Abstract

We study the non-Markovian dynamics of a qubit system coupled respectively to coherent state, squeezing vacuum state, and mixed state environments through dephasing interaction. Special attention is paid to the problem of environmental coherence and excitation on the effect of non-Markovianity of system dynamics. Some nontrivial and unexpected results are found. The number of environmental excitations serves to enhance the non-Markovianity of system dynamics, but the enhancement slows down with the increasing of the variance of excitation number. However, environmental coherence can play dual effects, which enhances in some cases and suppresses in other cases the non-Markovianity of system dynamics.

© 2015 Optical Society of America

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References

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2015 (4)

N. Tang, Z. L. Fan, and H. S. Zeng, “Improving the quality of noisy spatial quantum channels,” Quant. Inform. Comput. 15, 0568–0581 (2015).

S. C. Hou, S. L. Liang, and X. X. Yi, “Non-Markovianity and memory effects in quantum open systems,” Phys. Rev. A 91, 012109 (2015).
[Crossref]

D. Chruściński and F. A. Wudarski, “Non-Markovianity degree for random unitary evolution,” Phys. Rev. A 91, 012104 (2015).
[Crossref]

H. Mäkelä, “Bounds for the divisibility-based and distinguishability-based non-Markovianity measures,” Phys. Rev. A 91, 012108 (2015)
[Crossref]

2014 (5)

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quanrum non-Markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77, 094001 (2014).
[Crossref]

C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits,” Phys. Rev. A 89, 024101 (2014).
[Crossref]

D. Chruściński and S. Maniscalco, “Degree of non-Markovianity of quantum evolution,” Phys. Rev. Lett. 112, 120404 (2014).
[Crossref]

E. M. Laine, H. P. Breuer, and J. Piilo, “Nonlocal memory effects allow perfect teleportation with mixed states,” Scientfic Reports 4, 4620 (2014).

B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,” Scientfic Reports 4, 5720 (2014).

2013 (4)

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
[Crossref]

H. S. Zeng, Y. P. Zheng, N. Tang, and G. Y. Wang, “Correlation quantum beats induced by non-Markovian effect,” Quantum Inform. Process 12, 1637–1650 (2013).
[Crossref]

P. Haikka, T. H. Johnson, and S. Maniscalco, “Non-Markovianity of local dephasing channels and time-invariant discord,” Phys. Rev. A 87, 010103(R) (2013).
[Crossref]

F. Giraldi and F. Petruccione, “Survival of coherence for open quantum systems in the thermal baths,” Phys. Rev. A 88, 042102 (2013).
[Crossref]

2012 (6)

H. S. Zeng, N. Tang, Y. P. Zheng, and T. T. Xu, “Non-Markovian dynamics for an open two-level system without rotating wave approximation: indivisibility versus backflow of information,” Eur. Phys. J. D 66, 225 (2012).
[Crossref]

E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
[Crossref] [PubMed]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

S. Luo, S. Fu, and H. Song, “Quantifying non-Markovianity via correlations,” Phys. Rev. A 86, 044101 (2012).
[Crossref]

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
[Crossref]

2011 (6)

H. S. Zeng, N. Tang, Y. P. Zheng, and G. Y. Wang, “Equivalence of the measures of non-Markovianity for open two-level systems,” Phys. Rev. A 84, 032118 (2011).
[Crossref]

R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable quantum key distribution in non-Markovian channels,” Phys. Rev. A 83, 042321 (2011).
[Crossref]

R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Phys. Rev. A 83, 052128 (2011).
[Crossref]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
[Crossref]

2010 (4)

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

Y. H. Ji and J. J. Hu, “Entanglement and decoherence of coupled superconductor qubits in a non-Markovian environment,” Chin. Phys. B 19, 060304 (2010).
[Crossref]

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Phys. Rev. Lett. 105, 050403 (2010).
[Crossref] [PubMed]

X. M. Lu, X. G. Wang, and C. P. Sun, “Quantum Fisher information flow and non-Markovian processes of open systems,” Phys. Rev. A 82, 042103 (2010).
[Crossref]

2009 (3)

Y. Kubota and K. Nobusada, “Applicability of site-basis time-evolution equation for thermalization of exciton states in a quantum dot array,” J. Phys. Soc. Jpn 78, 114603 (2009).
[Crossref]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

