Abstract

The technique of shortcuts to adiabaticity (STA) has attracted broad attention due to their possible applications in quantum information processing and quantum control. However, most studies published so far have been only focused on Hermitian systems under the rotating-wave approximation (RWA). In this paper, we propose a modified shortcuts to adiabaticity technique to realize population transfer for a non-Hermitian system without RWA. We work out an exact expression for the control function and present examples consisting of two-and three-level systems with decay to show the theory. The results suggest that the shortcuts to adiabaticity technique presented here is robust for fast passages. We also find that the decay has small effect on the population transfer in the three-level system. To shed more light on the physics behind this result, we reduce the quantum three-level system to an effective two-level one with large detunings. The shortcuts to adiabaticity technique of effective two-level system is studied. Thereby the high-fidelity population transfer can be implemented in non-Hermitian systems by our method, and it works even without RWA.

© 2017 Optical Society of America

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References

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2017 (1)

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]

2016 (9)

X. B. Huang, Y. H. Chen, and Z. Wang, “Fast generation of three-qubit Greenberger-Horne-Zeilinger state based on the Lewis-Riesenfeld invariants in coupled cavities,” Sci. Rep. 6, 25707 (2016).
[Crossref] [PubMed]

M. Okuyama and K. Takahashi, “From Classical Nonlinear Integrable Systems to Quantum Shortcuts to Adiabaticity,” Phys. Rev. Lett. 117, 070401 (2016).
[Crossref] [PubMed]

S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016).
[Crossref]

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]

K. Paul and A. K. Sarma, “High-fidelity entangled Bell states via shortcuts to adiabaticity,” Phys. Rev. A 94, 052303 (2016).
[Crossref]

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, 052109 (2016).
[Crossref]

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

X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
[Crossref]

X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
[Crossref]

2015 (6)

X. Shi and L. F. Wei, “High-efficiency single-photon Fock state production by transitionless quantum driving, Laser,” Phys. Lett. 12, 015204 (2015).

S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
[Crossref] [PubMed]

S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015).
[Crossref]

J. W. Chen and L. F. Wei, “Implementation speed of deterministic population passages compared to that of Rabi pulses,” Phys. Rev. A 91, 023405 (2015).
[Crossref]

K. Paul and A. K. Sarma, “Shortcut to adiabatic passage in a waveguide coupler with a complex-hyperbolic-secant scheme,” Phys. Rev. A 91, 053406 (2015).
[Crossref]

M. B. Kenmoe, S. E. M. Tchouobiap, C. K. Sadem, A. B. Tchapda, and L. C. Fai, “Non-adiabatic and adiabatic transitions at level crossing with decay: two-and three-level systems,” J. Phys. A 48, 095303 (2015).
[Crossref]

2014 (8)

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

A. Kiely and A. Ruschhaupt, “Inhibiting unwanted transitions in population transfer in two-and three-level quantum systems,” J. Phys. B 47, 115501 (2014).
[Crossref]

S. Masuda and S. A. Rice, “Rapid Coherent Control of Population Transfer in Lattice Systems,” Phys. Rev. A 89, 033621 (2014).
[Crossref]

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

T. Opatrný and K. Mømer, “Partial suppression of nonadiabatic transitions,” New J. Phys. 16, 015025 (2014).
[Crossref]

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

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

Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

2013 (6)

K. Takahashi, “How fast and robust is the quantum adiabatic passage?” J. Phys. A: Math. Theor. 46, 315304 (2013).
[Crossref]

B. T. Torosov and N. V. Vitanov, “Composite Stimulated Raman Adiabatic Passage,” Phys. Rev. A 87, 043418 (2013).
[Crossref]

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

K. Takahashi, “Transitionless quantum driving for spin systems,” Phys. Rev. E 87, 062117 (2013).
[Crossref]

L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum Routing of Single Photons with a Cyclic Three-Level System,” Phys. Rev. Lett. 111, 103604 (2013).
[Crossref]

C. Hang, G. Huang, and V. V. Konotop, “PT-Symmetry with a System of Three-Level Atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

2012 (8)

