Download Diapositiva 1 - Applied Quantum Mechanics group

Non-Markovian Open
Quantum Systems
Sabrina Maniscalco
Quantum 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 interaction
of a quantum system with its environment
Quantum Mechanics
closed systems
unitary dynamics
reversible dynamics
i
d
dt
 Hˆ 
Schrödinger Equation

dˆ
i
 Hˆ , ˆ
dt

Liouville – von Neumann Equation
open quantum systems
reduced density operator
ˆ S t   TrE ˆT t 
master equation
dˆ
 Lˆ
dt
non-unitary and
irreversible dynamics
Motivation
Fundamentals of Quantum Theory
Quantum Technologies
Quantum Measurement Theory
Border between quantum
and classical descriptions
Decoherence
Entanglement between the degrees
of freedom of the system and those
of the enviornment
PROBLEM: All new quantum
technologies rely on quantum
coherence
quantum computation
quantum cryptography
quantum communication
Qubit
Approaches to the dynamics of
open quantum systems
Derivation of an equation of motion for the reduced density matrix of the
system of interest: The master equation
microscopic approach
phenomenological approach
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 methods
numerical methods
damping basis method1
Monte Carlo wave function (and variants)3,4
green function
algebraic superoperatorial
Influence functional – path integral5
methods2
3
1
H.-J. Briegel, B.-G. Englert, PRA 47, 3311 (1993)
2
F. Intravaia, S.M., A. Messina, PRA 67, 042108 (2003)
J. Dalibard, Y. Castin and K. Molmer, Phys. Rev.Lett. 68, 580 (1992)
J. Piilo, S.M., A. Messina, F. Petruccione, PRE 71, 056701 (2005)
5 U. Weiss, Quantum Dissipative Systems 2nd ed. (World Scientific,
Singapore, 1999).
4
Approximations
weak coupling approximation
weak coupling between system and environment
perturbative approaches
(expansions in the coupling constant)
Markovian approximation
assumes that the reservoir correlation time is much smaller
than the relaxation time of the open system
coarse-graining in time
changes in the reservoir due to the interaction with the system
do not feed back into the system
Lindblad master equation 1,2
Weak coupling + Markovian approximation + RWA or secular approx.
 density matrix
of the reduced
system
d
1


 iH ,     V j V j†   ,V j†V j 
dt
2

j 
[
ADVANTAGES


]
simulation techniques
important properties
Semigroup property:
Positivity:
1
2
Vj and Vj†transition or jump
operators depending on the
specific physical system
t t    t  t  
t : iff  0  0  t  0  0,   BH 
G. Lindblad, Comm. Math. Phys. 48, 119 (1976)
V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)
 t   t  0
dynamical map
 t  e Lt
Markovian dynamical map
Positivity and
Complete positivity
POSITIVITY
    0, 
COMPLETE POSITIVITY
probabilistic interpretation of
the density matrix
 t : B(H)  B(H) completely positive iff
 t  I n : B(H  H n )  B(H  H n ) is positive n  N
Less intuitive property!
Violation of CP incompatible with the assumption of
a total closed system (for factorized initial condition)
Complete positivity of the dynamical map Λt guarantees that the eigenvalues of
any entangled state ρS+Sn of S + Sn remain positive at any time
S
system
No dynamical
coupling
Sn
n-level
system
Importance of CP
A consistent physical description of an open quantum system
must be not only positive but also completely positive 1
The Lindblad form of the master equation is the
only possible form of first-order linear differential
equation for a completely positive semigroup
having bounded generator
[
This is valid whenever
TOT     ENV
What about non-Lindblad-type master equations?
]
CP is not guaranteed and unphysical situations, showing that the model
we are using is not appropriate, may show up in the dynamics
1
K. Kraus, States, Effects and Operations, Fundamental Notions of Quantum Theory (Academic, Berlin, 1983).
Non-Markovian master equations
Non-Markovian master equations need not be
in the Lindblad form, and usually they are not.
No man’s land
[ Why to study them ? ]
 photonic band-gap materials 1
 quantum dots
 atom lasers 2
 non-Markovian quantum information
processing 3
QUANTUM
NANOTECHNOLOGIES 4,5,6,7
4
5
6
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., ibid. 70, 010501 (2004)
Non-Lindblad ME
A. Micheli et al, Single Atom Transistor in a 1D
Optical Lattice, Phys. Rev. Lett. 93,140408 (2004)
L. Tian et al, Interfacing Quantum-Optical and
Solid-State Qubits, Phys. Rev. Lett. 92, 247902 (2004)
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)
Damped harmonic oscillator
or Quantum Brownian Motion in a harmonic potential
Paradigmatic model of the theory of open quantum systems
Ubiquitous model
may be solved exactly
Quantum Optics, Quantum Field Theory, Solid State Physics
Microscopic model
system
H sys
1
 †
  0  a a  
2

