Non-Non-MarkovianMarkovian Open OpenQuantum SystemsQuantum Systems

Sabrina ManiscalcoQuantum Optics Group, Department of Physics

University of Turku

Contents

Motivation Theoretical approaches to the study of

open systems dynamics Markovian approximation and

Lindblad Master Equation Non-Markovian systems Examples: Quantum Brownian Motion and two-level atom

Open Quantum Systems

The theory of open quantum systems describes the interactionof a quantum system with its environment

Quantum MechanicsQuantum Mechanicsclosed systems

unitary dynamicsreversible dynamics

[ ]!!ˆ,ˆ

ˆH

dt

di =h

Liouville – von Neumann Equation

!=!

Hdt

di ˆh

Schrödinger Equation

open quantum systemsopen quantum systems

reduced density operator

( ) ( )[ ]tTrtTES!! ˆˆ =

non-unitary and irreversible dynamics

master equation

!!

ˆˆL

dt

d=

Motivation

Fundamentals of Quantum Theory

Quantum Technologies

Quantum Measurement TheoryQuantum Measurement Theory

Border between quantumand classical descriptions

DecoherenceDecoherence

Entanglement between the degrees of freedom of the system and those

of the enviornment

PROBLEM: All new quantum technologies rely on quantumcoherence Qubit

quantum computationquantum cryptography

quantum communication

damping basis method1

green function algebraic superoperatorial methods2

1 H.-J. Briegel, B.-G. Englert, PRA 47, 3311 (1993)2 F. Intravaia, S.M., A. Messina, PRA 67, 042108 (2003)

Approaches to the dynamics ofopen quantum systems

microscopic approachmicroscopic approach phenomenological approachphenomenological approach

Derivation of an equation of motion for the reduced density matrix of the system of interest: The master equation

Microscopic hamiltonian for the total closed system (model of environment and interaction)

Trace over the environmental degrees of freedom Approximations

Solution of the master equation

analytical methodsanalytical methods numerical methodsnumerical methods

Monte Carlo wave function (and variants)3,4

Influence functional – path integral5

3 J. Dalibard, Y. Castin and K. Molmer, Phys. Rev.Lett. 68, 580 (1992)4 J. Piilo, S.M., A. Messina, F. Petruccione, PRE 71, 056701 (2005)5 U. Weiss, Quantum Dissipative Systems 2nd ed. (World Scientific, Singapore, 1999).

Approximations

weak coupling approximation

Markovian approximation

weak coupling between system and environment

perturbative approaches (expansions in the coupling constant)

assumes that the reservoir correlation time is much smallerthan the relaxation time of the open system

changes in the reservoir due to the interaction with the system do not feed back into the system

coarse-graining in time

Lindblad master equation 1,2

[ ] { }! "#

$%&

'(+(=

j

jjjj VVVVHidt

d ††,

2

1, )))

)

Weak coupling + Markovian approximation + RWA or secular approx.

Vj and Vj†transition or jump

operators depending on the specific physical system

ρ density matrix of the reducedsystem

ADVANTAGES[ ] simulation techniques

important properties( ) ( )0!!

tt "=

dynamical mapLt

te=!

Markovian dynamical map

SemigroupSemigroup property: property: [ ] [ ]!!tttt ""+ ##=# o

Positivity: ( ) ( )[ ] ( ) HB 0,0 00 iff : !"#$%#$ &&&tt

1 G. Lindblad, Comm. Math. Phys. 48, 119 (1976)2 V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)

Positivity andComplete positivity

POSITIVITY

COMPLETE POSITIVITY Nn positive is )HB(H )HB(H :

iff positive completely B(H) B(H) :

nn !"#$##%

$%

nt

t

I

Complete positivity of the dynamical map Λt guarantees that the eigenvalues of any entangled stateentangled state ρS+Sn of S + Sn remain positive at any time

Ssystem

Snn-level systemNo dynamical

coupling

Less intuitive property!

probabilistic interpretationof the density matrix!"#!! ,0$

Violation of CP incompatible with the assumption ofa total closed system (for factorized initial condition)

A consistent physical description of an open quantum systemmust be not only positive but also completely positive 1

The Lindblad form of the master equation is theonly possible form of first-order linear differentialequation for a completely positive semigroup having bounded generator

Importance of CP

1 K. Kraus, States, Effects and Operations, Fundamental Notions of Quantum Theory (Academic, Berlin, 1983).

This is valid whenever

ENVTOT!!! "=

CP is not guaranteed and unphysical situations, showing that the modelwe are using is not appropriate, may show up in the dynamics

What about non-Lindblad-type master equations?[ ]

1 S. John and T. Quang, Phys. Rev. Lett. 78, 1888 (1997)2 J.J. Hope et al., Phys. Rev. A 61, 023603 (2000)3 R. Alicki et al., Phys. Rev A 65, 062101 (2002); R. Alicki et al., ib id. 70, 010501 (2004)

Non-Markovian master equations

Non-Markovian master equations need not be in the Lindblad form, and usually they are not.

