Abstract

Dynamics of an open system is vividly influenced by the structure of environments. This paper studies in detail the dynamics of a two-level atom in the presence of an overall environment composed of two hierarchies. The first hierarchy is just a single lossy cavity while the second hierarchy consists of a number of other lossy cavities. The atom is coupled directly to the first hierarchy but indirectly to the second one via the couplings between the two hierarchies. We show that even when the coupling between the atom and the first hierarchy is weak the atom’s dynamics can become non-Markovian if the number of cavities in the second hierarchy or/and the coupling between the two hierarchies are large enough. We also analyze the case when the coupling between the atom and the first hierarchy is strong and show that the non-Markovian dynamics exhibits different patterns depending on both the number of cavities in the second hierarchy and the coupling between the two hierarchies.

© 2015 Optical Society of America

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References

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    [Crossref]
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  56. 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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    [Crossref]

2014 (9)

A. D’Arrigo, R. LoFranco, G. Benenti, E. Paladino, and G. Falci, “Recovering entanglement by local operations,” Ann. Phys. 350, 211–224 (2014).
[Crossref]

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

C. Addis, B. Bylicka, D. Chruściński, and S. Maniscalco, “Comparative study of non-Markovianity measures in exactly solvable one- and two-qubit models,” Phys. Rev. A 90, 052103 (2014).
[Crossref]

B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective,” Sci. Rep. 4, 5720 (2014).
[Crossref] [PubMed]

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112, 110401 (2014).
[Crossref] [PubMed]

C. -K. Chan, G. -D. Lin, S. F. Yelin, and M. D. Lukin, “Quantum interference between independent reservoirs in open quantum systems,” Phys. Rev. A 89, 042117 (2014).
[Crossref]

Tony J. G. Apollaro, S. Lorenzo, C. D. Franco, F. Plastina, and M. Paternostro, “Competition between memory-keeping and memory-erasing decoherence channels,” Phys. Rev. A 90, 012310 (2014).
[Crossref]

Z. X. Man, N. B. An, and Y. J. Xia, “Non-Markovianity of a two-level system transversally coupled to multiple bosonic reservoirs,” Phys. Rev. A 90, 062104 (2014).
[Crossref]

T. Ma, Y. Chen, T. Chen, S. R. Hedemann, and T. Yu, “Crossover between non-Markovian and Markovian dynamics induced by a hierarchical environment,” Phys.Rev. A 90, 042108 (2014).

2013 (8)

E. A. Chekhovich, M. N. Makhonin, A. I. Tartakovskii, A. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Vandersypen, “Nuclear spin effects in semiconductor quantum dots,” Nat. Mater. 12, 494–504 (2013).
[Crossref] [PubMed]

S. Lorenzo, F. Plastina, and M. Paternostro, “Geometrical characterization of non-Markovianity,” Phys. Rev. A 88, 020102 (2013).
[Crossref]

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. L. Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref] [PubMed]

B. H. Liu, D. Y. Cao, Y. F. Huang, C. F. Li, G. C. Guo, E. M. Laine, H. P. Breuer, and J. Piilo, “Photonic realization of nonlocal memory effects and non-Markovian quantum probes,” Sci. Rep. 3, 1781 (2013).
[Crossref]

M. del Rey, A.W. Chin, S. F. Huelga, and M. B. Plenio, “Exploiting structured environments for efficient energy transfer: the phonon antenna mechanism,” J. Phys. Chem. Lett. 4, 903–907 (2013).
[Crossref]

R. LoFranco, B. Bellomo, S. Maniscalco, and G. Compagno, “Dynamics of quantum correlations in two-qubit systems within non-Markovian environments,” Int. J. Mod. Phys. B 27, 1345053 (2013).
[Crossref]

D. M. Kennes, O. Kashuba, M. Pletyukhov, H. Schoeller, and V. Meden, “Oscillatory dynamics and non-Markovian memory in dissipative quantum systems,” Phys. Rev. Lett. 110, 100405 (2013).
[Crossref] [PubMed]

A. Bermudez, T. Schaetz, and M. B. Plenio, “Dissipation-assisted quantum information processing with trapped ions,” Phys. Rev. Lett. 110, 110502 (2013).
[Crossref] [PubMed]

2012 (11)

