Muga G. Time in Quantum Mechanics - Vol. 2
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274 J. Martorell et al.
91. J.G. Muga, J.P. Palao, B. Navarro, I.L. Egusquiza, Phys. Rep. 395, 357 (2004) 265
92. J.G. Muga, F. Delgado, A. del Campo, G. Garc
´
ıa-Calder
´
on, Phys. Rev. A 73, 052112 (2006) 244, 250, 253,
2
93. H. Nakazato, M. Namiki, S. Pascazio, Int. J. Mod. Phys. B10, 247 (1996) 243
94. R.G. Newton, Ann. Phys. (NY) 14, 333 (1961) 244, 261, 264
95. R.G. Newton, Scattering Theory of Waves and Particles, 2nd edn (Dover, Mineola, 2002),
Chap. 19 244, 261, 264
96. T.D. Nghiep, V.T. Hahn, N.N. Son, Nucl. Phys. B (Proc. Suppl.) 66, 533 (1998) 242
97. C.A. Nicolaides, Phys. Rev. A 66, 022118 (2002) 242
98. C.A. Nicolaides, D.R. Beck, Phys. Rev. Lett. 38, 683, 1037 (1977) 242, 264
99. C.A. Nicolaides, T. Mercouris, J. Phys. B 29, 1151 (1996) 262
100. V.R. Nikitenko, D. Hertel, H. B
¨
assler, Chem. Phys. Lett. 348, 89 (2001) 243, 267
101. N.N. Nikolaev, Sov. Phys. Usp. 11, 522 (1968) 242, 262
102. I.S. Nikolaev, P. Lodahl, A.F. Van Driel, A.F. Koenderink, W.L. Vos, Phys. Rev. B 75, 115302
(2007) 243
103. E.B. Norman, S.B. Gazes, S.G. Crane, D.A. Bennett, Phys. Rev. Lett. 60, 2246 (1988) 242, 261
104. E.B. Norman, B. Sur, K.T. Lesko, R.M. Larimer, D.J. DePaolo, T.L. Owens, Phys. Lett. B
357, 521 (1995) 242
105. D. Novkovic et al., Nucl. Inst. Methods A 566, 477 (2006) 262
106. H.M. Nussenzveig, Causality and Dispersion Relations (Academic Press, New York, 1972) 260
107. T.F. O’Malley, Phys. Rev. 137, A1668 (1965) 255, 257
108. J. Okolowicz, M. Ploszajczak, I. Rotter, Phys. Rep. 374, 271 (2003) 248
109. D.S. Onley, A. Kumar, Am. J. Phys. 60, 432 (1992) 246
110. M. Ozawa, Phys. Lett. A 356, 411 (2006) 266
111. R.E. Paley, N. Wiener, Fourier Transforms in the Complex Domain (American Mathematical
Society Colloquium Publication 19, 1934) 245
112. R.E. Parrott, J. Lawrence, Europhys. Lett. 57, 632 (2002) 242, 263
113. A. Peres, Ann. Phys. (NY) 129, 33 (1980) 247
114. J. Petzold, Zeit. Phys. 155, 422 (1959) 244
115. P.M. Radmore, P.L. Knight, Phys. Lett. A 102, 180 (1984) 243
116. M. Razavy, Quantum Theory of Tunnelling (World Scientific, Singapore, 2003), p. 18. 239
117. M. Razavy, Can. J. Phys. 57, 1731 (1979) 271
118. E.J. Robinson, Phys. Rev. A 33, 1461 (1986) 263
119. E.J. Robinson, J. Phys. B: At. Mol. Opt. Phys. 27, L305 (1994) 243
120. R.T. Robiscoe, Phys. Lett. A 100, 407 (1984) 262
121. C. Rothe, S.I. Hintschich, A.P. Monkman, Phys. Rev. Lett. 96, 163601 (2006) 242, 267
122. L.C. Ryff, Eurphys. Lett. 39, 231 (1997) 266
123. K. Rzazewski, M. Lewenstein, J.H. Eberly, J. Phys. B: At. Mol. Phys. 15, L661 (1982) 264
124. J.J. Sakurai, Modern Quantum Mechanics. Rev. Ed. (Addison-Wesley, New York, 1994),
Supplement II, pp. 481–486 243
125. D.V. Savin, V.V. Sokolov, H.J. Sommers, Phys. Rev. E 67, 026215 (2003) 253
126. L.S. Schulman, B. Gaveau, J. Phys. A 28, 7359 (1995) 243
127. L.S. Schulman, C.R. Doering, B. Gaveau, J. Phys. A 91, 2053 (1991) 243
128. L.S. Schulman, D. Tolkunov, E. Mihokova, Phys. Rev. Lett. 96, 065501 (2006) 243
129. P.H. Sher, J.M. Smith, P.A. Dalgarno, R.J. Warburton, X. Chen, P.J. Dobson, S.M. Daniels,
N.L. Pickett, P. O’Brien, Appl. Phys. Lett. 92, 101111 (2008) 267
130. K.M. Sluis, E.A. Gislason, Phys. Rev. A 43, 4581 (1991) 247
131. J.R. Taylor, Scattering Theory (J. Wiley, New York, 1972) 253, 254
132. D. Tomono et al., Nucl. Inst. Methods A 503, 283 (2003) 262
