Symmetries and conserved quantities in Lindblad master equations

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 89, Issue 2, 2014, Pages

  Albert, Victor V ^a   Jiang, Liang ^a  

^a YALE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARBITRARY STATE; CONSERVED QUANTITY; DENSITY MATRIX; INITIAL CONDITIONS; INITIAL STATE; LINDBLAD MASTER EQUATION; QUANTUM INFORMATION; TIME INDEPENDENTS;

QUANTUM OPTICS; QUANTUM THEORY;

FEEDBACK CONTROL;

EID: 84897696628     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.89.022118     Document Type: Article

References (81)

1. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17, 821 (1976); JMAPAQ 0022-2488 10.1063/1.522979 1b G. Lindblad, Commun. Math. Phys. 48, 119 (1976). CMPHAY 0010-3616 10.1007/BF01608499
2. F. Breuer and H.-P. Petruccione, Theory of Open Quantum Systems (Oxford University Press, New York, 2002).
3. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2009).
4. K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. Girvin, and I. Siddiqi, Phys. Rev. Lett. 109, 183602 (2012); PRLTAO 0031-9007 10.1103/PhysRevLett.109. 183602 4b Z. Leghtas, U. Vool, S. Shankar, M. Hatridge, S. M. Girvin, M. H. Devoret, and M. Mirrahimi, Phys. Rev. A 88, 023849 (2013). PLRAAN 1050-2947 10.1103/PhysRevA.88.023849
5. A. Sarlette, Z. Leghtas, M. Brune, J. M. Raimond, and P. Rouchon, Phys. Rev. A 86, 012114 (2012). PLRAAN 1050-2947 10.1103/PhysRevA.86.012114
6. J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, Nature (London) 470, 486 (2011). NATUAS 0028-0836 10.1038/nature09801
7. A. Tomadin, S. Diehl, M. D. Lukin, P. Rabl, and P. Zoller, Phys. Rev. A 86, 033821 (2012). PLRAAN 1050-2947 10.1103/PhysRevA.86.033821
8. C.-E. Bardyn, M. A. Baranov, C. V. Kraus, E. Rico, A. İmamoǧlu, P. Zoller, and S. Diehl, New J. Phys. 15, 085001 (2013). 10.1088/1367-2630/15/8/085001
9. S. Diehl, E. Rico, M. A. Baranov, and P. Zoller, Nat. Phys. 7, 971 (2011); NJOPFM 1745-2473 10.1038/nphys2106 9b C. V. Kraus, S. Diehl, P. Zoller, and M. A. Baranov, New J. Phys. 14, 113036 (2012). 1367-2630 10.1088/1367-2630/14/11/113036
10. C.-E. Bardyn, M. A. Baranov, E. Rico, A. Imamoglu, P. Zoller, and S. Diehl, Phys. Rev. Lett. 109, 130402 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.109.130402
11. O. Viyuela, A. Rivas, and M. A. Martin-Delgado, Phys. Rev. B 86, 155140 (2012). PRBMDO 1098-0121 10.1103/PhysRevB.86.155140
12. A. Rivas, O. Viyuela, and M. A. Martin-Delgado, Phys. Rev. B 88, 155141 (2013). PRBMDO 1098-0121 10.1103/PhysRevB.88.155141
13. T. Prosen, Phys. Rev. Lett. 106, 217206 (2011); PRLTAO 0031-9007 10.1103/PhysRevLett.106.217206 13b T. Prosen, Phys. Rev. Lett. 107, 137201 (2011); PRLTAO 0031-9007 10.1103/PhysRevLett.107.137201 13c T. Prosen Phys. Rev. Lett. 112, 030603 (2014). PRLTAO 0031-9007 10.1103/PhysRevLett.112.030603
14. V. Popkov and M. Salerno, J. Stat. Mech.-Theory E (2013) P02040; 10.1088/1742-5468/2013/02/P02040 14b J. J. Mendoza-Arenas, S. Al-Assam, S. R. Clark, and D. Jaksch, J. Stat. Mech. (2013) P07007; 10.1088/1742-5468/2013/07/ P07007 14c E. Ilievski and T. Prosen, arXiv:1402.0342.
15. V. Popkov and R. Livi, New J. Phys. 15, 023030 (2013). NJOPFM 1367-2630 10.1088/1367-2630/15/2/023030
16. Z. Cai and T. Barthel, Phys. Rev. Lett. 111, 150403 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.150403
17. S. Diehl, A. Tomadin, A. Micheli, R. Fazio, and P. Zoller, Phys. Rev. Lett. 105, 015702 (2010). PRLTAO 0031-9007 10.1103/PhysRevLett.105.015702
