Study of Correlations and Non-Markovianity in Open Quantum Systems

universit degli studi di milano n.w
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Explore correlations and non-Markovian behavior in dephasing open quantum systems, focusing on two-time correlation functions, Quantum Regression Theorem, and a case study on the Pure Dephasing Spin-Boson System within the realm of Open Quantum Systems Theory.

  • Quantum Systems
  • Markovianity
  • Correlations
  • Open Systems
  • Quantum Theory
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  1. Universit degli Studi di Milano STUDY OF CORRELATIONS AND NON-MARKOVIANITY IN DEPHASING OPEN QUANTUM SYSTEMS Giacomo GUARNIERI Supervisor: Bassano VACCHINI Co-Supervisor: Matteo PARIS PhD school of Physics, XXIX Cicle

  2. CONTENTS CONTENTS Open Quantum Systems Theory Open Quantum Systems Theory Quantum non Quantum non- -Markovianity Markovianity Two Two- -time correlation functions and time correlation functions and Quantum Regression Theorem Quantum Regression Theorem A case of study: A case of study: The Pure Dephasing Spin The Pure Dephasing Spin- -Boson System Boson System

  3. CONTENTS CONTENTS Open Quantum Systems Theory Open Quantum Systems Theory Quantum non-Markovianity Two-time correlation functions and Quantum Regression Theorem A case of study: The Pure Dephasing Spin-Boson System

  4. Open Open Quantum Quantum Systems Theory Systems Theory System System ?? HT= HS

  5. Open Quantum Systems Theory Open Quantum Systems Theory Environment Environment ?? System System ?? HT= HS + HE

  6. Open Quantum Systems Theory Open Quantum Systems Theory Environment Environment ?? System System ?? Interaction Interaction Interaction Interaction U(t,0)= exp [iHTt] HT= HS + HE + HI

  7. Open Quantum Systems Theory Open Quantum Systems Theory ? PROPERTIES OF PROPERTIES OF ? COMPLETE POSITIVITY (CP) COMPLETE POSITIVITY (CP) TRACE PRESERVING (T) TRACE PRESERVING (T)

  8. CONTENTS CONTENTS Open Quantum Systems Theory Quantum non Quantum non- -Markovianity Markovianity Two-time correlation functions and Quantum Regression Theorem A case of study: The Pure Dephasing Spin-Boson System

  9. Quantum non Quantum non- -Markovianity Markovianity The trace distance The trace distance measures the distinguishability between states measures the distinguishability between states 1? ,?? 2? ? ?? Physical interpretation: evolution of information flow between S and E Physical interpretation: evolution of information flow between S and E Environment Environment D(t; 1,2) System 1 System 1 System 2 System 2 Time Time (arbitrary units) (arbitrary units)

  10. Quantum non Quantum non- -Markovianity Markovianity The trace distance The trace distance measures the distinguishability between states measures the distinguishability between states 1? ,?? 2? ? ?? Physical interpretation: evolution of information flow between S and E Physical interpretation: evolution of information flow between S and E Environment Environment D(t; 1,2) System 1 System 1 System 2 System 2 Time Time (arbitrary units) (arbitrary units)

  11. Quantum non Quantum non- -Markovianity Markovianity The trace distance The trace distance measures the distinguishability between states measures the distinguishability between states 1? ,?? 2? ? ?? Physical interpretation: evolution of information flow between S and E Physical interpretation: evolution of information flow between S and E Environment Environment D(t; 1,2) System 1 System 1 System 2 System 2 Time Time (arbitrary units) (arbitrary units)

  12. Quantum non Quantum non- -Markovianity Markovianity Backflow Backflow of information of information Memory effects Memory effects non non- -Markovian dynamics Markovian dynamics ? = ??? ? 1? , ? 2? ??? ? ?? 1,2 H.P. Breuer, E. M. Laine, J. Piilo, PRL 103, 210401 (2009) B. H. Liu et al. , Nat. Phys. 7, 931 (2011)

  13. CONTENTS CONTENTS Open Quantum Systems Theory Quantum non-Markovianity Two Two- -time correlation functions and time correlation functions and Quantum Regression Theorem Quantum Regression Theorem A case of study: The Pure Dephasing Spin-Boson System

  14. Two Two- -time correlation functions and Quantum Regression Theorem time correlation functions and Quantum Regression Theorem Mean Values : Mean Values : = ????? ?,0 ?? ?,0 ?? ? ? Two Two- -time time CF: CF: = ????? ?1,0 ? ? ?1,0 ? ?2,0 ? ? ?2,0 ?? ? ?1? ?2

  15. Two Two- -time correlation functions and Quantum Regression Theorem time correlation functions and Quantum Regression Theorem Mean Values : Mean Values : = ????? ?,0 ?? ?,0 ?? ? ? = ??? ? ?(?) ??0 ? (?,0) = ? ?0 with with ?(?) = ???? ?,0 EASY ! EASY ! Two Two- -time time CF: CF: = ????? ?1,0 ? ? ?1,0 ? ?2,0 ? ? ?2,0 ?? ? ?1? ?2 = ??? ? X?(?1,?2) X?(?1,?2) = ??? ? ?1,?2 ? ???1? (?1,?2) ?1,?2 B ??1 with with VERY DIFFICULT! VERY DIFFICULT!

