Non- Non- Markovian Markovian Open Open Quantum Systems Quantum Systems Sabrina Maniscalco Quantum Optics Group, Department of Physics University of Turku
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).