18. M. Foss-Feig, K. R. A. Hazzard, J. J. Bollinger, and A. M. Rey, Phys. Rev. A 87, 042101 (2013). PLRAAN 1050-2947 10.1103/PhysRevA.87.042101
19. S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and P. Zoller, Nat. Phys. 4, 878 (2008); NJOPFM 1745-2473 10.1038/nphys1073 19b W. Yi, S. Diehl, A. J. Daley, and P. Zoller, New J. Phys. 14, 055002 (2012). 1367-2630 10.1088/1367-2630/14/5/055002
20. A. Pechen and H. Rabitz, Phys. Rev. A 73, 062102 (2006); PLRAAN 1050-2947 10.1103/PhysRevA.73.062102 20b B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller, Phys. Rev. A 78, 042307 (2008); PLRAAN 1050-2947 10.1103/PhysRevA.78.042307 20c F. Ticozzi and L. Viola, Automatica 45, 2002 (2009); ATCAA9 0005-1098 10.1016/j.automatica.2009.05.005 20d W. Wang, L. C. Wang, and X. X. Yi, Phys. Rev. A 82, 034308 (2010); PLRAAN 1050-2947 10.1103/PhysRevA.82.034308 20e A. Pechen, Phys. Rev. A 84, 042106 (2011); PLRAAN 1050-2947 10.1103/PhysRevA.84.042106 20f F. Ticozzi and L. Viola, Philos. Trans. R. Soc., A 370, 5259 (2012). PTRMAD 1364-503X 10.1098/rsta.2011.0485 (Pubitemid 43839385)
21. F. Verstraete, M. M. Wolf, and J. I. Cirac, Nat. Phys. 5, 633 (2009). 1745-2473 10.1038/nphys1342
22. S. G. Schirmer and X. Wang, Phys. Rev. A 81, 062306 (2010). PLRAAN 1050-2947 10.1103/PhysRevA.81.062306
23. S. Sauer, C. Gneiting, and A. Buchleitner, Phys. Rev. Lett. 111, 030405 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.030405
24. M. Müller, S. Diehl, G. Pupillo, and P. Zoller, in Advances in Atomic, Molecular, and Optical Physics (Academic Press, San Diego, 2012), Vol. 61, pp. 1-80.
25. W. H. Zurek, Phys. Rev. D 24, 1516 (1981); RMPHAT 0556-2821 10.1103/PhysRevD.24.1516 25b W. H. Zurek, Phys. Rev. D 26, 1862 (1982); 0556-2821 10.1103/PhysRevD.26.1862 25c W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003). 0034-6861 10.1103/RevModPhys.75.715
26. P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). PRLTAO 0031-9007 10.1103/PhysRevLett.79.3306 26b P. Zanardi, Phys. Rev. A 57, 3276 (1998). PLRAAN 1050-2947 10.1103/PhysRevA.57.3276
27. D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998). PRLTAO 0031-9007 10.1103/PhysRevLett.81.2594
28. J. Kempe, D. Bacon, D. A. Lidar, and K. B. Whaley, Phys. Rev. A 63, 042307 (2001). PLRAAN 1050-2947 10.1103/PhysRevA.63.042307 (Pubitemid 33722121)
29. D. A. Lidar and K. B. Whaley, in Irreversible Quantum Dynamics, edited by F. Benatti and R. Floreanini,. Lecture Notes in Physics Vol. 622 (Springer, Berlin, Heidelberg, 2003), Chap. 5, p. 83.
30. E. Knill, R. Laflamme, and L. Viola, Phys. Rev. Lett. 84, 2525 (2000). PRLTAO 0031-9007 10.1103/PhysRevLett.84.2525
31. P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, Science 290, 498 (2000); SCIEAS 0036-8075 10.1126/science.290.5491.498 31b D. A. R. Dalvit, J. Dziarmaga, and W. H. Zurek, Phys. Rev. A 62, 013607 (2000); PLRAAN 1050-2947 10.1103/PhysRevA.62.013607 31c L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. Laflamme, and D. G. Cory, Science 293, 2059 (2001); SCIEAS 0036-8075 10.1126/science.1064460 31d D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Science 291, 1013 (2001). SCIEAS 0036-8075 10.1126/science.1057357
32. F. Ticozzi and L. Viola, IEEE Trans. Autom. Control 53, 2048 (2008). IETAA9 0018-9286 10.1109/TAC.2008.929399
33. R. Blume-Kohout, H. K. Ng, D. Poulin, and L. Viola, Phys. Rev. A 82, 062306 (2010). PLRAAN 1050-2947 10.1103/PhysRevA.82.062306