S. A. Reyes, F. A. Olivares, and L. Morales-Molina, “Landau-Zener-Stückelberg interferometry in PT-symmetric optical waveguides,” J. Phys. A: Math. Theor. 45, 444027 (2012).
[Crossref]

J. Larson, “Absence of Vacuum Induced Berry Phases without the Rotating Wave Approximation in Cavity QED,” Phys. Rev. Lett. 108, 033601 (2012).
[Crossref] [PubMed]

Z. Sun, J. Ma, X. Wang, and F. Nori, “Photon-assisted Landau-Zener transition: Role of coherent superposition states,” Phys. Rev. A 86, 012107 (2012).
[Crossref]

R. Uzdin, U. Günther, S. Rahav, and N. Moiseyev, “Time-dependent Hamiltonians with 100 0.000000e+000volution speed efficiency,” J. Phys. 45, 415304 (2012).

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

N. Wiebe and N. S. Babcock, “Improved error-scaling for adiabatic quantum evolutions,” New J. Phys. 14, 013024 (2012).
[Crossref]

J. Sun, S. F. Lu, and F. Liu, “Speedup in adiabatic evolution based quantum algorithms,” Sci. China-Phys. Mech. Astron. 55, 1630 (2012).
[Crossref]

S. Ibáñez, 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]

2011 (4)

S. Masuda and K. Nakamura, “Acceleration of adiabatic quantum dynamics in electromagnetic fields,” Phys. Rev. A 84, 043434 (2011).
[Crossref]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature 474, 589 (2011).
[Crossref] [PubMed]

S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011).
[Crossref]

2010 (8)

I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010).
[Crossref]

A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
[Crossref] [PubMed]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
[Crossref]

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010).
[Crossref]

A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
[Crossref] [PubMed]

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]

2009 (5)

M. V. Berry, “Transitionless quantum driving,” J. Rhys. A: Math. Theor 42, 365303 (2009).

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009).
[Crossref]

G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A 80, 013417 (2009).
[Crossref]

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

T. Liu, K. L. Wang, and M. Feng, “The generalized analytical approximation to the solution of the single-mode spin-boson model without rotating-wave approximation,” Europhys. Lett. 86, 54003 (2009).
[Crossref]

2008 (1)

L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
[Crossref] [PubMed]

2007 (4)

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 957 (2007).
[Crossref]

O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449, 588(2007).
[Crossref] [PubMed]

E. K. Irish, “Generalized rotating-wave approximation for arbitrarily large coupling,” Phys. Rev. Lett. 99, 173601 (2007).
[Crossref] [PubMed]

E. K. Irish, “Generalized Rotating-Wave Approximation for Arbitrarily Large Coupling,” Phys. Rev. Lett. 99, 173601 (2007).
[Crossref] [PubMed]

2006 (1)

R. Schilling, M. Vogelsberger, and D. A. Garanin, “Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions,” J. Phys. A: Math. Gen. 39, 13727 (2006).
[Crossref]

2005 (2)

R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
[Crossref] [PubMed]

Y. X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical Selection Rules and Phase-Dependent Adiabatic State Control in a Superconducting Quantum Circuit,” Phys. Rev. Lett. 95, 087001 (2005).
[Crossref] [PubMed]

2003 (3)

A. B. Klimov, I. Sainz, and S. M. Chumakov, “Resonance expansion versus the rotating-wave approximation,” Phys. Rev. A 68, 063811 (2003).
[Crossref]

A. Mostafazadeh, “Exact PT-symmetry is equivalent to Hermiticity,” J. Phys. A 36, 7081 (2003).
[Crossref]

M. Demirplak and S. A. Rice, “Adiabatic Population Transfer with Control Fields,” J. Phys. Chem. A 107, 9937 (2003).
[Crossref]

2002 (1)

A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian,” J. Math. Phys. 43, 205 (2002).
[Crossref]

2001 (1)

N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001).
[Crossref] [PubMed]

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 (1999).
[Crossref]

1998 (2)

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]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
[Crossref]

1997 (1)

N. V. Vitanov and S. Stenholm, “Analytic properties and effective two-level problems in stimulated Raman adiabatic passage,” Phys. Rev. A 55, 648 (1997).
[Crossref]

1996 (1)

T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996).
[Crossref]

Abdumalikov, A. A.