environment
Htot= Hsys + Hres + Hint
n
coupling
H int    kn
n
H res   n bn†bn
Spectral density of the reservoir



bn  bn† a  a †
2mnn

kn2
J    
   n 
2
m

n
n n
Two approaches:
microscopic and phenomenologic 1
GOAL: To study the dynamics of the system oscillator,
in presence of coupling with the reservoir beyond
the Markovian approximation
Exact Master Equation
Time-convolutionless approach
Phenomenological master equation
containing a memory kernel
generalized master equation
local in time
d t 
 Lt  t 
dt
1
generalized master equation
non-local in time
d t 
  K t  t L t dt 
dt
0
H.-P. Breuer and F. Petruccione, "The theory of open quantum systems",
Oxford University Press, Oxford (2002)
t
1 J.P.
Exact Master Equation 1,2
(time convolutionless approach)

HU-PAZ-ZHANG MASTER EQUATION
d t 
s
  iHS  t  X S
dt
 Lt  t 
   t X P
2
S
t
0
t
t      sin 0 d
0
noise kernel

  t N  2  t 
TIME-DEPENDENT
SUPEROPERATORS
COEFFICIENTS
Time dependent coefficients have the form
of series in the coupling constant α
t      cos0 d
S
N
i
2
P X

S
 X  PS
to the second order in the
coupling constant we have
DAMPING TERM (dissipation)
DIFFUSION
TERMS
(decoherence)
t
 t      sin 0 d
0
dissipation kernel
Paz and W.H. Zurek, Environment-induced decoherence and the transition from quantum to classical, Proceedings of the 72 nd
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).

Secular approximated master
equation (and applications)
d t 
 t    t  †

a a  2aa †  a † a
dt
2
 t    t  †

aa   2a † a  aa †
2



t    t   0 LINDBLAD TYPE

Trapped ions
F. Intravaia, S. M., J. Piilo and A. Messina, "Quantum
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 ions ”,
Phys. Rev. A 69 (2004) 052101.
S. M., J. Piilo, F. Intravaia, F. Petruccione and A. Messina,
“Lindblad and non-Lindblad type dynamics of a
quantum Brownian particle”, Phys. Rev. A. 70 (2004) 032113.
S. M. "Revealing virtual processes in the phase space",
J.Opt.B: Quantum and Semiclass. Opt. 7 S398–S402 (2005)
otherwise
NON-LINDBLAD
TYPE
Linear amplifier
S.M., J. Piilo, N. Vitanov, and
S. Stenholm, “Transient dynamics
of quantum linear amplifiers”,
Eur. Phys. J. D 36, 329–338 (2005)
Exact solution1
[NO MARKOVIAN APPROX, NO WEAK COUPLING APPROX, NO RWA APPROX]
The Quantum Characteristic Function (QCF)2
    TrD   Tr exp a†   *a 
  , t   e
2
2

0 e
1
*






,
t
D

d

d

2 
  t  / 2 i0t
e


t
t
  t   e t   et t dt 
0
0
t   2  t dt 
1
  t  
 t  
F. Intravaia, S. M. and A. Messina, Phys. Rev. A 67, 042108 (2003)
S.M.Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, Oxford, 1997)
Example of non-Markovian
dynamics
The risks of working with
non-Lindblad Master Equations
Master equation with memory kernel
EXAMPLE: Phenomenological non-Markovian master equation
describing an harmonic oscillator interacting with
a zero T reservoir
d t 
  K t  t L t dt 
dt
0
t
Exponential memory kernel
2  t t 