M. A. Cirone, G. De Chiara, G. M. Palma, and A. Recati, “Collective decoherence of trapped atoms coupled to a Bose-Einstein condensate,” New J. Phys. 11, 103055 (2009).
[Crossref]

2008 (2)

B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302(R) (2008).
[Crossref]

M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Phys. Rev. Lett. 101, 150402 (2008).
[Crossref] [PubMed]

2004 (1)

J. Shao, “Decoupling quantum dissipation interaction via stochastic fields,” J. Chem. Phys. 120, 5053 (2004).
[Crossref] [PubMed]

1995 (1)

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[Crossref] [PubMed]

Addis, C.

C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits,” Phys. Rev. A 89, 024101 (2014).
[Crossref]

Ankerhold, J.

R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

Bellomo, B.

B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302(R) (2008).
[Crossref]

Brebner, G.

C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits,” Phys. Rev. A 89, 024101 (2014).
[Crossref]

Breuer, H. P.

E. M. Laine, H. P. Breuer, and J. Piilo, “Nonlocal memory effects allow perfect teleportation with mixed states,” Scientfic Reports 4, 4620 (2014).

E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
[Crossref] [PubMed]

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
[Crossref]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2007).
[Crossref]

Bylicka, B.

B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,” Scientfic Reports 4, 5720 (2014).

Calarco, T.

R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

Caycedo-Soler, F.

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
[Crossref]

Chen, G.

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

Chin, A. W.

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
[Crossref]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

Chruscinski, D.

D. Chruściński and F. A. Wudarski, “Non-Markovianity degree for random unitary evolution,” Phys. Rev. A 91, 012104 (2015).
[Crossref]

D. Chruściński and S. Maniscalco, “Degree of non-Markovianity of quantum evolution,” Phys. Rev. Lett. 112, 120404 (2014).
[Crossref]

B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,” Scientfic Reports 4, 5720 (2014).

D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Phys. Rev. A 83, 052128 (2011).
[Crossref]

Chumakov, S. M.

A. B. Klimov and S. M. Chumakov, A Group-Theoretical Approach to Quanyum Optics (WILEY-VCH Verlag GmbH & Co. KGaA, 2009).
[Crossref]

Cirac, J. I.

M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Phys. Rev. Lett. 101, 150402 (2008).
[Crossref] [PubMed]

Cirone, M. A.

M. A. Cirone, G. De Chiara, G. M. Palma, and A. Recati, “Collective decoherence of trapped atoms coupled to a Bose-Einstein condensate,” New J. Phys. 11, 103055 (2009).
[Crossref]

Compagno, G.

B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302(R) (2008).
[Crossref]

Cubitt, T. S.

M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Phys. Rev. Lett. 101, 150402 (2008).
[Crossref] [PubMed]

De Chiara, G.

P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
[Crossref]

M. A. Cirone, G. De Chiara, G. M. Palma, and A. Recati, “Collective decoherence of trapped atoms coupled to a Bose-Einstein condensate,” New J. Phys. 11, 103055 (2009).
[Crossref]

DiVincenzo, D. P.

D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51, 1015–1022 (1995).
[Crossref] [PubMed]

Eisert, J.

M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Phys. Rev. Lett. 101, 150402 (2008).
[Crossref] [PubMed]

Fan, Z. L.

N. Tang, Z. L. Fan, and H. S. Zeng, “Improving the quality of noisy spatial quantum channels,” Quant. Inform. Comput. 15, 0568–0581 (2015).

Fu, S.

S. Luo, S. Fu, and H. Song, “Quantifying non-Markovianity via correlations,” Phys. Rev. A 86, 044101 (2012).
[Crossref]

Giraldi, F.

F. Giraldi and F. Petruccione, “Survival of coherence for open quantum systems in the thermal baths,” Phys. Rev. A 88, 042102 (2013).
[Crossref]

Gong, M.

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

Gou, G. C.

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

Guo, G. C.

E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
[Crossref] [PubMed]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

Haikka, P.

C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits,” Phys. Rev. A 89, 024101 (2014).
[Crossref]

P. Haikka, T. H. Johnson, and S. Maniscalco, “Non-Markovianity of local dephasing channels and time-invariant discord,” Phys. Rev. A 87, 010103(R) (2013).
[Crossref]

P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
[Crossref]

Hou, S. C.