A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
[Crossref] [PubMed]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Ai, Q.

X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
[Crossref]

X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
[Crossref]

Arimondo, E.

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

Ashhab, S.

S. Ashhab and F. Nori, “Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states,” Phys. Rev. A 81, 042311 (2010).
[Crossref]

L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
[Crossref] [PubMed]

Astafiev, O.

A. A. Abdumalikov, O. Astafiev, A. M. Zagoskin, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Electromagnetically Induced Transparency on a Single Artificial Atom,” Phys. Rev. Lett. 104, 193601(2010).
[Crossref] [PubMed]

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

O. Astafiev, K. Inomata, A. O. Niskanen, T. Yamamoto, Yu. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449, 588(2007).
[Crossref] [PubMed]

Babcock, N. S.

N. Wiebe and N. S. Babcock, “Improved error-scaling for adiabatic quantum evolutions,” New J. Phys. 14, 013024 (2012).
[Crossref]

Bai, C. H.

S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 957 (2007).
[Crossref]

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
[Crossref]

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]

R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
[Crossref] [PubMed]

N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001).
[Crossref] [PubMed]

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]

T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996).
[Crossref]

Berry, M. V.

M. V. Berry, “Transitionless quantum driving,” J. Rhys. A: Math. Theor 42, 365303 (2009).

Blais, A.

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real Spectra in Non-Hermitian Hamiltonians Having P-T Symmetry,” Phys. Rev. Lett. 80, 5243(1998).
[Crossref]

Boradjiev, I. I.

I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010).
[Crossref]

Bourassa, J.

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Campbell, S.

S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
[Crossref] [PubMed]

Casanova, J.

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010).
[Crossref]

Cen, L. X.

L. F. Wei, J. R. Johansson, L. X. Cen, S. Ashhab, and F. Nori, “Controllable coherent population transfers in superconducting qubits for quantum computing,” Phys. Rev. Lett. 100, 113601 (2008).
[Crossref] [PubMed]

Chan, S.

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009).
[Crossref]

Chen, J. W.

J. W. Chen and L. F. Wei, “Implementation speed of deterministic population passages compared to that of Rabi pulses,” Phys. Rev. A 91, 023405 (2015).
[Crossref]

Chen, X.

S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015).
[Crossref]

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

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

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

S. Ibáñez, 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]

X. Chen, E. Torrontegui, and J. G. Muga, “Lewis-Riesenfeld invariants and transitionless quantum driving,” Phys. Rev. A 83, 062116 (2011).
[Crossref]

S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011).
[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]

Chen, Y. H.

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, 052109 (2016).
[Crossref]

X. B. Huang, Y. H. Chen, and Z. Wang, “Fast generation of three-qubit Greenberger-Horne-Zeilinger state based on the Lewis-Riesenfeld invariants in coupled cavities,” Sci. Rep. 6, 25707 (2016).
[Crossref] [PubMed]

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

Chumakov, S. M.

A. B. Klimov, I. Sainz, and S. M. Chumakov, “Resonance expansion versus the rotating-wave approximation,” Phys. Rev. A 68, 063811 (2003).
[Crossref]

Danilin, S.

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]

De Chiara, G.

S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
[Crossref] [PubMed]

Demirplak, M.

M. Demirplak and S. A. Rice, “Adiabatic Population Transfer with Control Fields,” J. Phys. Chem. A 107, 9937 (2003).
[Crossref]

Deng, F. G.

X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, “Physically feasible three-level transitionless quantum driving with multiple Schrödinger dynamics,” Phys. Rev. A 93, 052324 (2016).
[Crossref]

X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
[Crossref]

Deppe, F.

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

Du, Y. X.

Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

Ekers, A.

R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
[Crossref] [PubMed]

Fai, L. C.

M. B. Kenmoe, S. E. M. Tchouobiap, C. K. Sadem, A. B. Tchapda, and L. C. Fai, “Non-adiabatic and adiabatic transitions at level crossing with decay: two-and three-level systems,” J. Phys. A 48, 095303 (2015).
[Crossref]

Fazio, R.