U. Hoeppe, C. Wolff, J. Küchenmeister, J. Niegemann, M. Drescher, H. Benner, and K. Busch, “Direct observation of non-Markovian radiation dynamics in 3D bulk photonic crystals,” Phys. Rev. Lett. 108, 043603 (2012).
[Crossref] [PubMed]

E. Barnes, Lukasz Cywiński, and S. Das Sarma, “Nonperturbative master equation solution of central spin de-phasing dynamics,” Phys. Rev. Lett. 109, 140403 (2012).
[Crossref]

R. LoFranco, A. D’Arrigo, G. Falci, G. Compagno, and E. Paladino, “Entanglement dynamics in superconducting qubits affected by local bistable impurities,” Phys. Scr. T147, 014019 (2012).
[Crossref]

Z. X. Man, A. Smirne, Y.J. Xia, and B. Vacchini, “Quantum interference induced by initial system-environment correlations,” Phys. Letts. A 376, 2477–2483 (2012).
[Crossref]

R. LoFranco, B. Bellomo, E. Andersson, and G. Compagno, “Revival of quantum correlations without system-environment back-action,” Phys. Rev. A 85, 032318 (2012).
[Crossref]

S. F. Huelga, A. Rivas, and M. B. Plenio, “Non-Markovianity-assisted steady state entanglement,” Phys. Rev. Lett. 108, 160402 (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]

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. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, and A. Morello, “A single-atom electron spin qubit in silicon,” Nature (London) 489, 541–545 (2012).
[Crossref]

A. M. Tyryshkin and et al., “Electron spin coherence exceeding seconds in high-purity silicon,” Nat. Mater. 11, 143–147 (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)

C.-K. Chan and L. J. Sham, “Precision of electromagnetic control of a quantum system,” Phys. Rev. A 84, 032116 (2011).
[Crossref]

B. Bellomo, G. Compagno, R. LoFranco, A. Ridolfo, and S. Savasta, “Dynamics and extraction of quantum discord in a multipartite open system,” Int. J. Quant. Inf. 9, 1665–1676 (2011).
[Crossref]

K. H. Madsen, S. Ates, T. Lund-Hansen, A. Löffler, S. Reitzenstein, A. Forchel, and P. Lodahl, “Observation of non-Markovian dynamics of a single quantum dot in a micropillar cavity,” Phys. Rev. Lett. 106, 233601 (2011).
[Crossref] [PubMed]

C. F. Li, J. S. Tang, Y. L. Li, and G. C. Guo, “Experimentally witnessing the initial correlation between an open quantum system and its environment,” Phys. Rev. A 83, 064102 (2011).
[Crossref]

P. Rebentrost and A. Aspuru-Guzik, “Exciton-phonon information flow in the energy transfer process of photosynthetic complexes,” J. Chem. Phys. 134, 101103 (2011).
[Crossref]

T. J. G. Apollaro, C. Di Franco, F. Plastina, and M. Paternostro, “Memory-keeping effects and forgetfulness in the dynamics of a qubit coupled to a spin chain,” Phys. Rev. A 83, 032103 (2011).
[Crossref]

2010 (4)

W. M. Zhang, M. H. Wu, C. U. Lei, and H. N. Xiong, “Non-Markovian dynamics of a microcavity coupled to a waveguide in photonic crystals,” Opt. Express 18, 18407–18418 (2010).
[Crossref] [PubMed]

J. Dajka and J. Luczka, “Distance growth of quantum states due to initial system-environment correlations,” Phys. Rev. A 82, 012341 (2010).
[Crossref]

A. Smirne, H.P. Breuer, J. Piilo, and B. Vacchini, “Initial correlations in open-systems dynamics: the Jaynes-Cummings model,” Phys. Rev. A 82, 062114 (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]

2009 (3)

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]

L. Mazzola, S. Maniscalco, J. Piilo, K. A. Suominen, and B. M. Garraway, “Pseudomodes as an effective description of memory: non-Markovian dynamics of two-state systems in structured reservoirs,” Phys. Rev. A 80, 012104 (2009).
[Crossref]

J. Piilo, K. Härkönen, S. Maniscalco, and K.-A. Suominen, “Open system dynamics with non-Markovian quantum jumps,” Phys. Rev. A 79, 062112 (2009).
[Crossref]

2008 (3)

