscalco and F. Petruccione, “Non-Markovian dynamics of a qubit,” Phys. Rev. A 73(1), 012111 (2006).
[Crossref]

2004 (2)

M. V. Berry, “Physics of nonhermitian degeneracies,” Czech. J. Phys. 54(10), 1039–1047 (2004).
[Crossref]

P. A. Ivanov, N. V. Vitanov, and K. Bergmann, “Effect of dephasing on stimulated Raman adiabatic passage,” Phys. Rev. A 70(6), 063409 (2004).
[Crossref]

2003 (3)

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107(46), 9937–9945 (2003).
[Crossref]

M. Demirplak and S. A. Rice, “Adiabatic population transfer with control fields,” J. Phys. Chem. A 107(46), 9937–9945 (2003);.
[Crossref]

J. Okolowicz, M. Ploszajczak, and I. Rotter, “Dynamics of quantum systems embedded in a continuum,” Phys. Rep. 374(4), 271–383 (2003).
[Crossref]

2002 (2)

M. Kasevich, “Coherence with atoms,” Science 298(5597), 1363–1368 (2002).
[Crossref] [PubMed]

M. Demirplak and S. A. Rice, “Optical control of molecular dynamics in a liquid,” J. Chem. Phys. 116(18), 8028–8035 (2002).
[Crossref]

2001 (1)

N. V. Vitanov, M. Fleischhauer, B. W. Shore, and K. Bergmann, “Coherent manipulation of atoms and molecules by sequential laser pulses,” Adv. At. Mol. Opt. Phys. 46, 55–190 (2001).
[Crossref]

1999 (1)

T. Yu, L. Diósi, N. Gisin, and W. T. Strunz, “Non-Markovian quantum-state diffusion: Perturbation approach,” Phys. Rev. A 60(1), 91 (1999);;T. Yu, “Non-Markovian quantum trajectories versus master equations: finite-temperature heat bath,” Phys. Rev. A 69(6), 062107 (2004).
[Crossref]

1998 (4)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70(1), 101 (1998).
[Crossref]

N. Moiseyev, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Phys. Rep. 302(5), 212–293 (1998).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243 (1998);;C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, “Faster than Hermitian quantum mechanics,” Phys. Rev. Lett. 98(4), 040403 (2007).
[Crossref] [PubMed]

K. Bergmann, H. Theuer, and B. W. Shore, “Coherent population transfer among quantum states of atoms and molecules,” Rev. Mod. Phys. 70(3), 1003 (1998).
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X. Chen, I. Lizuain, A. Ruschhaupt, D. Guery-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two-and three-level atoms,” Phys. Rev. Lett. 105(12), 123003 (2010).
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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,” Nature Phys. 8(2), 147–152 (2012).
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K. Kotru, D. L. Butts, J. M. Kinast, and R. E. Stoner, “Large-Area Atom Interferometry with Frequency-Swept Raman Adiabatic Passage,” Phys. Rev. Lett. 115(10), 103001 (2015).
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Liang, Z. T.

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 superadiabatic passage with cold atoms,” arXiv:1601.06058.

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X. Chen, I. Lizuain, A. Ruschhaupt, D. Guery-Odelin, and J. G. Muga, “Shortcut to adiabatic passage in two-and three-level atoms,” Phys. Rev. Lett. 105(12), 123003 (2010).
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Figures (10)