S. Campbell, G. De Chiara, M. Paternostro, G. M. Palma, and R. Fazio, “Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model,” Phys. Rev. Lett. 114, 177206 (2015).
[Crossref] [PubMed]

Fedorov, A.

A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
[Crossref] [PubMed]

Feng, M.

T. Liu, K. L. Wang, and M. Feng, “The generalized analytical approximation to the solution of the single-mode spin-boson model without rotating-wave approximation,” Europhys. Lett. 86, 54003 (2009).
[Crossref]

Feofanov, A. K.

A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
[Crossref] [PubMed]

Ficek, Z.

S. Chan, M. D. Reid, and Z. Ficek, “Entanglement evolution of two remote and non-identical Jaynes-Cummings atoms,” J. Phys. B 42, 065507(2009).
[Crossref]

Forn-Díaz, P.

A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
[Crossref] [PubMed]

Gambetta, J. M.

J. Bourassa, J. M. Gambetta, A. A. Abdumalikov, O. Astafiev, Y. Nakamura, and A. Blais, “Ultrastrong coupling regime of cavity QED with phase-biased flux qubits,” Phys. Rev. A 80, 032109 (2009).
[Crossref]

Garanin, D. A.

R. Schilling, M. Vogelsberger, and D. A. Garanin, “Nonadiabatic transitions for a decaying two-level system: geometrical and dynamical contributions,” J. Phys. A: Math. Gen. 39, 13727 (2006).
[Crossref]

Garcia Fernandez, R.

R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
[Crossref] [PubMed]

Garcia-Ripoll, J. J.

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

García-Ripoll, J. J.

J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep Strong Coupling Regime of the Jaynes-Cummings Model,” Phys. Rev. Lett. 105, 263603 (2010).
[Crossref]

Giannelli, L.

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

Gross, R.

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

Guéry-Odelin, D.

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]

Günther, U.

R. Uzdin, U. Günther, S. Rahav, and N. Moiseyev, “Time-dependent Hamiltonians with 100 0.000000e+000volution speed efficiency,” J. Phys. 45, 415304 (2012).

Halfmann, T.

N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, “Laser-induced population transfer by adiabatic passage techniques,” Annu. Rev. Phys. Chem. 52, 763 (2001).
[Crossref] [PubMed]

T. Halfmann and K. Bergmann, “Coherent Population Transfer and Dark Resonances in SO2,” J. Chem. Phys. 104, 7068 (1996).
[Crossref]

Hang, C.

C. Hang, G. Huang, and V. V. Konotop, “PT-Symmetry with a System of Three-Level Atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Harmans, C. J. P. M.

A. Fedorov, A. K. Feofanov, P. Macha, P. Forn-Díaz, C. J. P. M. Harmans, and J. E. Mooij, “Strong Coupling of a Quantum Oscillator to a Flux Qubit at Its Symmetry Point,” Phys. Rev. Lett. 105, 060503 (2010).
[Crossref] [PubMed]

He, S.

S. He, S. L. Su, D. Y. Wang, W. M. Sun, C. H. Bai, A. D. Zhu, H. F. Wang, and S. Zhang, “Efcient shortcuts to adiabatic passage for three-dimensional entanglement generation via transitionless quantum driving,” Sci. Rep. 6, 30929 (2016).
[Crossref]

Hocke, F.

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

Huang, B. H.

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, 052109 (2016).
[Crossref]

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

Huang, G.

C. Hang, G. Huang, and V. V. Konotop, “PT-Symmetry with a System of Three-Level Atoms,” Phys. Rev. Lett. 110, 083604 (2013).
[Crossref]

Huang, W.

Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
[Crossref]

Huang, X. B.

X. B. Huang, Y. H. Chen, and Z. Wang, “Fast generation of three-qubit Greenberger-Horne-Zeilinger state based on the Lewis-Riesenfeld invariants in coupled cavities,” Sci. Rep. 6, 25707 (2016).
[Crossref] [PubMed]

Huebl, H.

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

Hümmer, T.