133. E. Torrontegui, J.G. Muga, J. Martorell, D.W.L. Sprung, arXiv:0903.4156 244, 251, 264, 270
134. R.A. Valle
´
e, K. Baert, B. Kolaric, M. Van der Auweraer, K. Clays, Phys. Rev. B 76, 045113
(2007) 243
135. W. Van Dijk, Y. Nogami, Phys. Rev. Lett. 90, 028901 (2003) 244
136. W. Van Dijk, Y. Nogami, Phys. Rev. C 65, 024608 (2002) 244
9 Quantum Post-exponential Decay 275
137. S. Wallentowitz, P.E. Toschek, Phys. Rev. A 72, 046101 (2005) 266, 267
138. S. Wallentowitz, P.E. Toschek, Phys. Lett. A 367, 420 (2007) 266
139. V.F. Weisskopf, E. Wigner, Zeit. Phys. 63, 54 (1930) 240
140. V.F. Weisskopf, E. Wigner, Zeit. Phys. 65, 18 (1930) 240
141. R.G. Winter, Phys. Rev. 123, 1503 (1961) 244, 253, 261
142. R.G. Winter, Phys. Rev. 126, 1152 (1962) 242
143. J. Zakrzewski, K. Rzazewski, M. Lewenstein, J. Phys. B: At. Mol. Phys. 17, 729 (1984) 264
Chapter 10
Timescales in Quantum Open Systems:
Dynamics of Time Correlation Functions
and Stochastic Quantum Trajectory Methods
in Non-Markovian Systems
Daniel Alonso and In
´
es de Vega
Past experience, if not forgotten, is a guide for the future
Chinese proverb
10.1 Introduction
The dynamics of a system in interaction with another system, the later considered
as a reservoir, is studied in many different domains in physics. This approach is
useful not only to address fundamental questions like quantum decoherence and the
measurement problem [1] but also to deal with practical and theoretical problems
appearing in the emerging fields of nanotechnology [2, 3] and quantum computing
[4–6], as well as in systems of ultra-cold atoms [7]. In many of these cases, the basic
approximation is the Markov assumption in which there is a clear separation of the
typical timescales associated with the system and the reservoir or environment. This
separation of timescales, together with other assumptions like the weak coupling
between the system and the reservoir, has been central in the development of sev-
eral fields, in particular in quantum optics [8, 9]. However, in the last few years,
new systems where the Markov approximation is no longer valid have received
increasing attention. This has produced a growing interest in developing a theory
of non-Markovian quantum open systems.
The interaction of systems with structured environments, e.g., atoms coupled
to the radiation field within a photonic crystal or a cavity [10], the interaction of
systems with spin baths [11] (e.g., quantum dots in a bath of atomic nuclei) or with
D. Alonso (B)
Departamento de F
´
ısica Fundamental y Experimental, Electr
´
onica y Sistemas, Instituto
Universitario de Estudios Avanzados (IUdEA)enF
´
ısica At
´
omica, Molecular y Fot
´
onica,
Universidad de La Laguna, La Laguna 38203, Tenerife, Spain, dalonso@ull.es
I. de Vega
Max Planck Institute of Quantum Optics, Garching 85748, Germany, ines.devega@mpq.mpg.de
Alonso, D., de Vega, I.: Timescales in Quantum Open Systems: Dynamics of Time Correlation
Functions and Stochastic Quantum Trajectory Methods in Non-Markovian Systems. Lect. Notes
Phys. 789, 277–301 (2009)
DOI 10.1007/978-3-642-03174-8
10
c
Springer-Verlag Berlin Heidelberg 2009
278 D. Alonso and I. de Vega
environments at low temperatures, the non-Markovian dynamics of entanglement
[12], or the dynamics of a continuous atom laser based on focusing to an indepen-
dently formed atomic condensate [13], are just examples of current problems where
understanding non-Markovian features is crucial.