Fig. 1
Fig. 1 Three-level Λ system driven by two coherent fields, a pump (Stokes) field with Rabi frequency Ω p s ). The two fields have the same detuning Δ, where the time dependence has been omitted for convenience.
Fig. 2
Fig. 2 (a) Effective level scheme of the STIRAP process driven by three effective coherent fields Ω1(t), Ω2(t), and Ω3(t). (b) The energy difference between the adiabatic states |a+(t)〉 and |a0(t)〉 (|a0(t)〉 and |a(t)〉) is given by λ+ (−λ).
Fig. 3
Fig. 3 (a) The resultant energy level diagram of the three-level Λ system when the dissipation effects of the ground states [see Eq. (16)] are considered. (b) The energy difference between the adiabatic states |a+(t)〉 and |a0(t)〉 (|a0(t)〉 and |a(t)〉) in this case is given by Ω++ − Ω0000 − Ω−−).
Fig. 4
Fig. 4 Evolution in time of the pump and Stokes pulses and the detuning with different dephasing rates within different STIRAP time intervals. For (a), (b) and (c) the dephasing rate γ = 1: (a) tf = 1.5T, (b) tf = 2T, (c) tf = 2.5T. For (d), (e) and (f) the dephasing rate γ = 2: (d) tf = 1.5T, (e) tf = 2T, (f) tf = 2.5T. The shapes of the pump and Stokes pulses are based on Eq. (26) with Ω = 1, while the detuning Δ(t) are ploted according to Eq. (22) by numerical calculations with the boundary condition Δ(ti) = Δ(tf) = 0.
Fig. 5
Fig. 5 The time evolution of population P1 (P2, P3) for the bare state |1〉 (|2〉, |3〉) with the pump and Stokes pulses with different imperfect initial states for the traditional STIRAP process without detuning. The pump and Stokes pulses are based on Eqs. (2627), while the initial state is based on Eq. (23). For (a) and (b), the initial states are perfectly populated in |1〉 and the dephasing effects are ignored, the delays τ0 between pulses are 0.5 and 1, respectively. For (c) and (d), the dephasing rates γ are 1 and the delays τ0 between pulses are 0.5, the initial state in (c) is perfectly populated in |1〉, whereas there exists a slight deviation ε = 0.05 from |1〉 in (d).
Fig. 6
Fig. 6 The time evolution of population P1 (P2, P3) for the bare state |1〉 (|2〉, |3〉) with the pump and Stokes pulses and the detuning which have been shown in Fig. 4 with different imperfect initial states.
Fig. 7
Fig. 7 The relative population P 3 r as a function of t and with three sets of parameters. The parameters in (a) and (d), (b) and (e), (c) and (f) are identical, corresponding to (γ = 1, tf = 1.5T), (γ = 1, tf = 2T), (γ = 2, tf = 1.5T), respectively. The relative population P 3 r in (a), (b) and (c) with 3D surface, while in (d), (e) and (f) with two-dimensional (2D) plot.
Fig. 8
Fig. 8 The relative population P 3 r and the population P3 vs two relative deviations in the pluses δΩ/Ω and δτ. The other parameters are Ω = 1, τ = 0.5, γ = 1, tf = 1.5T, ε = −0.038, and the detuning with the form of Δ a as shown in Table I.
Fig. 9
Fig. 9 The relative population P 3 r and the population P3 vs two relative deviations in the detuning δΔ00 and δω. The other parameters are Ω = 1, τ = 0.5, γ = 1, tf = 1.5T, ε = −0.038, and the detuning with the form of Δ a as shown in Table 1.
Fig. 10
Fig. 10 The relative population P 3 r and the population P3 vs the decay rates Γ1 and Γ2. The other parameters are the same as shown in the caption of Fig. 6 (a).

Tables (2)

Tables Icon

Table 1 Formal solution for the detuning Δ n (t) (n = a, b, c, d, e, f), where Δ n (t) is the detuning in the Fig. 4(n). The units for Δ0,1 and ω are 1/T, while that for μ0,1 and ν0,1 are T.

Tables Icon

Table 2 Samples of relative population P 3 r and the population P3 with corresponding δΩ/Ω, δτ, δΔ00, and δω.

Equations (29)