J. Piilo, S. Maniscalco, K. Härkönen, and K.-A. Suominen, “Non-Markovian quantum jumps,” Phys. Rev. Lett. 100, 180402 (2008).
[Crossref] [PubMed]

B. Bellomo, R. Lo Franco, and G. Compagno, “Entanglement dynamics of two independent qubits in environments with and without memory,” Phys. Rev. A 77, 032342 (2008).
[Crossref]

R. Hanson, V. V. Dobrovitski, A. E. Feiguin, O. Gywat, and D. D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science 320, 352–355 (2008).
[Crossref] [PubMed]

2007 (5)

R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, “Spins in few-electron quantum dots,” Rev. Mod. Phys. 79, 1217 (2007).
[Crossref]

B. Bellomo, R. Lo Franco, and G. Compagno, “Non-Markovian effects on the dynamics of entanglement,” Phys. Rev. Lett. 99, 160502 (2007).
[Crossref] [PubMed]

J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C.M. Marcus, and M. D. Lukin, “Relaxation, dephasing, and quantum control of electron spins in double quantum dots,” Phys. Rev. B 76, 035315 (2007).
[Crossref]

H. Lee, Y.-C. Cheng, and G. R. Fleming, “Coherence dynamics in photosynthesis: protein protection of excitonic coherence,” Science 316, 1462–1465 (2007).
[Crossref] [PubMed]

R. LoFranco, G. Compagno, A. Messina, and A. Napoli, “Generating and revealing a quantum superposition of electromagnetic-field binomial states in a cavity,” Phys. Rev. A 76, 011804 (2007).
[Crossref]

2006 (1)

W. Yao, R.-B. Liu, and L. J. Sham, “Theory of electron spin decoherence by interacting nuclear spins in a quantum dot,” Phys. Rev. B 74, 195301 (2006).
[Crossref]

2005 (1)

L. S. Cederbaum, E. Gindensperger, and I. Burghardt, “Short-time dynamics through conical intersections in macrosystems,” Phys. Rev. Lett. 94, 113003 (2005).
[Crossref] [PubMed]

2004 (1)

W. A. Coish and D. Loss, “Hyperfine interaction in a quantum dot: non-Markovian electron spin dynamics,” Phys. Rev. B 70, 195340 (2004).
[Crossref]

1998 (1)

1997 (1)

B. M. Garraway, “Nonperturbative decay of an atomic system in a cavity,” Phys. Rev. A 55, 2290 (1997).
[Crossref]

Addis, C.

C. Addis, B. Bylicka, D. Chruściński, and S. Maniscalco, “Comparative study of non-Markovianity measures in exactly solvable one- and two-qubit models,” Phys. Rev. A 90, 052103 (2014).
[Crossref]

An, N. B.

Z. X. Man, N. B. An, and Y. J. Xia, “Non-Markovianity of a two-level system transversally coupled to multiple bosonic reservoirs,” Phys. Rev. A 90, 062104 (2014).
[Crossref]

Andersson, E.

J. S. Xu, K. Sun, C. F. Li, X. Y. Xu, G. C. Guo, E. Andersson, R. L. Franco, and G. Compagno, “Experimental recovery of quantum correlations in absence of system-environment back-action,” Nat. Commun. 4, 2851 (2013).
[Crossref] [PubMed]

R. LoFranco, B. Bellomo, E. Andersson, and G. Compagno, “Revival of quantum correlations without system-environment back-action,” Phys. Rev. A 85, 032318 (2012).
[Crossref]

Apollaro, T. J. G.

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Nature (London) (1)

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H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems (Oxford University Press, 2002).

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, 2000).

A. Orieux, A. d’Arrigo, G. Ferranti, R. Lo Franco, G. Benenti, E. Paladino, G. Falci, F. Sciarrino, and P. Mataloni, “Experimental on-demand recovery of entanglement by local operations within non-Markovian dynamics,” Sci. Rep. (to be published).