T. Niemczyk, F. Deppe, H. Huebl, E. P. Menzel, F. Hocke, M. J. Schwarz, J. J. Garcia-Ripoll, D. Zueco, T. Hümmer, E. Solano, A. Marx, and R. Gross, “Circuit quantum electrodynamics in the ultrastrong-coupling regime,” Nature Phys. 6, 772 (2010).
[Crossref]

Ibánez, S.

S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011).
[Crossref]

Ibáñez, S.

S. Ibáñez, Y. C. Li, X. Chen, and J. G. Muga, “Pulse design without the rotating-wave approximation,” Phys. Rev. A 92, 062136 (2015).
[Crossref]

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

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

S. Ibáñez, 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]

Inomata, K.

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

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Nature (2)

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Nature Phys. (1)

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X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, “Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm,” New J. Phys. 18, 023001 (2016).
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Phys. Lett. (1)

X. Shi and L. F. Wei, “High-efficiency single-photon Fock state production by transitionless quantum driving, Laser,” Phys. Lett. 12, 015204 (2015).

Phys. Rev. A (26)

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Y. X. Du, Z. T. Liang, W. Huang, H. Yan, and S. L. Zhu, “Experimental observation of double coherent stimulated Raman adiabatic passages in three-level Λ systems in a cold atomic ensemble,” Phys. Rev. A 90, 023821 (2014).
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B. T. Torosov and N. V. Vitanov, “Composite Stimulated Raman Adiabatic Passage,” Phys. Rev. A 87, 043418 (2013).
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K. Paul and A. K. Sarma, “Shortcut to adiabatic passage in a waveguide coupler with a complex-hyperbolic-secant scheme,” Phys. Rev. A 91, 053406 (2015).
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I. I. Boradjiev and N. V. Vitanov, “Stimulated Raman adiabatic passage with unequal couplings: Beyond two-photon resonance,” Phys. Rev. A 81, 053415 (2010).
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S. Ibánez, S. Martinez-Garaot, X. Chen, E. Torrontegui, and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Phys. Rev. A 84, 023415 (2011).
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G. S. Vasilev, A. Kuhn, and N. V. Vitanov, “Optimum pulse shapes for stimulated Raman adiabatic passage,” Phys. Rev. A 80, 013417 (2009).
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Y. X. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori, “Optical Selection Rules and Phase-Dependent Adiabatic State Control in a Superconducting Quantum Circuit,” Phys. Rev. Lett. 95, 087001 (2005).
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L. Zhou, L. P. Yang, Y. Li, and C. P. Sun, “Quantum Routing of Single Photons with a Cyclic Three-Level System,” Phys. Rev. Lett. 111, 103604 (2013).
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R. Garcia Fernandez, A. Ekers, L. P. Yatsenko, N. V. Vitanov, and K. Bergmann, “Control of Population Flow in Coherently Driven Quantum Ladders,” Phys. Rev. Lett. 95, 043001 (2005).
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[Crossref] [PubMed]

Science (1)

O. Astafiev, A. M. Zagoskin, A. A. Abdumalikov, Yu. A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, and J. S. Tsai, “Resonance Fluorescence of a Single Artificial Atom,” Science 327, 840 (2010).
[Crossref] [PubMed]

Other (1)

N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University, Cambridge, 2011).

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Figures (5)