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i c ˙ ( t ) = H 0 ( t ) c ( t ) ,
H 0 ( t ) = 1 2 ( 0 Ω p ( t ) 0 Ω p ( t ) 2 Δ ( t ) Ω s ( t ) 0 Ω s ( t ) 0 ) .
λ + ( t ) = Ω 0 ( t ) cot ϕ ( t ) 2 , λ 0 ( t ) = 0 , λ ( t ) = Ω 0 tan ϕ ( t ) 2 ,
Ω 0 ( t ) = Ω p ( t ) 2 + Ω s ( t ) 2 , tan ϕ ( t ) = Ω 0 ( t ) Δ ( t ) + Δ ( t ) 2 + Ω 0 ( t ) 2 ,
ϕ ˙ ( t ) = Ω ˙ 0 ( t ) Δ ( t ) Ω 0 ( t ) Δ ˙ ( t ) 2 ( Δ ( t ) 2 + Ω 0 ( t ) 2 ) .
| a + ( t ) = sin θ ( t ) sin ϕ ( t ) | 1 + cos ϕ | 2 + cos θ ( t ) sin ϕ ( t ) | 3 , | a 0 ( t ) = cos θ ( t ) | 1 sin θ ( t ) | 3 , | a ( t ) = sin θ ( t ) cos ϕ ( t ) | 1 sin ϕ ( t ) | 2 + cos θ ( t ) cos ϕ ( t ) | 3 ,
tan θ ( t ) = Ω p ( t ) Ω s ( t ) , θ ˙ ( t ) = Ω ˙ p ( t ) Ω s ( t ) Ω p ( t ) Ω ˙ s ( t ) Ω 0 ( t ) 2 .
c ( t ) = R ( t ) a ( t ) ,
R ( t ) = ( sin θ ( t ) sin ϕ ( t ) cos θ ( t ) sin θ ( t ) cos ϕ ( t ) cos ϕ ( t ) 0 sin ϕ ( t ) cos θ ( t ) sin ϕ ( t ) sin θ ( t ) cos θ ( t ) cos ϕ ( t ) ) .
i a ˙ ( t ) = H 0 a a ( t ) ,
H 0 a = ( λ + ( t ) Ω 1 * ( t ) Ω 3 * ( t ) Ω 1 ( t ) 0 Ω 2 * ( t ) Ω 3 ( t ) Ω 2 ( t ) λ ( t ) ) ,
Ω 1 ( t ) = i θ ˙ ( t ) sin ϕ ( t ) , Ω 2 ( t ) = i θ ˙ ( t ) cos ϕ ( t ) , Ω 3 ( t ) = i ϕ ˙ ( t ) .
| Ω 1 ( t ) | | λ + ( t ) | , | Ω 2 ( t ) | | λ ( t ) | , | Ω 3 ( t ) | | λ + ( t ) λ ( t ) | ,
tan θ ( t ) t t i 0 , tan θ ( t ) t t f ,
| a 0 ( t i ) = | 1 | a 0 ( t f ) = | 3 .
H I = i Γ ( t ) 2 [ | 1 1 | | 3 3 | ] ,
H Γ ( t ) = H 0 + H I = 2 ( i Γ ( t ) Ω p ( t ) 0 Ω p ( t ) 2 Δ ( t ) Ω s ( t ) 0 Ω s ( t ) i Γ ( t ) ) .
H Γ ( t ) a = ( Ω + + ( t ) Ω + 0 ( t ) Ω + ( t ) Ω 0 + ( t ) Ω 00 ( t ) Ω 0 ( t ) Ω + ( t ) Ω 0 ( t ) Ω ( t ) ) ,
Ω + + ( t ) = λ + ( t ) + i Γ cos 2 θ ( t ) sin 2 ϕ ( t ) 2 , Ω + 0 ( t ) = i θ ˙ ( t ) sin ϕ ( t ) i Γ sin 2 θ ( t ) sin ϕ ( t ) 2 , Ω + ( t ) = i ϕ ˙ ( t ) + i Γ cos 2 θ ( t ) sin 2 ϕ ( t ) 4 , Ω 0 + ( t ) = i θ ˙ ( t ) sin ϕ ( t ) i Γ sin 2 θ ( t ) sin ϕ ( t ) 2 , Ω 00 ( t ) = i Γ cos 2 θ ( t ) 2 , Ω 0 ( t ) = i θ ˙ ( t ) cos ϕ ( t ) i Γ sin 2 θ ( t ) cos ϕ ( t ) 2 , Ω ( t ) = λ ( t ) + i Γ cos 2 θ ( t ) cos 2 φ ( t ) 2 , Ω 0 ( t ) = i θ ˙ ( t ) cos ϕ ( t ) i Γ sin 2 θ ( t ) cos ϕ ( t ) 2 , Ω + ( t ) = i ϕ ˙ ( t ) + i Γ cos 2 θ ( t ) sin 2 ϕ ( t ) 4 .
Ω + 0 ( t ) , Ω 0 ( t ) , Ω + ( t ) , Ω + ( t ) ,
2 θ ˙ ( t ) sin 2 θ ( t ) = Γ ( t ) = Ω ˙ p ( t ) Ω p ( t ) Ω ˙ s ( t ) Ω s ( t ) = d d t ln Ω p ( t ) Ω s ( t ) ,
Δ ˙ ( t ) = Ω ˙ 0 ( t ) Ω 0 ( t ) Δ ( t ) + Γ ( t ) cos 2 θ ( t ) .
| ψ ( 0 ) = 1 2 2 | 1 + ( | 2 + | 3 ) ,
Ω p ( t ) Ω s ( t ) = C exp ( Γ t ) ,
Ω p ( t ) = Ω peak f ( t τ 0 / 2 T ) , Ω s ( t ) = α Ω peak f ( t + τ 0 / 2 T ) ,
Ω p ( t ) = Ω peak exp ( [ ( t τ 0 / 2 ) / T ] 2 ) , Ω s ( t ) = Ω peak exp ( [ ( t + τ 0 / 2 ) / T ] 2 ) ,
Ω peak = Ω T , Γ = γ T , τ 0 = γ T 2 = τ T ,
ρ ˙ = i [ H 0 ( t ) , ρ ] j = 1 , 2 , 3 [ L j ρ L j 1 2 ( L j L j ρ + ρ L j L j ) ] ,
L 1 = Γ 1 ( | 1 2 | + | 3 2 | ) , L 2 = Γ 2 | 2 2 | , L 3 = Γ ( | 1 1 | | 3 3 | ) ,

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