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Figures (6)

Fig. 1
Fig. 1 A two-level system (circle) is coupled with strength Ω0 to a mode (large ellipse) m0 which decays to a memoryless reservoir (rectangle) with a rate Γ0. The mode m0 is further coupled simultaneously with strengths Ω12,…, ΩN to modes m1,m2,…, mN (small ellipses) which also decay to their respective memoryless reservoirs (rectangles) with rates Γ12,…,ΓN.
Fig. 2
Fig. 2 The non-Markovianity N as a function of the number N of identical cavities mn (n = 1,2,…,N) for different values of the m0-mn coupling strength Ω (= Ωnn) but fixed values of the s-m0 coupling strength Ω0 = Γ0/5 and the cavity mn decay rate Γ (= Γnn) = Γ0.
Fig. 3
Fig. 3 Phase diagram in the N-Ω/Γ0 plane for the crossover between Markovian and non-Markovian dynamics in the weak s-m0 coupling regime with Ω00 = 0.2 while Γ = Γ0.
Fig. 4
Fig. 4 The trace distance evolution showing (a) type I pattern for the case of strong s-m0 coupling with Ω0 = 0.3Γ0 in the absence of any additional cavity and (b) type II pattern for the case of weak s-m0 coupling with Ω0 = 0.2Γ0 in the presence of two identical additional cavities with Ω = Ω1 = Ω2 = Γ0 and Γ = Γ1 = Γ2 = 0.5Γ0.
Fig. 5
Fig. 5 Evolution of (a) the trace distance, (b) the population |c0(t)|2 of the cavity m0 and (c) the witness Wn(t), Eq. (13), for the case when the s-m0 coupling is weak with Ω0 = 0.2Γ0 while the cavity m0 is coupled, with the same strength Ω = Ω1 = Ω2 = Γ0, to two identical cavities m1,m2, which decay with the same rate Γ = Γ1 = Γ2 = 0.5Γ0.
Fig. 6
Fig. 6 The non-Markovianity N as a function of the coupling constant Ω/Γ0 = (Ωn0,∀n) between the cavity m0 and the identical secondary cavities mn (n = 1,2,…,N). The circles represent N = 1 and the triangles N = 2. The white (black) color stands for type I (type II) pattern of non-Markovian dynamics and the black-white color specifies the coexistence of these two types of patterns. The symbols with a cross denote N = 0. The other parameters used are Ω00 = 0.5 and Γn = Γ0 (n = 1,2).

Equations (13)

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H 0 = ω s 2 σ z + ω c a a + n = 1 N ω n b n b n ,
H I = Ω 0 ( σ + a + σ a ) + n = 1 N Ω n ( a b n + a b n ) .
ρ ˙ ( t ) = i [ H , ρ ( t ) ] Γ 0 2 [ a a ρ ( t ) 2 a ρ ( t ) a + ρ ( t ) a a ] n = 1 N Γ n 2 [ b n b n ρ ( t ) 2 b n ρ ( t ) b n + ρ ( t ) b n b n ] ,
ρ s , m ( t ) = ( 1 λ ( t ) ) | ψ ( t ) s , m ψ ( t ) | + λ ( t ) | 00 0 s , m 00 0 | ,
| ψ ˜ ( t ) s , m 1 λ ( t ) | ψ ( t ) s , m = h ˜ ( t ) | 10 0 s , m + c ˜ 0 ( t ) | 01 0 s , m + + c ˜ N ( t ) | 00 1 s , m ,
ρ s , m ( t ) = | ψ ˜ ( t ) s , m ψ ˜ ( t ) | + λ ( t ) | 00 0 s , m 00 0 | .
i d h ˜ ( t ) d t = ω s h ˜ ( t ) + Ω 0 c ˜ 0 ( t ) ,
i d c ˜ 0 ( t ) d t = ( ω c i 2 Γ 0 ) c ˜ 0 ( t ) + Ω 0 h ˜ ( t ) + n = 1 N Ω n c ˜ n ( t ) ,
i d c ˜ n ( t ) d t = ( ω n i 2 Γ n ) c ˜ n ( t ) + Ω n c ˜ 0 ( t ) , n = 1 , 2 , , N .
N = max ρ 1 ( 0 ) , ρ 2 ( 0 ) σ > 0 σ [ t , ρ 1 ( 0 ) , ρ 2 ( 0 ) ] d t ,
D [ ρ 1 ( t ) , ρ 2 ( t ) ] = 1 2 Tr | ρ 1 ( t ) ρ 2 ( t ) | ,
σ [ t , ρ 1 ( 0 ) , ρ 2 ( 0 ) ] = d | h ˜ ( t ) | d t .
W n ( t ) d | c ˜ n ( t ) | 2 d t + Γ n | c ˜ n ( t ) | 2 ,

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