S. C. Hou, S. L. Liang, and X. X. Yi, “Non-Markovianity and memory effects in quantum open systems,” Phys. Rev. A 91, 012109 (2015).
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Y. H. Ji and J. J. Hu, “Entanglement and decoherence of coupled superconductor qubits in a non-Markovian environment,” Chin. Phys. B 19, 060304 (2010).
[Crossref]

Huang, Y. F.

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
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Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quanrum non-Markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77, 094001 (2014).
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A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
[Crossref]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Phys. Rev. Lett. 105, 050403 (2010).
[Crossref] [PubMed]

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Y. H. Ji and J. J. Hu, “Entanglement and decoherence of coupled superconductor qubits in a non-Markovian environment,” Chin. Phys. B 19, 060304 (2010).
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P. Haikka, T. H. Johnson, and S. Maniscalco, “Non-Markovianity of local dephasing channels and time-invariant discord,” Phys. Rev. A 87, 010103(R) (2013).
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S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
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D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Phys. Rev. A 83, 052128 (2011).
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Y. Kubota and K. Nobusada, “Applicability of site-basis time-evolution equation for thermalization of exciton states in a quantum dot array,” J. Phys. Soc. Jpn 78, 114603 (2009).
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Laine, E. M.

E. M. Laine, H. P. Breuer, and J. Piilo, “Nonlocal memory effects allow perfect teleportation with mixed states,” Scientfic Reports 4, 4620 (2014).

S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
[Crossref]

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
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E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
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B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
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H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
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E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
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J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
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B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
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J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
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S. C. Hou, S. L. Liang, and X. X. Yi, “Non-Markovianity and memory effects in quantum open systems,” Phys. Rev. A 91, 012109 (2015).
[Crossref]

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B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
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X. M. Lu, X. G. Wang, and C. P. Sun, “Quantum Fisher information flow and non-Markovian processes of open systems,” Phys. Rev. A 82, 042103 (2010).
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Luo, S.

S. Luo, S. Fu, and H. Song, “Quantifying non-Markovianity via correlations,” Phys. Rev. A 86, 044101 (2012).
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H. Mäkelä, “Bounds for the divisibility-based and distinguishability-based non-Markovianity measures,” Phys. Rev. A 91, 012108 (2015)
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D. Chruściński and S. Maniscalco, “Degree of non-Markovianity of quantum evolution,” Phys. Rev. Lett. 112, 120404 (2014).
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B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,” Scientfic Reports 4, 5720 (2014).

C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits,” Phys. Rev. A 89, 024101 (2014).
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P. Haikka, T. H. Johnson, and S. Maniscalco, “Non-Markovianity of local dephasing channels and time-invariant discord,” Phys. Rev. A 87, 010103(R) (2013).
[Crossref]

P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
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R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable quantum key distribution in non-Markovian channels,” Phys. Rev. A 83, 042321 (2011).
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B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302(R) (2008).
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McEndoo, S.

P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
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R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

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Y. Kubota and K. Nobusada, “Applicability of site-basis time-evolution equation for thermalization of exciton states in a quantum dot array,” J. Phys. Soc. Jpn 78, 114603 (2009).
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R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable quantum key distribution in non-Markovian channels,” Phys. Rev. A 83, 042321 (2011).
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Palma, G. M.

P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
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M. A. Cirone, G. De Chiara, G. M. Palma, and A. Recati, “Collective decoherence of trapped atoms coupled to a Bose-Einstein condensate,” New J. Phys. 11, 103055 (2009).
[Crossref]

Paris, M. G. A.

R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable quantum key distribution in non-Markovian channels,” Phys. Rev. A 83, 042321 (2011).
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F. Giraldi and F. Petruccione, “Survival of coherence for open quantum systems in the thermal baths,” Phys. Rev. A 88, 042102 (2013).
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H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2007).
[Crossref]

Piilo, J.

E. M. Laine, H. P. Breuer, and J. Piilo, “Nonlocal memory effects allow perfect teleportation with mixed states,” Scientfic Reports 4, 4620 (2014).

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
[Crossref]

E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
[Crossref] [PubMed]

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

Plenio, M. B.