Fig. 1
Fig. 1 Real (red solid line) and imaginary (blue dashed line) parts of Ω a (a). The population evolution is driven by Ω a (b), and [ImΩ a ] (c). The parameters chosen are Γ = 2π × 0.5Mhz, Ω0 = 2π × 5Mhz, δ = 2π × 300Mhz, tf = 2 × 10−10s, t0 = 10−10s.
Fig. 2
Fig. 2 The evolution of the population for a ndecaying two-level system with decay rate Γ = 2π × 0.5MHz. The population transfer is implemented by the shortcuts to adiabaticity technique under RWA [Figs. 2(a), and 2(b)], and implemented by the shortcuts to adiabaticity technique beyond RWA [Fig. 2(c)]. Note that all the counter-rotating terms are neglected in Fig. 2(a), but are considered in Fig. 2(b). The population of level |1〉 (dashed red), and |2〉 (solid blue) are shown. The other parameters: Ω0 = 2π × 5MHz, δ = 2π × 300MHz, tf = 2 × 10−10s, t0 = 10−10s, ωL = 2π × 10GHz.
Fig. 3
Fig. 3 Transition probability P1→3 of the fast stimulated Raman by the shortcuts to adiabaticity under and beyond RWA. Fig. 3(a) is plotted by using the shortcuts to adiabaticity technique under RWA, but consider the CR terms. Fig. 3(b) is plotted by using the shortcuts to adiabaticity technique beyond RWA. We set the parameters: tf = 30ns, T = tf/6, τ = tf/10, Ω0 = 2π×200MHz, Δ = 2π×200MHz, Γ = 2π×100MHz, ωp = 100MHz, ωs = 80MHz. The dashed red, dotted green, and solid blue lines describe the time-dependent population of the state |1〉, |2〉, and |3〉, respectively.
Fig. 4
Fig. 4 Fidelity as a function of Γ, Ω0 and Δ. The fidelity is defined as the population on the state |3〉 with initial state |1〉. The population transfer is realized by using the STA technique under RWA but with counter-rotating terms [Figs. 4(a) and 4(c)], and without RWA [Figs. 4(b) and 4(d)]. In Figs. 4(a) and 4(b), the parameters chosen are, tf = 30ns, Ω0 = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ωp = 100MHz, ωs = 80MHz. In Figs. 4(c) and 4(d), the parameters are set as: tf = 30ns, Δ = 2π × [100, 800]MHz, Γ = 2π × [100, 600]MHz, ωp = 100MHz, ωs = 80MHz. We can clearly find that the fidelity without RWA is robust against the parameter fluctuations.
Fig. 5
Fig. 5 The population evolution of the effective two-level system by shortcuts to adiabaticity (a), and the three-level system by shortcuts to adiabaticity (b). Here the population of level |1〉 (dashed red), and |3〉 (solid blue) is presented. We set the parameters as: tf = 30ns, T = tf /6, τ = tf/10, Ω0 = 2π × 0.16GHz, Δ = 2π × 2.5GHz, Γ = 2π × 0.16GHz, ωp = 0.1GHz, ωs = 0.08GHz. We can find that the population evolution given by the effective two-level model and by the three-level model is almost the same.

Equations (33)