Non-Lindblad ME

No man’s land

photonic band-gap materials 1

quantum dots atom lasers 2

non-Markovian quantum information processing 3

QUANTUMQUANTUMNANOTECHNOLOGIES NANOTECHNOLOGIES 4,5,6,74,5,6,7

4 A. Micheli et al, Single Atom Transistor in a 1D Optical Lattice, Phys. Rev. Lett. 93,140408 (2004)

5 L. Tian et al, Interfacing Quantum-Optical and Solid-State Qubits, Phys. Rev. Lett. 92, 247902 (2004)

6 L. Florescu et al, Theory of a one-atom laser in a photonic band-gap microchip, Phys. Rev. A 69, 013816 (2004)

7 L. Tian and P. Zoller, Coupled ion-nanomechanical systems, Phys. Rev. Lett. 93, 266403 (2004)

Why to study them ?[ ]

Ubiquitous model

Damped harmonic oscillatoror Quantum Brownian Motion in a harmonic potential

Paradigmatic model of the theory of open quantum systems

may be solved exactly

Quantum Optics, Quantum Field Theory, Solid State Physics

Htot= Hsys + Hres + Hint

Microscopic modelMicroscopic model

!"

#$%

&+=2

1†

0 aaHsys 'h

system

!=n

nnnresbbH†"h

environment

coupling

( )( )††

int2

aabbm

kH

n

nn

nn

n++= !

"#

h( ) ( )! "=

n

n

nn

n

m

kJ ##$

##

2

2

Spectral density of the reservoir

1 H.-P. Breuer and F. Petruccione, "The theory of open quantum systems",Oxford University Press, Oxford (2002)

Two approaches:microscopic and phenomenologic 1

Exact Master EquationTime-convolutionless approach

Phenomenological master equationcontaining a memory kernel

GOAL: To study the dynamics of the system oscillator, in presence of coupling with the reservoir beyond the Markovian approximation

generalized master equationlocal in time

( )( ) ( )ttL

dt

td!

!=

generalized master equationnon-local in time

( )( ) ( ) tdtLttK

dt

tdt

!!!"= # $$

0

Time dependent coefficients have the form of series in the coupling constant α

dissipation kernelnoise kernel

TIME-DEPENDENT COEFFICIENTSSUPEROPERATORS

Exact Master Equation 1,2

(time convolutionless approach)

1 J.P. Paz and W.H. Zurek, Environment-induced decoherence and the transition from quantum to classical, Proceedings of the 72nd Les Houches Summer School on Condensed Matter Waves, July-August 1999, quant-ph/0010011.2 F. Intravaia, S.M., A. Messina, Eur. Phys. J. B 32, 109 (2003).

HU-PAZ-ZHANG MASTER EQUATION

( )( )( ) ( ) ( )( )[ ]( )

( ) ( )ttL

ttttidt

td SSSs

!

!"!

=

++#$%$$= 2NPXXH2

S

( )SS

2PXXPN

!!" "= i

( ) ( ) ( ) !!"!# dt

t

0

0

cos$=%

( ) ( ) ( ) !!"!# dt

t

0

0

sin$=%

DIFFUSIONDIFFUSIONTERMSTERMS((decoherencedecoherence))

( ) ( ) ( ) !!"!µ# dt

t

0

0

sin$=

DAMPING TERM (DAMPING TERM (dissipationdissipation))

to the second order in thecoupling constant we have

Secular approximated masterequation (and applications)

( ) ( ) ( )[ ]

( ) ( )[ ]†††

†††

22

22

aaaaaatt

aaaaaatt

dt

td

!!!"

!!!"!

+##$

#

+#+$

#= ( ) ( ) 0>±! tt " LINDBLAD TYPE

NON-LINDBLAD TYPE

otherwise

Trapped ions

F. Intravaia, S. M., J. Piilo and A. Messina, "Quantum"Quantumtheory of heating of a single trapped ion",theory of heating of a single trapped ion",Phys. Lett. A 308 (2003) 6.