10.2 Atoms in a Structured Environment, an Example
of Non-Markovian Interaction
One of the most basic concepts in physics is the spontaneous emission of an atom. It
is known that spontaneous emission from an excited atom can be modified by plac-
ing the atom in a structured environment [14, 15]. Such environments are physically
realized by placing the atoms in cavities or within photonic crystals. In both cases,
the photonic density of states of the radiation field is severely modified. On the one
hand, the boundary conditions that a cavity imposes to the radiation field modify its
photonic density of states, which is no longer a growing function of the frequency,
but becomes a Lorentzian centered in the cavity resonant frequency. This resonant
frequency is related to the cavity length. Following the well-known Fermi Golden
rule, the spontaneous emission rate of an atom is proportional to the photonic density
of states of the field in the central emission frequency. Consequently, the emission
of an atom within a cavity can be suppressed if the central emission frequency is
off-resonant with respect to the cavity resonant frequency, i.e., with respect to the
center of the Lorentzian. On the other hand, photonic crystals present a periodicity
in the refraction index, which produces strong Bragg scattering for photons within
a certain frequency range. Hence, these photons are not present within the crystal,
and therefore the photonic density of states is 0 for the corresponding frequency
range. As a consequence, the photonic density of states typically presents a band-
gap profile consisting of regions where it varies abruptly (the bands) and regions
where it is 0 (the gaps). The spontaneous emission of an atom near a band-gap edge
[16, 17] differs from the exponential decay and exhibits oscillations, see Chap. 9
in this volume. Furthermore, even at zero temperature the steady-state population
may remain finite for the excited state [18]. A similar phenomenon, the so-called
limited quantum decay, is discussed in [19] for the case of a single level coupled to
a continuum which is bounded in energy.
As noted before, a key element in the interaction of an atom with the electro-
magnetic field is the density of electromagnetic modes of such field for the atomic
transition frequency. However, it is also important to consider the profile of the
density of states around such a frequency. The electromagnetic field may be viewed
as a reservoir or environment in interaction with the atom. If the density of photons
around the atomic transition frequency is smooth then the interaction between the
atom and the reservoir is, to a good approximation, Markovian. This means that
there is a timescale associated with the dynamics of the atom that is slow and also
a fast timescale associated with the correlation function of the environment. This
in turn leads to an atomic dynamics that does not depend on its previous history.
10 Time Scales and Correlations in Non-Markovian Quantum Open Systems 279
The spontaneous decay in this case is almost exponential with a characteristic decay
rate and the spectrum of the photons emitted exhibits a Lorentzian profile. A typical
example is the interaction of an atom with a free electromagnetic field.
On the other hand, if the photonic density of states varies on a frequency range
that is comparable with the spontaneous emission rate, the dynamics of the atom in
contact with the reservoir may show noticeable non-Markovian effects. In such case
the decay is nonexponential and the spectrum of the emitted light does not present
a Lorentzian profile [20]. For non-Markovian interactions, the timescale associated
with the environmental correlation function is not fast enough in comparison to that
of the atomic dynamics, and the atomic dynamics depends on its previous history.
We may find other examples where non-Markovian effects may be important,
as in the dynamics of qubits [21]. In this chapter we shall focus on atoms in non-
Markovian environments and will particularize to atoms in photonic crystals.
10.3 Two Complementary Descriptions of the Dynamics
of a Quantum Open System
The description of the dynamics of a quantum open system is usually done in a
reduced scheme in which the degrees of freedom of the environment are averaged
out. There are different ways of tackling the problem: we may use the reduced den-
sity matrix of the system that evolves according to some master equation or we
may consider some Schr
¨
odinger type of equation that evolves vectors in the Hilbert
space of the system conditioned to the state of the environment. In the last context,
a frequent approach is based on the so-called Stochastic Schr
¨
odinger equations.
Alternatively, other Monte Carlo methods have been developed, as the quantum
jump approach (see Chap. 6). Along the chapter, we will consider a large system
described by a Hamiltonian H, which consists of the addition of a system Hamilto-
nian H
S
, an environment Hamiltonian H
E
, and an interaction Hamiltonian between
the system and the environment H
I
. In the examples discussed here, the system is an
atom, the environment is an electromagnetic field, and their interaction is described
by assuming the dipolar approximation.