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quanrum non-Markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77, 094001 (2014).
[Crossref]

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
[Crossref]

A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
[Crossref]

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Phys. Rev. Lett. 105, 050403 (2010).
[Crossref] [PubMed]

Prior, J.

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
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M. A. Cirone, G. De Chiara, G. M. Palma, and A. Recati, “Collective decoherence of trapped atoms coupled to a Bose-Einstein condensate,” New J. Phys. 11, 103055 (2009).
[Crossref]

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Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quanrum non-Markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77, 094001 (2014).
[Crossref]

D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Phys. Rev. A 83, 052128 (2011).
[Crossref]

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Phys. Rev. Lett. 105, 050403 (2010).
[Crossref] [PubMed]

Rosenbach, R.

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
[Crossref]

Schmidt, R.

R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
[Crossref]

Shao, J.

J. Shao, “Decoupling quantum dissipation interaction via stochastic fields,” J. Chem. Phys. 120, 5053 (2004).
[Crossref] [PubMed]

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J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

Song, H.

S. Luo, S. Fu, and H. Song, “Quantifying non-Markovianity via correlations,” Phys. Rev. A 86, 044101 (2012).
[Crossref]

Stockburger, J. T.

R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

Sun, C. P.

X. M. Lu, X. G. Wang, and C. P. Sun, “Quantum Fisher information flow and non-Markovian processes of open systems,” Phys. Rev. A 82, 042103 (2010).
[Crossref]

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J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

Tang, N.

N. Tang, Z. L. Fan, and H. S. Zeng, “Improving the quality of noisy spatial quantum channels,” Quant. Inform. Comput. 15, 0568–0581 (2015).

H. S. Zeng, Y. P. Zheng, N. Tang, and G. Y. Wang, “Correlation quantum beats induced by non-Markovian effect,” Quantum Inform. Process 12, 1637–1650 (2013).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and T. T. Xu, “Non-Markovian dynamics for an open two-level system without rotating wave approximation: indivisibility versus backflow of information,” Eur. Phys. J. D 66, 225 (2012).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and G. Y. Wang, “Equivalence of the measures of non-Markovianity for open two-level systems,” Phys. Rev. A 84, 032118 (2011).
[Crossref]

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R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable quantum key distribution in non-Markovian channels,” Phys. Rev. A 83, 042321 (2011).
[Crossref]

Wang, G. Y.

H. S. Zeng, Y. P. Zheng, N. Tang, and G. Y. Wang, “Correlation quantum beats induced by non-Markovian effect,” Quantum Inform. Process 12, 1637–1650 (2013).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and G. Y. Wang, “Equivalence of the measures of non-Markovianity for open two-level systems,” Phys. Rev. A 84, 032118 (2011).
[Crossref]

Wang, X. G.

X. M. Lu, X. G. Wang, and C. P. Sun, “Quantum Fisher information flow and non-Markovian processes of open systems,” Phys. Rev. A 82, 042103 (2010).
[Crossref]

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S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
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D. Chruściński and F. A. Wudarski, “Non-Markovianity degree for random unitary evolution,” Phys. Rev. A 91, 012104 (2015).
[Crossref]

Xu, J. S.

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

Xu, T. T.

H. S. Zeng, N. Tang, Y. P. Zheng, and T. T. Xu, “Non-Markovian dynamics for an open two-level system without rotating wave approximation: indivisibility versus backflow of information,” Eur. Phys. J. D 66, 225 (2012).
[Crossref]

Yi, X. X.

S. C. Hou, S. L. Liang, and X. X. Yi, “Non-Markovianity and memory effects in quantum open systems,” Phys. Rev. A 91, 012109 (2015).
[Crossref]

Zeng, H. S.

N. Tang, Z. L. Fan, and H. S. Zeng, “Improving the quality of noisy spatial quantum channels,” Quant. Inform. Comput. 15, 0568–0581 (2015).

H. S. Zeng, Y. P. Zheng, N. Tang, and G. Y. Wang, “Correlation quantum beats induced by non-Markovian effect,” Quantum Inform. Process 12, 1637–1650 (2013).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and T. T. Xu, “Non-Markovian dynamics for an open two-level system without rotating wave approximation: indivisibility versus backflow of information,” Eur. Phys. J. D 66, 225 (2012).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and G. Y. Wang, “Equivalence of the measures of non-Markovianity for open two-level systems,” Phys. Rev. A 84, 032118 (2011).
[Crossref]

Zheng, Y. P.