34. B. Buča and T. Prosen, New J. Phys. 14, 073007 (2012). NJOPFM 1367-2630 10.1088/1367-2630/14/7/073007
35. D. Manzano and P. I. Hurtado, arXiv:1310.7370.
36. A. Frigerio, Lett. Math. Phys. 2, 79 (1977); LMPHDY 0377-9017 10.1007/BF00398571 36b H. Spohn, Lett. Math. Phys. 2, 33 (1977); LMPHDY 0377-9017 10.1007/BF00420668 36c D. E. Evans, Commun. Math. Phys. 54, 293 (1977). CMPHAY 0010-3616 10.1007/BF01614091
37. K. Alicki and R. Lendi, Quantum Dynamical Semigroups and Applications (Springer-Verlag, Berlin/Heidelberg, 2007); 37b A. Rivas and S. Huelga, Open Quantum Systems (Springer, Berlin/Heidelberg, 2012).
38. F. Fagnola and R. Rebolledo, Infinite Dimens. Anal. Quantum Prob. Relat. Top. 11, 467 (2008); 0219-0257 10.1142/S0219025708003142 38b A. Dhahri, F. Fagnola, and R. Rebolledo, Infinite Dimens. Anal. Quantum Prob. Relat. Top. 13, 413 (2010); 0219-0257 10.1142/S0219025710004176 38c P. Rouchon and A. Sarlette, arXiv:1302.6899.
39. H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer-Verlag, Berlin/Heidelberg, 1999).
40. B. Baumgartner, J. Phys. A: Math. Theor. 41, 395303 (2008). 1751-8113 10.1088/1751-8113/41/39/395303
41. G. Teschl, Ordinary Differential Equations and Dynamical Systems (American Mathematical Society, Providence, RI, 2012).
42. W. Weidlich, Z. Phys. 241, 325 (1971). EPJAFV 1434-6001 10.1007/BF01395429
43. H.-J. Briegel and B.-G. Englert, Phys. Rev. A 47, 3311 (1993); PLRAAN 1050-2947 10.1103/PhysRevA.47.3311 43b H.-J. Briegel, B.-G. Englert, C. Ginzel, and A. Schenzle, Phys. Rev. A 49, 5019 (1994); PLRAAN 1050-2947 10.1103/PhysRevA.49.5019 43c S. M. Barnett and S. Stenholm, J. Mod. Opt. 47, 2869 (2000). 10.1080/09500340008232202 (Pubitemid 23667249)
44. B.-G. Englert and G. Morigi, in Coherent Evolution in Noisy Environments, edited by A. Buchleitner and K. Hornberger, Lecture Notes in Physics Vol. 611 (Springer, Berlin/Heidelberg, 2002), Chap. 2, pp. 55-106.
45. B. Baumgartner, H. Narnhofer, and W. Thirring, J. Phys. A: Math. Theor. 41, 065201 (2008). 10.1088/1751-8113/41/6/065201
46. B. Baumgartner and H. Narnhofer, Rev. Math. Phys. 24, 1250001 (2012). RMPHEX 0129-055X 10.1142/S0129055X12500018
47. R. R. Puri, Mathematical Methods of Quantum Optics (Springer, Berlin/Heidelberg, 2001).
48. A. Barchielli and M. Gregoratti, Quantum Trajectories and Measurements in Continuous Time: The Diffusive Case (Springer, Berlin/Heidelberg, 2009).
49. A. V. Sokolov, A. A. Andrianov, and F. Cannata, J. Phys. A: Math. Gen. 39, 10207 (2006). JPHAC5 0305-4470 10.1088/0305-4470/39/32/S20 (Pubitemid 44159647)
50. A. Maassen van den Brink and K. Young, J. Phys. A: Math. Gen. 34, 2607 (2001). JPHAC5 0305-4470 10.1088/0305-4470/34/12/308
51. T. Prosen, New J. Phys. 10, 043026 (2008). NJOPFM 1367-2630 10.1088/1367-2630/10/4/043026
52. T. Prosen, J. Stat. Mech.-Theory E (2010) P07020. 10.1088/1742-5468/2010/ 07/P07020
53. T. Prosen and T. H. Seligman, J. Phys. A: Math. Theor. 43, 392004 (2010). 1751-8113 10.1088/1751-8113/43/39/392004
54. M. Jakob and S. Stenholm, Phys. Rev. A 69, 042105 (2004). PLRAAN 1050-2947 10.1103/PhysRevA.69.042105
55. R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Clarendon Press, Oxford, 1987).
56. V. E. Tarasov, Quantum Mechanics of Non-Hamiltonian and Dissipative Systems (Elsevier, Amsterdam, 2008).
57. A. B. Klimov and S. M. Chumakov, A Group-Theoretical Approach to Quantum Optics (Wiley, Weinheim, 2009).