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H S ( t ) = 2 [ Ω R ( t ) ( | 2 1 | + | 1 2 | ) ( e i ω L t + e i ω L t ) + ω 0 ( t ) ( | 2 2 | | 1 1 | ) ] ,
H L ( t ) = i U ˙ ( t ) U ( t ) = ω L 2 ( | 2 2 | | 1 1 | ) ,
U ( t ) = e i 0 t H L ( t ) d t = e i ω L t / 2 | 2 2 | e i ω L t / 2 | 1 1 | ,
H ( t ) = U ( t ) [ H S ( t ) H L ( t ) ] U ( t ) = 2 [ Δ ( t ) Ω ( t ) Ω ( t ) Δ ( t ) ] .
Ω ( t ) = Ω R ( t ) ( 1 + e 2 i ω L t ) ,
Δ ( t ) = ω 0 ( t ) ω L ,
H ˜ 0 ( t ) = 2 [ Δ ( t ) Ω ( t ) Ω ( t ) Δ ( t ) i Γ ] .
i t [ C 1 ( t ) C 2 ( t ) ] = H ˜ 0 ( t ) [ C 1 ( t ) C 2 ( t ) ] ,
| φ + ( t ) = sin β | 1 + cos β e i ω L t | 2 , | φ ( t ) = cos β e i ω L t | 1 sin β | 2 ,
β = 1 2 arctan 2 Ω R cos ω L t Δ i Γ / 2 ,
| φ ^ + ( t ) = sin β * | 1 + cos β * e i ω L t | 2 , | φ ^ ( t ) = cos β * e i ω L t | 1 sin β * | 2 ,
H ˜ CD ( t ) = i [ | t φ + ( t ) φ ^ + ( t ) | φ ^ + ( t ) | t φ + ( t ) | φ + ( t ) φ ^ + ( t ) | + | t φ ( t ) φ ^ ( t ) | φ ^ ( t ) | t φ ( t ) | φ ( t ) φ ^ ( t ) | ] ,
φ ^ ± ( t ) | t φ ± ( t ) = ± i ω L cos 2 β 2 , φ ^ ( t ) | t φ ± ( t ) = ( ± β ˙ 2 i ω L 2 sin β ) e ± i ω L t ,
H ˜ CD ( t ) = [ ω L 2 sin 2 β ( i β ˙ 2 + ω L 4 sin 2 β ) e i ω L t ( i β ˙ 2 + ω L 4 sin 2 β ) e i ω L t ω L 2 sin 2 β ] .
H 0 ( t ) = 2 [ Δ ( t ) Ω R ( t ) Ω R ( t ) Δ ( t ) i Γ ] .
H CD ( t ) = [ 0 Ω a ( t ) Ω a ( t ) 0 ] ,
θ ˙ = Ω ˙ R [ Δ ( t ) i Γ / 2 ] Ω R ( t ) ( Δ ˙ i Γ ˙ / 2 ) Ω R 2 ( t ) + [ Δ ( t ) i Γ / 2 ] 2 ,
Ω R ( t ) = Ω 0 sech [ π ( t t f / 2 ) 2 t 0 ] , Δ ( t ) = 2 δ 2 t 0 π tanh [ π ( t t f / 2 ) 2 t 0 ] .
i d d t c ( t ) = H ¯ 0 ( t ) c ( t ) ,
H ¯ 0 ( t ) = 2 [ 0 Ω ¯ p ( t ) 0 Ω ¯ p * ( t ) 2 Δ p i Γ Ω ¯ s ( t ) 0 Ω ¯ s * ( t ) 2 ( Δ p Δ s ) ] ,
| E 0 ( t ) = 1 Ξ 0 ( t ) [ Ω ¯ s ( t ) | 1 Ω ¯ p * ( t ) | 3 ] , | E + ( t ) = 1 Ξ 1 ( t ) [ Ω ¯ p ( t ) | 1 + ε + ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] , | E ( t ) = 1 Ξ 2 ( t ) [ Ω ¯ p ( t ) | 1 + ε ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] ,
E 0 ( t ) = 0 , E ± ( t ) = 2 ε ± ( t ) ,
ε ± ( t ) = ( Δ i Γ / 2 ) ± ( Δ i Γ / 2 ) 2 + Ξ 0 2 ( t ) .
| E ^ 0 ( t ) = 1 Ξ 0 ( t ) [ Ω ¯ s ( t ) | 1 Ω ¯ p * ( t ) | 3 ] , | E ^ + ( t ) = 1 Ξ 1 ( t ) [ Ω ¯ p ( t ) | 1 + ε ^ + ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] , | E ^ ( t ) = 1 Ξ 2 ( t ) [ Ω ¯ p ( t ) | 1 + ε ^ ( t ) | 2 + Ω ¯ s * ( t ) | 3 ] ,