H. S. Zeng, Y. P. Zheng, N. Tang, and G. Y. Wang, “Correlation quantum beats induced by non-Markovian effect,” Quantum Inform. Process 12, 1637–1650 (2013).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and T. T. Xu, “Non-Markovian dynamics for an open two-level system without rotating wave approximation: indivisibility versus backflow of information,” Eur. Phys. J. D 66, 225 (2012).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and G. Y. Wang, “Equivalence of the measures of non-Markovianity for open two-level systems,” Phys. Rev. A 84, 032118 (2011).
[Crossref]

Zou, X. B.

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

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M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
[Crossref]

Chin. Phys. B (1)

Y. H. Ji and J. J. Hu, “Entanglement and decoherence of coupled superconductor qubits in a non-Markovian environment,” Chin. Phys. B 19, 060304 (2010).
[Crossref]

Eur. Phys. J. D (1)

H. S. Zeng, N. Tang, Y. P. Zheng, and T. T. Xu, “Non-Markovian dynamics for an open two-level system without rotating wave approximation: indivisibility versus backflow of information,” Eur. Phys. J. D 66, 225 (2012).
[Crossref]

Europhys. Lett. (1)

J. S. Tang, C. F. Li, Y. L. Li, X. B. Zou, G. C. Gou, H. P. Breuer, E. M. Laine, and J. Piilo, “Measureing non-Markovianity of processes with controllable system-environment interaction,” Europhys. Lett. 97, 10002 (2012).
[Crossref]

J. Chem. Phys. (1)

J. Shao, “Decoupling quantum dissipation interaction via stochastic fields,” J. Chem. Phys. 120, 5053 (2004).
[Crossref] [PubMed]

J. Phys. Soc. Jpn (1)

Y. Kubota and K. Nobusada, “Applicability of site-basis time-evolution equation for thermalization of exciton states in a quantum dot array,” J. Phys. Soc. Jpn 78, 114603 (2009).
[Crossref]

Nat. Phys. (1)

B. H. Liu, L. Li, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems,” Nat. Phys. 7, 931–934 (2011).
[Crossref]

Nature Phys. (1)

A. W. Chin, J. Prior, R. Rosenbach, F. Caycedo-Soler, S. F. Huelga, and M. B. Plenio, “The role of non-equilibrium vibrational structures in electronic coherence and recoherence in pigmentCprotein complexes,” Nature Phys. 9, 113–118 (2013).
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New J. Phys. (1)

M. A. Cirone, G. De Chiara, G. M. Palma, and A. Recati, “Collective decoherence of trapped atoms coupled to a Bose-Einstein condensate,” New J. Phys. 11, 103055 (2009).
[Crossref]

Phys. Rev. A (15)

S. Wißmann, A. Karlsson, E. M. Laine, J. Piilo, and H. P. Breuer, “Optimal state pairs for non-Markovian quantum dynamics,” Phys. Rev. A 86, 062108 (2012).
[Crossref]

R. Vasile, S. Olivares, M. G. A. Paris, and S. Maniscalco, “Continuous-variable quantum key distribution in non-Markovian channels,” Phys. Rev. A 83, 042321 (2011).
[Crossref]

X. M. Lu, X. G. Wang, and C. P. Sun, “Quantum Fisher information flow and non-Markovian processes of open systems,” Phys. Rev. A 82, 042103 (2010).
[Crossref]

S. Luo, S. Fu, and H. Song, “Quantifying non-Markovianity via correlations,” Phys. Rev. A 86, 044101 (2012).
[Crossref]

S. C. Hou, S. L. Liang, and X. X. Yi, “Non-Markovianity and memory effects in quantum open systems,” Phys. Rev. A 91, 012109 (2015).
[Crossref]

H. S. Zeng, N. Tang, Y. P. Zheng, and G. Y. Wang, “Equivalence of the measures of non-Markovianity for open two-level systems,” Phys. Rev. A 84, 032118 (2011).
[Crossref]