S. M., J. Piilo, F. Intravaia, F. Petruccione and A. Messina,““Simulating Quantum Brownian Motion with single trapped ionsSimulating Quantum Brownian Motion with single trapped ions””,,Phys. Rev. A 69 (2004) 052101.S. M., J. Piilo, F. Intravaia, F. Petruccione and A. Messina,““LindbladLindblad and non- and non-LindbladLindblad type dynamics of a type dynamics of aquantum Brownian particlequantum Brownian particle””,, Phys. Rev. A. 70 (2004) 032113.S. M. "Revealing virtual processes in the phase space""Revealing virtual processes in the phase space",,J.Opt.B: Quantum and Semiclass. Opt. 7 S398–S402 (2005)

Linear amplifier

S.M., J. Piilo, N. Vitanov, andS. Stenholm, ““Transient dynamics Transient dynamics of quantum linear amplifiersof quantum linear amplifiers””,, Eur. Phys. J. D 36, 329–338 (2005)

( ) ( )[ ] ( )[ ]aaTrDTr*†exp !!"!"!# $%=

The Quantum Characteristic Function (QCF)2

Exact solution1

[NO MARKOVIAN APPROX, NO WEAK COUPLING APPROX, NO RWA APPROX]

1 F. Intravaia, S. M. and A. Messina, Phys. Rev. A 67, 042108 (2003)2 S.M.Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, Oxford, 1997)

( ) ( ) ( )!= *,

2

1""""#

$% ddDtt

( ) ( )! ""=#t

tdtt

0

2 $ ( ) ( ) ( ) ( )! ""#=#"$$%

$

t

tttdteet

0

( ) ( ) ( )[ ]!"!" #! titt

eeet 0

22/

0,$%$&$ %=

Example of non-Markoviandynamics

The risks of working with The risks of working with non-non-LindbladLindblad Master Equations Master Equations

Master equation with memory kernel

EXAMPLE: Phenomenological non-Markovian master equationdescribing an harmonic oscillator interacting witha zero T reservoir

( )( ) ( ) tdtLttK

dt

tdt

!!!"= # $$

0

aaaaaaL†††

2 !!!! ""=

Markovian Liouvillian

Exponential memory kernel

( ) ttegttK

!""=!"

#2

g coupling strengthg decay constant of system-reservoir correlations

Non-Lindblad Master Equation: CP and positivityare not guaranteed!

Solution using the QCF

What is the QCF?

( ) ( )[ ] 2/2

,!

!"!#peDTrp =

Fourier transform

p = 0Wigner function

p = 1P function

p = -1Q function

defining properties

( ) 10 ==!" ( ) 1!"#

Quantum Characteristic Function( ) ( )0, =! p"#"#

useful property

( )!"!!

nm

nm

d

d

d

daa ##

$

%&&'

()##

$

%&&'

(=

*

†

p = p = 0 symmetric orderingp = -1 antinormal ordering

p = 1, normal ordering

Solution using the QCF

( ) ( ) ( )( )[ ] 2/2/2 2

32.1sin37.032.1cos1,!" ""!"!#

$$ +$= eeExample: g / g = 1t = g t

defining properties

( ) 10 ==!" ( ) 1!"#

The defining properties are always satisfied

No problem with the dynamics

Density matrix solution

The quantum characteristic function contain all the informationnecessary to reconstruct the density matrix, and therefore is analternative complete description of the state of the system1

( ) ( ) ( )!= *,

2

1""""#

$% ddDtt

( ) ( ) 1111

=== ntnt !!

we look at the time evolution of thepopulation of the initial state 1=n

( ) !"

#$%

&'

'+'= (

ttett

sin2

cos2/

11

)* )

Example: g / g = 1t = g t

1 S.M.Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, Oxford, 1997)2 S.M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).

problem of positivity firstly noted by Barnett&Stenholm2

Limits in the QCF description

The QCF and the density matrix are not “operatively” equivalent descriptions of the dynamics, in the sense that the QCF may fail in discriminating whennon-physical conditions (negativity of the density matrix eigenvalues) show up.

defining properties

( ) 10 ==!" ( ) 1!"#

The defining properties of the QCFare only necessary conditions

The additional condition to be imposed on the QCF in order to ensurethe positivity of the density matrix does not seem to have a simple form

S. M., "Limits in the quantum characteristic function description of open quantum systems", Phys. Rev. A 72, 024103 (2005)

Non-Markovian dynamics of a qubit

Phenomenological model: Non-Markovian master equation with exponential memory

( ) ( )! "=

t

ttLtkdtdt

d

0

''' ##

( )

!"

#$%

&''+

!"

#$%

&''+=

+'+''+

'+'++'

()()(()((*

()()(()((*)

2

1

2

1

2

1

2

11

0

0

N

NL

tetk

!! "=)(

memory kernel

Markovian Liouvillian

( ) ( )[ ]!"rr

#+= twIt2

1

Solution in terms of the Bloch vector

Bloch vector

( ) ( ) ( ) Twtww

rrrr+!="# 00:

( )( )

( )!!!

"

#

$$$

%

&

='

tR

tR

tR

,00

0,20

00,2

(

(

(

( ) ( )[ ]!!!

"

#

$$$

%

&

'+

='

1,12

0

0

1tRN

T

(

( ) [ ] [ ] [ ]{ }241cosh241sinh41,212 !!!" !