K t  t   g e
Markovian Liouvillian
L  2aa †  a † a  a † a
g coupling strength
 decay constant of system-reservoir
correlations
Non-Lindblad Master Equation: CP and positivity
are not guaranteed!
Solution using the QCF
What is the QCF?
  , p   TrD e
p = 1, normal ordering
p
2
/2
Fourier transform
p=
p = 0 symmetric ordering
p = -1 antinormal ordering
Quantum Characteristic Function
p=1
P function
p=0
Wigner function
p = -1
Q function
      , p  0
defining properties
    1
   0  1
useful property
†m
a an
 d 
  
 d 
m
n
 d 
  *    
 d 
Solution using the QCF
Example: g /  = 1
=t

  ,   1   e
2
 / 2
cos1.32   0.37 sin 1.32 e
  /2
2
defining properties
   0  1
    1
The defining properties are
always satisfied
No problem with the dynamics
Density matrix solution
The quantum characteristic function contain all the information
necessary to reconstruct the density matrix, and therefore is an
alternative complete description of the state of the system1
1
*




 t  


,
t
D

d

d

2 
11 t   e
t / 2
we look at the time evolution of the
population of the initial state n  1
11t   n  1  t  n  1



sin t 
 cos t 
2


Example: g /  = 1
=t
problem of positivity firstly noted by Barnett&Stenholm2
1
2
S.M.Barnett and P.M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, Oxford, 1997)
S.M. Barnett and S. Stenholm, Phys. Rev. A 64, 033808 (2001).
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 when
non-physical conditions (negativity of the density matrix eigenvalues) show up.
The defining properties of the QCF
are only necessary conditions
defining properties
   0  1
    1
The additional condition to be imposed on the QCF in order to ensure
the 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
Markovian Liouvillian
d
  dt ' k t 'L t  t '
dt 0
t
1
1


L   0  N  1              
2
2


1
1


  0 N               
2
2


memory kernel
t
k (t )  e
Solution in terms of the Bloch vector
 t  
1
I  w t    
2
Bloch vector




 : w0  wt   w0  T
  R,    e
 2
1  4R
1 2

0
0 
  R 2, t 



0
 R 2, t 
0 

0
0
 R, t 



0




T 
0


1
 2 N  1  R, t   1


sinh 1  4 R  2  cosh 1  4 R  2

  t
R  0 
Positivity and Complete Positivity
THE BLOCH SPHERE
 wz  2 N  11  R, t   1
 wy 
 wx 


 1

 

  R, t 
  R 2, t 
  R 2, t 


2
2
2
Condition for positivity: The dynamical map  maps a density matrix into another density
matrix if and only if the Bloch vector describing the initial state is transformed into a
vector contained in the interior of the Bloch sphere, i.e. the Bloch ball.
4R≤1
ATHIS
largely
RESULT
unexplored
PROVIDES
region: The
THEpositivity
EXPLICIT
– complete
CONDITIONS
positivity
OF
region
VALIDITY OF A
Are
PARADIGMATIC
there non-Markovian
PHENOMENOLOGICAL
systems which are MODEL
positive but
OF not
THECP?
THEORY OF OPEN
QUANTUM SYSTEMS, NAMELY THE SPIN-BOSON MODEL WITH
EXPONENTIAL
MEMORY.
Applying
the criterion
for CP demonstrated in [1], and using the analytical solutions
we have derived for our model, we have proved that, in the case of exponential
memory and for the non-Markovian model here considered positivity is a necessary
and sufficient condition for complete positivity [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
Summary
Open quantum systems
Markovian master equation
Non-Markovian master equation
Lindblad form
non-Lindblad form
Beyond the Lindblad form
Two approaches: microscopic and phenomenological
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
References
Quantum Brownian Motion
F. 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) .
Trapped ion simulator
S. Maniscalco, J. Piilo, F. Intravaia, F. Petruccione and A. Messina, “Simulating Quantum Brownian Motion
with single trapped ions ”, Phys. Rev. A 69, 052101 (2004) .
Lindblad non-Lindblad border and the existence of a continuous measurement
interpretation for non-Markovian stochastic processes
S. 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).
Positivity and complete positivity
Non-Markovian wavefunction method
S. Maniscalco, "Limits in the quantum characteristic function
description of open quantum systems", Phys. Rev. A 72, 024103
J. Piilo, S. Maniscalco, A. Messina, and F. Petruccione
(2005)
"Scaling of Monte Carlo wavefunction simulations for
non-Markovian systems", Phys. Rev. E 71 056701 (2005). S. Maniscalco and F. Petruccione “Non-Markovian dynamics of a
qubit”, Phys. Rev. A. 73, 012111 (2006)
S. Maniscalco “Complete positivity of the spin-boson model with
exponential memory”, submitted for publication