F. Giraldi and F. Petruccione, “Survival of coherence for open quantum systems in the thermal baths,” Phys. Rev. A 88, 042102 (2013).
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P. Haikka, S. McEndoo, G. De Chiara, G. M. Palma, and S. Maniscalco, “Quantifying, characterizing, and controlling information flow in ultracold atomic gases,” Phys. Rev. A 84, 031602(R) (2011).
[Crossref]

D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Phys. Rev. A 83, 052128 (2011).
[Crossref]

D. Chruściński and F. A. Wudarski, “Non-Markovianity degree for random unitary evolution,” Phys. Rev. A 91, 012104 (2015).
[Crossref]

H. Mäkelä, “Bounds for the divisibility-based and distinguishability-based non-Markovianity measures,” Phys. Rev. A 91, 012108 (2015)
[Crossref]

B. Bellomo, R. Lo Franco, S. Maniscalco, and G. Compagno, “Entanglement trapping in structured environments,” Phys. Rev. A 78, 060302(R) (2008).
[Crossref]

P. Haikka, T. H. Johnson, and S. Maniscalco, “Non-Markovianity of local dephasing channels and time-invariant discord,” Phys. Rev. A 87, 010103(R) (2013).
[Crossref]

C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits,” Phys. Rev. A 89, 024101 (2014).
[Crossref]

Phys. Rev. Lett. (8)

E. M. Laine, H. P. Breuer, J. Piilo, C. F. Li, and G. C. Guo, “Nonlocal memory effects in the dynamics of open quantum systems,” Phys. Rev. Lett. 108, 210402 (2012).
[Crossref] [PubMed]

J. S. Xu, C. F. Li, M. Gong, X. B. Zou, C. H. Shi, G. Chen, and G. C. Guo, “Experimental demonstration of photonic entanglement collapse and revival,” Phys. Rev. Lett. 104, 100502 (2010).
[Crossref] [PubMed]

D. Chruściński and S. Maniscalco, “Degree of non-Markovianity of quantum evolution,” Phys. Rev. Lett. 112, 120404 (2014).
[Crossref]

R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quantum systems: Cooperative effects of driving and dissipation,” Phys. Rev. Lett. 107, 130404 (2011).
[Crossref] [PubMed]

H. P. Breuer, E. M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Phys. Rev. Lett. 103, 210401 (2009).
[Crossref]

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Phys. Rev. Lett. 105, 050403 (2010).
[Crossref] [PubMed]

M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Phys. Rev. Lett. 101, 150402 (2008).
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A. W. Chin, S. F. Huelga, and M. B. Plenio, “Quantum metrology in non-Markovian environments,” Phys. Rev. Lett. 109, 233601 (2012).
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Quant. Inform. Comput. (1)

N. Tang, Z. L. Fan, and H. S. Zeng, “Improving the quality of noisy spatial quantum channels,” Quant. Inform. Comput. 15, 0568–0581 (2015).

Quantum Inform. Process (1)

H. S. Zeng, Y. P. Zheng, N. Tang, and G. Y. Wang, “Correlation quantum beats induced by non-Markovian effect,” Quantum Inform. Process 12, 1637–1650 (2013).
[Crossref]

Rep. Prog. Phys. (1)

Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quanrum non-Markovianity: characterization, quantification and detection,” Rep. Prog. Phys. 77, 094001 (2014).
[Crossref]

Scientfic Reports (2)

E. M. Laine, H. P. Breuer, and J. Piilo, “Nonlocal memory effects allow perfect teleportation with mixed states,” Scientfic Reports 4, 4620 (2014).

B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,” Scientfic Reports 4, 5720 (2014).

Other (3)

H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2007).
[Crossref]

A. B. Klimov and S. M. Chumakov, A Group-Theoretical Approach to Quanyum Optics (WILEY-VCH Verlag GmbH & Co. KGaA, 2009).
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M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University, 1997).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 (a) Non-Markovianity as a function of average excitation number for four kinds of excitation number statistics (For binomial distribution we set N = 10). (b) Time evolution of γ(t) for Poisson statistical environment. Where we assume the mode-independent coupling strength |g(ω)| = 1, the central frequency of spectrum ω 0 = 100 and width λ = 2.
Fig. 2
Fig. 2 (a) Non-Markovianity as a function of the average excitation number for CS and DCS environments. (b) Time evolution of γ(t). Where we set |g(ω)| = 1, ω 0 = 100 and λ = 2.
Fig. 3
Fig. 3 (a) Non-Markovianity as a function of environmental coherence. (b) Time evolution of γ(t) for different coherence η = 0, 1. Where we set |g(ω)| = 1 and ω = 1, which are assumed mode-independent. Other parameters are chosen as ω 0 = 100 and λ = 2.
Fig. 4
Fig. 4 Non-Markovianity as a function of the average excitation number for SV and DSV environments, θ = 0 for (a) and θ = π for (b). Where we set |g(ω)| = 1, ω 0 = 100 and λ = 2.