10.3.1 Reduced Density Matrix Approach
The state of the system at time t can be characterized by its reduced density matrix,
ρ
S
(t), that is obtained by tracing out the degrees of freedom of the environment from
the density matrix of the whole system ρ(t), i.e., ρ
S
(t) = Tr
E
(ρ(t)), with Tr
E
being
the trace over the environmental degrees of freedom. The reduced density matrix
depends on time and evolves according to some mapping L
t
that when applied to
and initial state ρ
S
(0) gives the state at later time ρ
S
(t). In the Markov case such
mapping reduces to a master equation of Lindblad form ( = 1) [22, 23]:
280 D. Alonso and I. de Vega
dρ
S
(t)
dt
=−i
[
H
s
,ρ
S
(t)
]
+
1
2
n
(
1
L
n
ρ
S
(t), L
†
n
2
+
1
L
n
,ρ
S
(t)L
†
n
2
) ≡ L
t
ρ
S
(t) ,
(10.1)
where L
n
are operators in the Hilbert space of the system that model the coupling
of the system to its environment.
In the non-Markovian case, the Lindblad equation does not rule the dynamics
of ρ
S
, and a more general master equation is needed. An example is the Redfield
equation that has been proven to be valid for long times and within the weak cou-
pling limit [24, 25] by considering a slippage or controlled modification of the initial
condition, ρ
S
(0) [26, 27]. In general, non-Markovian master equations contain some
memory effects in the system–environment interaction which are absent in the Lind-
blad equation. Apart from the Redfield equation, some other non-Markovian master
equation approaches are discussed in [28–30] where memory effects are taken into
account. Another set of time-convolutionless master equations is found in [31, 32].
Other useful methods rely on the computation of the evolution of the Heisenberg
equations of motion [33, 34].
With the solution of the master equation it is then possible to compute the expec-
tation values of system observables. If A is one observable of the system its expec-
tation value is then given by
A(t)=Tr
S
(ρ
S
(t)A) . (10.2)
In this particular sense the dynamics of the atom is solved within the approxima-
tions made.
10.3.2 Stochastic Schr
¨
odinger Equation Approach
Complementary to the master equation approach, different methods have been
developed in many fields that consider the evolution of vectors in the Hilbert
space of the system conditioned to the dynamics of the environment. In such
schemes the environment acts as a source of quantum noise, z
t
[8, 35–38], affecting
stochastically the evolution of the system. If ψ(z, t) is the time-dependent system
vector, and its evolution is driven by a quantity z
t
that depends on the coordinates
of the environment z, one can obtain the reduced density matrix of the system as
ρ
S
(t) = M
1
|ψ
t
(z
∗
)
ψ
t
(z)|
]
, (10.3)
where M is the average over the environmental noise. Such stochastic equations
are said to give an unraveling of the reduced density matrix ρ
S
(t). In the Markov
case there are several unravelings for the Lindblad master equation. Each unravel-
ing corresponds to a particular way of making decomposition (10.3) of the reduced
density matrix. We may found in this context the quantum jump or Monte Carlo
wave function approach [8, 39–43] that involves jumps occurring at random times.
10 Time Scales and Correlations in Non-Markovian Quantum Open Systems 281
Alternatively, the quantum state diffusion approach considers a continuous time
evolution [44–47]. These two approaches or unravelings have provided powerful
methods to numerically compute the evolution of the system quantum mean values.
Moreover, some of them provide us with an interpretation of a particular measur-
ing scheme. In particular, different measurement schemes of the emitted light in a
homodyne or heterodyne detection lead to different stochastic Schr
¨
odinger equa-
tions with distinct diffusive terms [48, 49]. A particular example is the continu-
ous measurements on atoms described by diffusive type of Schr
¨
odinger equation
[50]. The relation of continuous measurements and the stochastic schemes for state
vectors in the Markovian case have been extensively discussed, see for instance
[51, 36].
For non-Markovian interactions, several groups have developed a theory of
stochastic Schr
¨
odinger equations for state vectors [52–54, 26, 27, 55–58, 33, 1].
A different method, which is based on using a pair of state vectors rather than a
single one, was proposed in [59, 36] and further developed in [60]. Nonetheless, the
physical interpretation of non-Markovian Schr
¨
odinger equations is still on debate
[61–63].