Π 0 ( t ) = | E 0 ( t ) E ^ 0 ( t ) | = 1 Ξ 0 2 ( t ) [ | Ω ¯ s ( t ) | 2 0 Ω ¯ p ( t ) Ω ¯ s ( t ) 0 0 0 Ω ¯ p * ( t ) Ω ¯ s * ( t ) 0 | Ω ¯ p ( t ) | 2 ] , Π 1 ( t ) = | E 1 ( t ) E ^ 1 ( t ) | = 1 Ξ 1 2 ( t ) [ | Ω ¯ p ( t ) | 2 Ω ¯ p ( t ) ε + ( t ) Ω ¯ p ( t ) Ω ¯ s ( t ) Ω ¯ p * ( t ) ε + ( t ) ε + 2 ( t ) Ω ¯ s ( t ) ε + ( t ) Ω ¯ p * ( t ) Ω ¯ s * ( t ) Ω ¯ s ( t ) ε + ( t ) | Ω ¯ s ( t ) | 2 ] , Π 2 ( t ) = | E 2 ( t ) E ^ 2 ( t ) | = 1 Ξ 2 2 ( t ) [ | Ω ¯ p ( t ) | 2 Ω ¯ p ( t ) ε ( t ) Ω ¯ p ( t ) Ω ¯ s ( t ) Ω ¯ p * ( t ) ε ( t ) ε 2 ( t ) Ω ¯ s ( t ) ε ( t ) Ω ¯ p * ( t ) Ω ¯ s * ( t ) Ω ¯ s ( t ) ε ( t ) | Ω ¯ s ( t ) | 2 ] .
H ¯ CD = i j k Π j ( t ) t H ¯ 0 ( t ) Π k ( t ) E k ( t ) E j ( t ) = i Ξ 1 2 ( t ) Ξ 2 2 ( t ) [ | Ω ¯ p ( t ) | 2 B + C D Ω ¯ p ( t ) A 2 i Γ Ω ¯ s ( t ) G Ω ¯ p ( t ) Ω ¯ s ( t ) B + C F Ω ¯ p * ( t ) A * 2 i Γ Ω ¯ s * ( t ) G * Ξ 0 2 ( t ) B Ω ¯ s ( t ) A * 2 i Γ Ω ¯ p ( t ) G * Ω ¯ p * ( t ) Ω ¯ s * ( t ) B * C F * Ω ¯ s * ( t ) A + 2 i Γ Ω ¯ p * ( t ) G | Ω ¯ s | 2 B C D ] ,
A = ( 2 Δ i Γ ) [ Ω ¯ p * ( t ) Ω ¯ ˙ p ( t ) + Ω ¯ s ( t ) Ω ¯ ˙ s * ( t ) ] , B = Ω ¯ p * ( t ) Ω ¯ ˙ p * ( t ) + Ω ¯ s ( t ) Ω ¯ ˙ s * ( t ) H . C . , C = [ | ε + ( t ) | 2 + | ε ( t ) | 2 + 2 Ξ 0 2 ( t ) ] / Ξ 0 2 ( t ) , D = | Ω ¯ p ( t ) | 2 [ Ω ¯ s * ( t ) Ω ¯ ˙ s * ( t ) H . C . ] + | Ω ¯ s ( t ) | 2 [ ( Ω ¯ p * ( t ) Ω ¯ ˙ p ( t ) H . C . ) ] , F = Ω ¯ p 2 ( t ) [ Ω ¯ s ( t ) Ω ¯ p * ( t ) Ω ¯ p * ( t ) Ω ¯ ˙ s * ( t ) ] + Ω ¯ s 2 ( t ) [ Ω ¯ s * ( t ) Ω ¯ ˙ p ( t ) Ω ¯ p ( t ) Ω ¯ ˙ s * ( t ) ] , G = Ω ¯ s * ( t ) Ω ¯ ˙ p ( t ) Ω ¯ p ( t ) Ω ¯ ˙ s * ( t ) .
H CD ( t ) = i [ 0 A ( t ) C ( t ) A ( t ) 0 B ( t ) C ( t ) B ( t ) 0 ] ,
A ( t ) = sin θ ( t ) ϕ ˙ ( t ) , B ( t ) = cos θ ( t ) ϕ ˙ ( t ) , C ( t ) = θ ˙ ( t ) ,
H eff ( t ) = 2 [ Δ eff ( t ) Ω ef f ( t ) Ω eff * ( t ) Δ eff ( t ) ] ,
Δ eff ( t ) = Ω ¯ p ( t ) Ω ¯ p ( t ) 2 Δ i Γ , Ω eff ( t ) = | Ω ¯ p ( t ) | 2 | Ω ¯ s ( t ) | 2 4 Δ 2 i Γ .
H CD ( t ) = i 2 M ( t ) [ 0 P ( t ) P * ( t ) Q ( t ) ] ,
P ( t ) = Ω ˙ eff ( t ) Δ eff ( t ) Δ ˙ eff ( t ) Ω eff ( t ) , Q ( t ) = Ω ˙ eff * ( t ) Ω eff ( t ) Ω ˙ eff ( t ) Ω eff * , M ( t ) = | Ω eff | 2 + Δ eff 2 .

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