  16. Two Two- -time correlation functions and Quantum Regression Theorem time correlation functions and Quantum Regression Theorem Classical World Classical World of Stochastic Stochastic Processes Quantum World Quantum World of Processes Quantum Quantumnon non- -Markovian Markovian dynamics dynamics Markovian process Markovian process iff iff & Mean Values : Mean Values : ? ? relation relation between between Two Two- -time time CF: CF: ? ?1? ?2

  17. Two Two- -time correlation functions and Quantum Regression Theorem time correlation functions and Quantum Regression Theorem Mean Values : Mean Values : = ????? ?,0 ?? ?,0 ?? ? ? Quantum Regression Theorem! Quantum Regression Theorem! Two Two- -time time CF: CF: = ????? ?1,0 ? ? ?1,0 ? ?2,0 ? ? ?2,0 ?? ? ?1? ?2 M. Lax, Phys. Rev 172, 350 (1968)

  18. Two Two- -time correlation functions and Quantum Regression Theorem time correlation functions and Quantum Regression Theorem Basis for the space of operators Basis for the space of operators ?? = ?,? ??? ?=1, ,?2,?? ?? ? ????? = ??????? Closed and linear system Closed and linear system of evolution equations of evolution equations for mean values for mean values ? ? ???1??(t2= ???????1???2

  19. Two Two- -time correlation functions and Quantum Regression Theorem time correlation functions and Quantum Regression Theorem Basis for the space of operators Basis for the space of operators ?? = ?,? ??? ?=1, ,?2,?? ?? ? ????? = ??????? Closed and linear system Closed and linear system of evolution equations of evolution equations for mean values for mean values ? ? ???1??(t2= ???????1???2 Main assumption for the validity of QRT: Main assumption for the validity of QRT: ??? = ?? ?0 In most physical situation this assumption is never strictly satisfied!! In most physical situation this assumption is never strictly satisfied!! Search for regions of parameters where Search for regions of parameters where ??? ?? ?0

  20. Question to answer Question to answer ? ? Quantum Quantum Regression Regression Theorem Theorem Quantum non Quantum non- - Markovianity Markovianity

  21. CONTENTS CONTENTS Open Quantum Systems Theory Quantum non-Markovianity Two-time correlation functions and Quantum Regression Theorem A case of study: A case of study: The Pure Dephasing Spin The Pure Dephasing Spin- -Boson System Boson System

  22. Pure Dephasing Spin Pure Dephasing Spin- -Boson System Boson System Two Two- -level system coupled to an infinite number of bosonic modes level system coupled to an infinite number of bosonic modes The populations The populations of the system are constants of motion! of the system are constants of motion! 00, 11 G.G. , A. Smirne, B. Vacchini , PRA 90, 022110 (2014)

  23. Pure Dephasing Spin Pure Dephasing Spin- -Boson System Boson System The action of the total evolution operator can be explicitly evaluated The action of the total evolution operator can be explicitly evaluated ? = 0,1 and therefore the reduced system state has the analytic form and therefore the reduced system state has the analytic form where where is the is the DECOHERENCE FUNCTION DECOHERENCE FUNCTION. .

  24. Pure dephasing spin Pure dephasing spin- -boson model boson model If we assume... If we assume... Thermal Bath Thermal Bath Spectral Density Spectral Density Ohmicity Ohmicity Coupling strength Coupling strength Cutoff frequency Cutoff frequency the decoherence function, in the limit the decoherence function, in the limit + , becomes becomes G.G. , A. Smirne, B. Vacchini , PRA 90, 022110 (2014)

  25. Pure dephasing spin Pure dephasing spin- -boson model boson model Non Non- -Markovianity condition Markovianity condition for the pure dephasing spin for the pure dephasing spin- -boson boson ????? = = 2 such such that that for for ? > ????? the dynamics the dynamics Is Is non non- -Markovian Markovian for for ? < ????? the dynamics the dynamics is is Markovian Markovian Indipendently of Indipendently of !! !!

  26. Pure dephasing spin Pure dephasing spin- -boson model boson model Non Non- -Markovianity condition Markovianity condition for the pure dephasing spin for the pure dephasing spin- -boson boson ????? = = 2 such such that that for for ? > ????? the dynamics the dynamics Is Is non non- -Markovian Markovian for for ? < ????? the dynamics the dynamics is is Markovian Markovian Indipendently of Indipendently of !! !! The degree of non The degree of non- -Markovianity though Markovianity though depends on depends on

  27. Pure dephasing spin Pure dephasing spin- -boson model boson model Two Two- -time Correlation Functions time Correlation Functions & & Quantum Quantum Regression Theorem Regression Theorem ??? ?? ?0 To estimate the violations to To estimate the violations to the QRT by computing the the QRT by computing the RELATIVE ERROR RELATIVE ERROR S=1.5

  28. Pure dephasing spin Pure dephasing spin- -boson model boson model Two Two- -time Correlation Functions time Correlation Functions & & Quantum Quantum Regression Theorem Regression Theorem ??? ?? ?0 To estimate the violations to To estimate the violations to the QRT by computing the the QRT by computing the RELATIVE ERROR RELATIVE ERROR S=1.5 < scrit!

  29. Pure dephasing spin Pure dephasing spin- -boson model boson model CONCLUSIONS CONCLUSIONS Quantum Quantum non non- - Markovia Markovia nity nity Quantum Quantum Regression Regression Theorem Theorem Related to the Related to the EFFECTS OF EFFECTS OF CORRELATIONS CORRELATIONS between S and E between S and E at the the reduced the reduced dynamics Related to the Related to the ACTUAL CORRELATIONS ACTUAL CORRELATIONS between S and E between S and E at the the overall dynamics the overall dynamics at the level of level of at the level of level of dynamics G.G. , A. Smirne, B. Vacchini , PRA 90, 022110 (2014)

  30. THANK YOU THANK YOU

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