58. C. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin/Heidelberg, 2000).
59. G. Kuperberg, IEEE T. Inf. Theory 49, 1465 (2003). IETTAW 0018-9448 10.1109/TIT.2003.811917
60. M. T. Vaughn, Introduction to Mathematical Physics (Wiley, Weinheim, 2007).
61. G. Heckman, Lie Algebras in Mathematics and Physics (unpublished), p. 8, http://www.math.ru.nl/~heckman/Lie.pdf.
62. S. G. Schirmer, in Proceedings of the 40th IEEE Conference on Decision and Control (IEEE, Piscataway, NJ, 2001), Vol. 1, p. 298.
63. T. Prosen, Phys. Rev. Lett. 109, 090404 (2012). PRLTAO 0031-9007 10.1103/PhysRevLett.109.090404
64. T. Prosen, Phys. Rev. A 86, 044103 (2012). PLRAAN 1050-2947 10.1103/PhysRevA.86.044103
65. M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, arXiv:1312.2017.
66. H. D. Simaan and R. Loudon, J. Phys. A: Math. Gen. 8, 539 (1975); JPHAC5 0305-4470 10.1088/0305-4470/8/4/016 66b L. Gilles and P. L. Knight, Phys. Rev. A 48, 1582 (1993). PLRAAN 1050-2947 10.1103/PhysRevA.48.1582
67. H. D. Simaan and R. Loudon, J. Phys. A: Math. Gen. 11, 435 (1978). JPHAC5 0305-4470 10.1088/0305-4470/11/2/018
68. H. Voigt, A. Bandilla, and H. H. Ritze, Z. Phys. B: Cond. Mat. 36, 295 (1980); ZPBBDJ 0340-224X 10.1007/BF01325293 68b M. S. Zubairy, J. Math. Phys. 21, 2690 (1980); JMAPAQ 0022-2488 10.1063/1.524371 68c A. B. Klimov and J. L. Romero, J. Opt. B: Quantum Semiclassical Opt. 5, S316 (2003). JOBOFD 1464-4266 10.1088/1464-4266/5/3/363
69. R. Loudon, Opt. Commun. 49, 67 (1984); OPCOB8 0030-4018 10.1016/0030-4018(84)90092-0 69b G. S. Agarwal, Opt. Commun. 62, 190 (1987); OPCOB8 0030-4018 10.1016/0030-4018(87)90025-3 69c E. E. Hach III and C. C. Gerry, Phys. Rev. A 49, 490 (1994); PLRAAN 1050-2947 10.1103/PhysRevA.49.490 69d V. V. Dodonov and S. S. Mizrahi, J. Phys. A: Math. Gen. 30, 5657 (1997). JPHAC5 0305-4470 10.1088/0305-4470/30/16/010 (Pubitemid 14553748)
70. M. J. Everitt, T. P. Spiller, G. J. Milburn, R. D. Wilson, and A. M. Zagoskin, arXiv:1212.4795.
71. A. Nunnenkamp, K. Borkje, and S. M. Girvin, Phys. Rev. A 85, 051803 (2012); PLRAAN 1050-2947 10.1103/PhysRevA.85.051803 71b K. Borkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, Phys. Rev. Lett. 111, 053603 (2013). PRLTAO 0031-9007 10.1103/PhysRevLett.111.053603
72. A. Voje, A. Croy, and A. Isacsson, New J. Phys. 15, 053041 (2013). NJOPFM 1367-2630 10.1088/1367-2630/15/5/053041
73. G. Sarma and H. Mabuchi, New J. Phys. 15, 035014 (2013). NJOPFM 1367-2630 10.1088/1367-2630/15/3/035014
74. H. Gould and J. Quaintance, Math. Mag. 85, 177 (2012). MAMGA8 0025-570X 10.4169/math.mag.85.3.177
75. M. Petkovsek, H. Wilf, and D. Zeilberger, A=B (AK Peters, Ltd., Natick, MA, 1997), p. 34.
76. NIST Digital Library of Mathematical Functions, Release 1.0.6, http://dlmf.nist.gov.
77. V. E. Tarasov, J. Phys. A: Math. Gen. 35, 5207 (2002). JPHAC5 0305-4470 10.1088/0305-4470/35/25/305 (Pubitemid 37023764)
78. A. Shabani and D. A. Lidar, Phys. Rev. A 72, 042303 (2005). PLRAAN 1050-2947 10.1103/PhysRevA.72.042303 (Pubitemid 41776290)
79. P. G. Brooke, J. D. Cresser, and M. K. Patra, Phys. Rev. A 77, 062313 (2008). PLRAAN 1050-2947 10.1103/PhysRevA.77.062313
80. H. M. Wiseman and G. J. Milburn, Phys. Rev. Lett. 70, 548 (1993). PRLTAO 0031-9007 10.1103/PhysRevLett.70.548 (Pubitemid 23684263)
81. J. Wang and H. M. Wiseman, Phys. Rev. A 64, 063810 (2001). PLRAAN 1050-2947 10.1103/PhysRevA.64.063810