RRReR #+##=##

t!" = !!0

=R

Positivity and Complete Positivity

( ) ( )( ) ( )[ ]

( )1

,

1,12

,2,2

2122

=!"

#$%

& '+'+!

"

#$%

&+!

"

#$%

& '

tR

tRNw

tR

w

tR

w zyx

(

(

((

THE BLOCH SPHERE

Condition for positivity: The dynamical map Φ maps a density matrix into anotherdensity matrix if and only if the Bloch vector describing the initial state is transformedinto a vector contained in the interior of the Bloch sphere, i.e. the Bloch ball.

4 R ≤ 1

A largely unexplored region: The positivity – complete positivity regionAre there non-Markovian systems which are positive but not CP?

Applying the criterion for CP demonstrated in [1], and using the analytical solutionsApplying the criterion for CP demonstrated in [1], and using the analytical solutionswe have derived for our model, we have proved that, in the case of exponentialwe have derived for our model, we have proved that, in the case of exponentialmemory and for the non-memory and for the non-MarkovianMarkovian model here considered model here considered positivitypositivity is a is anecessary and sufficient condition for complete necessary and sufficient condition for complete positivitypositivity [2] [2]

[1] M.B. Ruskai, S. Szarek, and E. Werner, Lin. Alg. Appl. 347, 159 (2002).[2] S. M., “Complete positivity of the spin-boson model with exponential memory”, submitted for publication

THIS RESULT PROVIDES THE EXPLICIT CONDITIONS OF VALIDITY OF APARADIGMATIC PHENOMENOLOGICAL MODEL OF THE THEORY OFOPEN QUANTUM SYSTEMS, NAMELY THE SPIN-BOSON MODEL WITHEXPONENTIAL MEMORY.

non-Lindblad form

SummaryOpen quantum systems

Two approaches: microscopicmicroscopic and phenomenologicalphenomenological

Markovian master equation Non-Markovian master equation

Lindblad form

Paradigmatic model: QBM or damped harmonic oscillator

TCL (microscopic exact approach)Generalized master equation

Solution based on algebraic method

Applications: trapped ions, linear amplifier

Memory kernel

QCF and density matrix solutions

Positivity violation

Limits in the use of the QCF

Beyond the Lindblad form

ReferencesQuantum Quantum BrownianBrownian MotionMotionF. Intravaia, S. Maniscalco, J. Piilo and A. Messina, "Quantum theory of heating of a single trapped ion",Phys. Lett. A 308, 6 (2003).F. Intravaia, S. Maniscalco and A. Messina "Comparison between the rotating wave and Feynman-Vernon system-reservoir couplings in the non-Markovian regime", Eur. Phys. J. B 32, 109 (2003).F. Intravaia, S. Maniscalco and A. Messina, "Density Matrix operatorial solution of the non-Markovian Master Equation for Quantum Brownian Motion", Phys. Rev. A 67, 042108 (2003) .S. Maniscalco, F. Intravaia, J. Piilo and A. Messina, “Misbelief and misunderstandings on the non--Markovian dynamics of a damped harmonic oscillator”, J.Opt.B: Quantum and Semiclass. Opt. 6, S98 (2004) .

TrappedTrapped ionion simulator simulatorS. Maniscalco, J. Piilo, F. Intravaia, F. Petruccione and A. Messina, “Simulating Quantum Brownian Motion with single trapped ions ”, Phys. Rev. A 69, 052101 (2004) .

LindbladLindblad non-Lindbladnon-Lindblad borderborder and the and the existenceexistence of a of a continuouscontinuous measurementmeasurementinterpretationinterpretation forfor non-Markoviannon-Markovian stochasticstochastic processesprocessesS. Maniscalco, J. Piilo, F. Intravaia, F. Petruccione and A. Messina, Lindblad and non-Lindblad type dynamics of a quantum Brownian particle”, Phys. Rev. A. 70, 032113 (2004) .S. Maniscalco "Revealing virtual processes in the phase space", J.Opt.B: Quantum and Semiclass. Opt. 7 S398–S402 (2005).

PositivityPositivity and complete and complete positivitypositivity S. Maniscalco, "Limits in the quantum characteristic functiondescription of open quantum systems", Phys. Rev. A 72, 024103(2005)S. Maniscalco and F. Petruccione “Non-Markovian dynamics of aqubit”, Phys. Rev. A. 73, 012111 (2006)S. Maniscalco “Complete positivity of the spin-boson model withexponential memory”, submitted for publication

Non-MarkovianNon-Markovian wavefunctionwavefunction methodmethodJ. Piilo, S. Maniscalco, A. Messina, and F. Petruccione"Scaling of Monte Carlo wavefunction simulations fornon-Markovian systems", Phys. Rev. E 71 056701 (2005).