10.3.3 Derivation of a Non-Markovian Stochastic Schr
¨
odinger
Equation (NMSSE)
In this section we shall derive an NMSSE, as the one obtained in [52–54]. To that
end we shall consider a Hamiltonian of the following form:
H = H
S
+ H
I
+ H
E
= H
S
+
λ
(g
λ
La
†
λ
+ g
∗
λ
L
†
a
λ
) +
λ
ω
λ
a
†
λ
a
λ
, (10.4)
where = 1, H
E
is a set of harmonic oscillators described by their creation and
annihilation operators a
†
λ
, a
λ
, and with frequencies ω
λ
. In the interaction Hamilto-
nian, L is a system operator that describes the coupling to the environment and g
λ
are the coupling constants of the λth mode to the system. This Hamiltonian is well
suited to describe different systems, in particular the interaction between an atom
and an electromagnetic field when one-photon processes are dominant.
Let |Φ
t
be the state of the whole system in the interaction picture with respect
to the environment. In this representation the Schr
¨
odinger equation reads
i∂
t
|Φ
t
=(H
S
+ H
I
(t))|Φ
t
= H
S
|Φ
t
+
λ
(g
λ
Le
iω
λ
t
a
†
λ
+ g
∗
λ
L
†
e
−iω
λ
t
a
λ
)|Φ
t
. (10.5)
Since the environment is a set of harmonic oscillators, it is convenient to describe
it by using a coherent state basis in the Bargmann representation. A coherent state
of the oscillators |z=|z
1
z
2
···z
λ
···satisfies the relations a
λ
|z=z
λ
|z and
282 D. Alonso and I. de Vega
z|a
λ
=
∂
∂z
∗
λ
z|. The coherent states fulfill a resolution of the identity of the form
1 =
%
dμ(z)|zz|, where the measure dμ(z) is given by
dμ(z) =
d
2
z
1
e
−|z
1
|
2
π
d
2
z
2
e
−|z
2
|
2
π
···
d
2
z
λ
e
−|z
λ
|
2
π
···
=
4
m
d
2
z
m
π
e
−|z
m
|
2
. (10.6)
The state |Φ
t
of the total system may be expressed, using the identity in the
coherent state basis, as
|Φ
t
=
dμ(z)|z|ψ
t
(z
∗
) , (10.7)
where |ψ
t
(z
∗
)=z|Φ
t
is a vector in the Hilbert space of the system and it depends
on the state of the environment |z. The solution of the Schr
¨
odinger equation (10.5)
can be obtained if one finds the system state vector |ψ
t
(z
∗
). The dynamical equation
for |ψ
t
(z
∗
), which follows from Eq. (10.5), is
i∂
t
|ψ
t
(z
∗
)=H
S
|ψ
t
(z
∗
)+Lz
t
|ψ
t
(z
∗
)+L
†
t
0
dsα(t − s)
δ|ψ
t
(z
∗
)
δz
s
, (10.8)
where
z
t
=
λ
g
λ
z
∗
λ
e
iω
λ
t
(10.9)
is a time-dependent function that takes into account the rotation of the environment
oscillators weighted by their coupling to the system. This function acts as a driving
in (10.8). The larger is the coupling of a particular oscillator to the system, the
greater is its contribution to the driving of the system. In the third term of the RHS
of (10.8), a dissipative term containing a memory kernel α(t−s) =
|g
λ
|
2
e
−iω
λ
(t−s)
appears. The functional derivative
δ|ψ
t
(z
∗
)
δz
s
still has to be handled in order to use Eq.
(10.8). In some simple cases, this functional derivative may be computed explicitly,
but in general some approximations are needed. On the other hand, an important set
of relations that has been used along the derivation of (10.8) is
dμ(z)z
t
= M[z
t
] = 0 ,
M[z
t
z
s
] = 0, and M[z
∗
t
z
s
] = α(t −s) . (10.10)
This set of relations allows a stochastic interpretation of (10.8). Indeed, we may
think on z
t
as a colored complex Gaussian noise with mean and correlations given
by (10.10), where α(t − s) is the so-called environment correlation function.The
non-Markovian character of (10.8) is contained in the fact that α(t −s) is not of delta
10 Time Scales and Correlations in Non-Markovian Quantum Open Systems 283
distribution. Let us point out that Eq. (10.8), together with the representation of state
(10.7), corresponds to an equivalent representation of original problem (10.5).
