study of correlations and non-markovianity in dephasing open quantum systems università degli studi...

STUDY OF CORRELATIONS AND NON-MARKOVIANITY

IN DEPHASING OPEN QUANTUM SYSTEMS

Università degli Studi di Milano

Giacomo GUARNIERI Supervisor: Bassano VACCHINI

Co-Supervisor: Matteo PARIS

PhD school of Physics, XXIX Cicle

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

CONTENTS

System𝜌𝑆


HT = HS


HT = HS + HE

System𝜌𝑆

Environment𝜌𝐸


Interaction Interaction

HT = HS + HE + HI U(t,0)= exp [i HT t]

Environment𝜌𝐸

System𝜌𝑆


Λ (𝑡 )

PROPERTIES OF

COMPLETE POSITIVITY (CP)

TRACE PRESERVING (T)

Λ (𝑡 )





CONTENTS


𝐷 (𝜌𝑆1 (𝑡 ) ,𝜌𝑆

2 (𝑡 ) )

System 1

Time (arbitrary units)

D(t;ρ1,2) Environment

The trace distance measures the distinguishability between states

System 2

Physical interpretation: evolution of information flow between S and E

System 1

Environment

System 2



D(t;ρ1,2)



2 (𝑡 ) )


System 1

Environment

System 2




D(t;ρ1,2)


2 (𝑡 ) )



Backflow of information

Memory effects

non-Markovian dynamics

ρ1,2

Ω

𝑁 ( Λ )=𝑚𝑎𝑥∫ 𝑑𝑑𝑡

𝐷 ( ρ𝑆1 (𝑡 ) , ρ𝑆2 (𝑡 ) )𝑑𝑡

H.P. Breuer, E. M. Laine, J. Piilo, PRL 103, 210401 (2009)B. H. Liu et al. , Nat. Phys. 7, 931 (2011)





CONTENTS


⟨ 𝐴 (𝑡 ) ⟩=𝑇𝑟 𝑆𝐸 [𝑈 (𝑡 ,0 ) 𝐴𝑈 † (𝑡 ,0 ) ρ𝑆𝐸 ]

⟨ 𝐴 (𝑡 1 ) 𝐵 (𝑡 2 ) ⟩=𝑇𝑟 𝑆𝐸 [𝑈 † (𝑡 1 ,0 ) 𝐴𝑈 (𝑡 1 ,0 )𝑈 † (𝑡 2 ,0 )𝐵𝑈 (𝑡 2 ,0 ) ρ𝑆𝐸 ]

Mean Values :

Two-time CF:


⟨ 𝐴 (𝑡 ) ⟩=𝑇𝑟 𝑆𝐸 [𝑈 (𝑡 ,0 ) 𝐴𝑈 † (𝑡 ,0 ) ρ𝑆𝐸 ]


Mean Values :

Two-time CF:

¿𝑇𝑟𝑆 [ 𝐴 ρ𝑆(𝑡)]

¿𝑇𝑟𝑆 [ 𝐴 X𝑆(𝑡 1 ,𝑡 2) ] ≠

= = with

with

‘EASY’ !

VERY DIFFICULT!

?Two-time correlation functions and Quantum Regression Theorem

⟨ 𝐴 (𝑡 1 ) 𝐵 (𝑡 2 ) ⟩Two-time CF:

Quantum World

⟨ 𝐴 (𝑡 ) ⟩Mean Values :

Quantum non-Markovian dynamics

relationbetween

&

Classical World of Stochastic Processes

Markovian process iff


Quantum Regression Theorem!

⟨ 𝐴 (𝑡 ) ⟩=𝑇𝑟 𝑆𝐸 [𝑈 (𝑡 ,0 ) 𝐴𝑈 † (𝑡 ,0 ) ρ𝑆𝐸 ]


Mean Values :

Two-time CF:

M. Lax, Phys. Rev 172, 350 (1968)


𝑑𝑑𝑡 ⟨𝐴𝑖 (𝑡 ) ⟩=∑

𝑗

𝐺𝑖𝑗 ⟨ 𝐴 𝑗 (𝑡 ) ⟩

𝑑𝑑 τ

¿

Closed and linear systemof evolution equations

for mean values

{ 𝐴𝑖 (𝑡 ) }𝑖=1,… ,𝑁2 ,𝑇𝑟 [𝐴𝑖† 𝐴 𝑗 ]=δ 𝑖 , 𝑗Basis for the space of operators


𝑑𝑑𝑡 ⟨𝐴𝑖 (𝑡 ) ⟩=∑

𝑗

𝐺𝑖𝑗 ⟨ 𝐴 𝑗 (𝑡 ) ⟩

𝑑𝑑 τ

¿

Main assumption for the validity of QRT:

ρ𝑆𝐸 (𝑡 )=ρ𝑆 (𝑡 )⊗ ρ𝐸 (0 )

Closed and linear systemof evolution equations

for mean values

{ 𝐴𝑖 (𝑡 ) }𝑖=1,… ,𝑁2 ,𝑇𝑟 [𝐴𝑖† 𝐴 𝑗 ]=δ 𝑖 , 𝑗

In most physical situation this assumption is never strictly satisfied!!

ρ𝑆𝐸 (𝑡 )≈ ρ𝑆 (𝑡 )⊗ ρ𝐸 (0 )Search for regions of parameters where

Basis for the space of operators

Question to answer

Quantum non-

Markovianity

Quantum Regression Theorem

?





CONTENTS

Pure Dephasing Spin-Boson System

Two-level system coupled to an infinite number of bosonic modes

The populations of the system are constants of motion!ρ00 , ρ11

G.G. , A. Smirne, B. Vacchini , PRA 90, 022110 (2014)

Pure Dephasing Spin-Boson System

and therefore the reduced system state has the analytic form

where

The action of the total evolution operator can be explicitly evaluated

( 𝑗=0,1 )

is the DECOHERENCE FUNCTION.

Pure dephasing spin-boson model

OhmicityCoupling strength Cutoff frequency

Thermal BathIf we assume...

Spectral Density

the decoherence function, in the limit β → +∞, becomes



Non-Markovianity condition for the pure dephasing spin-boson

= 2 such that

for the dynamicsis Markovian

for the dynamics

Isnon-Markovian

Indipendently of λ !!


Non-Markovianity condition for the pure dephasing spin-boson

= 2 such that

for the dynamicsis Markovian

for the dynamics

Isnon-Markovian

Increasing s!

Indipendently of λ !!The degree of non-Markovianity though

depends on λ

Pure dephasing spin-boson modelTwo-time Correlation Functions

& Quantum Regression Theorem

To estimate the violations to the QRT

by computing theRELATIVE ERROR

S=1.5

ρ𝑆𝐸 (𝑡 )≠ ρ𝑆 (𝑡 )⊗ ρ𝐸 (0 )

Pure dephasing spin-boson modelTwo-time Correlation Functions

& Quantum Regression Theorem

S=1.5< scrit!

To estimate the violations to the QRT

by computing theRELATIVE ERROR

ρ𝑆𝐸 (𝑡 )≠ ρ𝑆 (𝑡 )⊗ ρ𝐸 (0 )


CONCLUSIONS

≠Quantum non-Markovianity

Quantum

Regression

Theorem

Related to theEFFECTS OF

CORRELATIONSbetween S and E at

the level of the reduced dynamics

Related to theACTUAL

CORRELATIONSbetween S and E at

the level of the overall dynamics


THANK YOU

study of correlations and non-markovianity in dephasing open quantum systems università degli studi...

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