Equations (53)

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H = ω 0 2 σ z + k ω k b k b k + k σ z g k ( b k + b k ) ,
ρ 11 ( t ) = ρ 11 ( 0 ) , ρ 00 ( t ) = ρ 00 ( 0 ) , ρ 10 ( t ) = ρ 10 ( 0 ) e Γ ( t ) , ρ 01 ( t ) = ρ 01 ( 0 ) e Γ * ( t ) ,
Γ ( t ) = k ln D ( α k ) ,
x t = 1 2 x 0 [ e Γ ( t ) + e Γ * ( t ) ] i 2 y 0 [ e Γ ( t ) e Γ * ( t ) ] , y t = i 2 x 0 [ e Γ ( t ) e Γ * ( t ) ] + 1 2 y 0 [ e Γ ( t ) + e Γ * ( t ) ] , z t = z 0 ,
σ ( t , ρ 1 , 2 ( 0 ) ) = d d t D ( ρ 1 ( t ) , ρ 2 ( t ) ) ,
𝒩 = max ρ 1 , 2 ( 0 ) σ > 0 d t σ ( t , ρ 1 , 2 ( 0 ) ) ,
ρ 1 , 2 ( 0 ) = 1 2 ( I ± r 0 σ ) ,
D ( ρ 1 , t ) , ρ 2 ( t ) ) = 1 2 ( Δ x t ) 2 + ( Δ y t ) 2 + ( Δ z t ) 2 ,
Δ x t = x 0 [ e Γ ( t ) + e Γ * ( t ) ] i y 0 [ e Γ ( t ) e Γ * ( t ) ] , Δ y t = i x 0 [ e Γ ( t ) e Γ * ( t ) ] + y 0 [ e Γ ( t ) + e Γ * ( t ) ] , Δ z t = 2 z 0 .
D ( ρ 1 ( t ) , ρ 2 ( t ) ) = { ( 1 z 0 2 ) e γ ( t ) + z 0 2 } 1 / 2 ,
σ ( t ) = 1 D ( 1 z 0 2 ) e γ ( t ) γ ˙ ( t ) ,
z 0 m 2 = e γ ( t ) + 1 e γ ( t ) 1 ,
𝒩 1 = γ ˙ ( t ) < 0 d t γ ˙ ( t ) e 3 2 γ ( t ) ,
𝒩 = max { 𝒩 1 , 𝒩 2 } = γ ˙ ( t ) < 0 d t γ ˙ ( t ) e 3 2 γ ( t ) .
ρ B = k n k = 0 P n k | n k n k | ,
n k | D ( α k ) | n k = e | α k | 2 / 2 L n k ( | α k | 2 ) ,
Γ ( t ) = k ln { n k = 0 P n k e | α k | 2 / 2 L n k ( | α k | 2 ) } .
L n ( x ) = m = 0 n ( 1 ) m ( n n m ) x m m ! ,
( p n ) = p ! n ! ( p n ) ! .
Γ ( t ) = Γ 0 ( t ) + Γ 1 ( t ) ,
Γ 0 ( t ) = 1 2 0 d ω D ( ω ) | α ω | 2 ,
Γ 1 ( t ) = 0 d ω D ( ω ) ln [ n = 0 P n L n ( | α ω | 2 ) ] ,
| α ω | 2 = 8 | g ( ω ) | 2 1 cos ( ω t ) ω 2 .
D ( ω ) = λ 2 π [ ( ω 0 ω ) 2 + λ 2 ] ,
Γ 0 ( t ) = 2 ( λ 2 + ω 0 2 ) 2 { ω 0 2 λ 2 + ( λ 2 ω 0 2 ) e λ t cos ( ω 0 t ) 2 ω 0 λ e λ t sin ( ω 0 t ) + λ ( λ 2 + ω 0 2 ) t } .
P n = e n ¯ n ¯ n n ! ,
P 2 n = ( cosh r 1 ) ( 1 2 tanh r ) 2 n ( 2 n ) ! ( n ! ) 2 P 2 n + 1 = 0 ,
P n = 1 n ¯ + 1 ( n ¯ n ¯ + 1 ) n ,
P n = ( N n ) n ¯ n ( N n ¯ ) N n N N ,
T ( t ) = 8 0 d ω D ( ω ) | g ( ω ) | 2 1 cos ( ω t ) ω 2 ,
T ˙ ( t ) = 8 0 d ω D ( ω ) | g ( ω ) | 2 sin ( ω t ) ω
𝒩 1 = ( 2 n ¯ + 1 ) T ˙ ( t ) < 0 T ˙ ( t ) d t + 3 2 ( 2 n ¯ + 1 ) 2 T ˙ ( t ) < 0 T ( t ) T ˙ ( t ) d t .
γ ( t ) = ( 2 n ¯ + 1 ) T ( t ) + 1 2 [ n ¯ + n ¯ 2 V ( n ) ] T 1 ( t ) ,
𝒩 2 = 𝒩 1 1 2 ( n ¯ + n ¯ 2 ) T ˙ ( t ) < 0 T ˙ 1 ( t ) d t + 1 2 V ( n ) T ˙ 1 ( t ) < 0 T ˙ 1 ( t ) d t .
ρ B ( 0 ) = k | β k β k | ,
β k | D ( α k ) | β k = exp { 1 2 | α k | 2 + iIm ( α k β k * ) } ,
Γ ( t ) = k [ 1 2 | α k | 2 iIm ( α k β k * ) ] .
Γ ( t ) = 0 d ω D ( ω ) { 1 2 | α ω | 2 iIm [ α ω β ω * ] } ,
γ cs ( t ) = 2 Γ 0 ( t ) ,
γ dcs ( t ) = 2 Γ 0 ( t ) + 2 Γ 1 ( t ) ,
Γ 1 ( t ) = 0 d ω D ( ω ) ln J 0 ( 2 | α ω | n ¯ ) .
ρ B p = k { n k = 0 P n k | n k n k | + n k m k η P n k m k | n k m k | } ,
P n k = e | β k | 2 | β k | 2 n k n k ! , P n k m k = e | β k | 2 β k n k β k * m k n k ! m k ! .
Γ ( t ) = k ln { n k P n k n k | D ( α k ) | n k + n k m k η P n k m k m k | D ( α k ) | n k } ,
m | D ( γ ) | n = { n ! m ! γ m n e | γ | 2 / 2 L n m n ( | γ | 2 ) , m n m ! n ! ( γ * ) n m e | γ | 2 / 2 L m n m ( | γ | 2 ) , n m ,
Γ ( t ) = Γ 0 ( t ) + 0 d ω D ( ω ) n ¯ ω 0 d ω D ( ω ) ln ( B + η B 1 + η B 2 ) .
B = n = 0 n ¯ ω n n ! L n ( | α ω | 2 ) , B 1 = m > n α ω m n m ! β ω n [ β ω * ] m L n m n ( | α ω | 2 ) , B 2 = n > m [ α ω * ] n m n ! β ω n [ β ω * ] m L m n m ( | α ω | 2 ) ,
L n α ( x ) = m = 0 n ( 1 ) m ( n + α n m ) x m m ! .
| ϕ = k S k ( ξ k ) | 0
S k ( ξ k ) = exp [ 1 2 ξ k * b k 2 1 2 ξ k b k 2 ] .
γ s v ( t ) = 0 d ω D ( ω ) | α ω | 2 × { cosh [ 2 r ( ω ) ] sinh [ 2 r ( ω ) ] cos [ ω t θ ( ω ) ] } ,
γ 0 , π ( t ) = 2 [ cosh ( 2 r ) ± sinh ( 2 r ) ] Γ 0 ( t ) sinh ( 2 r ) Γ 0 ( 2 t ) ,
γ d s v ( t ) = 2 Γ 0 ( t ) + 2 Γ 1 ( t ) ,

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