At this point we may proceed in two ways. If we have all the information of
the coupling constants and the frequencies of the harmonic oscillators of the envi-
ronment, we may sample at random and according to the measure dμ(z), a set of
coherent state coordinates z. With such set we construct z
t
and the environment
correlation function α(t − s). For a given initial state of the system we may then
try to solve (10.8). Such solution |ψ
t
(z
∗
) is associated with a single realization of
the coordinates z and represents a single quantum trajectory of the system. If we
sample again the coherent state measure dμ(z) we will obtain different trajectories.
Alternatively we may proceed phenomenologically, i.e., we assume that we know
the correlation function of the environment α(t − s), then we may synthesize a
colored complex Gaussian noise [26, 64] such that it has a correlation α(t −s) and so
properties (10.10) are fulfilled. Again, every different realization of the synthesized
noise would give a different trajectory.
With the set of trajectories |ψ
t
(z
∗
) obtained we may in principle construct the
solution |Φ
t
of (10.5). More of our interest is the reduced density matrix of the
system that will be given by (10.3). In particular, if A is an observable of the system,
its expectation value A(t)=Φ
t
|A|Φ
t
is
A(t)=Tr
S
(ρ
S
(t)A) = M[ψ
t
|A|ψ
t
] . (10.11)
Therefore, the full description of the reduced dynamics can be tackled through
the solution of (10.8).
As we mentioned above the functional derivative
δ|ψ
t
(z
∗
)
δz
s
is difficult to manipulate
in most of the cases. Nonetheless, it is possible to construct some particular forms
of it. One of them consists on writing
δ|ψ
t
(z
∗
)
δz
s
= O(t, s, z)|ψ
t
(z
∗
) and constructing
the operator O(t, s, z) [55, 65]. For instance, within the weak coupling limit, the
operator O(t, s, z) is given by a perturbative expansion on the coupling constant
that leads to an approximation, up to second order, of (10.8) of the form
i∂
t
|ψ
t
(z
∗
)= H
S
|ψ
t
(z
∗
)+Lz
t
|ψ
t
(z
∗
)+L
†
t
0
dsα(t − s)V
t−s
L|ψ
t
(z
∗
)
+higher orders . (10.12)
Here, we have defined the operator V
t
that acts on system operators A, B, C, ...
as V
t
ABC... = e
−iH
S
t
Ae
iH
S
t
BC.... Let us point out that higher order terms in the
expansion of O(t, s, z) in terms of the coupling constant may involve the noise z
t
as shown in [66]. Here the time under which the perturbative expansion is valid
decreases as the square of the coupling constant, as it is stated by the Van Hove
limit [67, 68]. A discussion in the context of a two-level system immersed in a
photonic crystal may be found in [69].
284 D. Alonso and I. de Vega
10.4 Dynamics of Multiple Time Correlation Functions
Many properties of physical systems are described by correlation functions. Among
others, spectral properties of light in interaction with matter rely on the knowledge
of multiple time correlation functions (MTCF). In the case of a system in interaction
with an environment and within the Born–Markov approximation, it is known that
the dynamics of multiple time correlation functions are ruled by the so-called quan-
tum regression theorem [70–72, 9]. Basically, the statement is that the dynamical
equations for the expectation values of system observables (one-time functions) are
the same as those of two-time correlation functions. This principle is also valid to
derive the dynamical equations of higher order correlations.
10.4.1 Two-Time Correlation Functions in the Markov Case:
Quantum Regression Formula
Let us discuss in more detail the quantum regression theorem. Our aim is to compute
the expectation value of a set of system observables A
1
, A
2
, .... According to the dis-
cussion in Sect. 10.3 this may be done through the knowledge of the reduced density
matrix of the systems, ρ
S
(t). Alternatively we may try to compute the expectation
A
1
(t)
at any time studying its dynamics, i.e., the dynamical equation it satisfies.
This equation may be obtained from Eq. (10.1) and integrated along with the initial
condition
A
1
(0)
. The resulting linear equation is
d
A
i
(t)
dt
=
j
L
ij
(t)
"
A
j
(t)
. (10.13)
From the quantum regression theorem it follows that the two-time correlation func-
tions have the same equations of motion, i.e.,
d
A
i
(t +τ )A
k
(t)
dτ
=
j
L
ij
(t)
"
A
j
(t +τ )A
k
(t)
. (10.14)
This set of equations is rather convenient to compute correlations. The conditions
of its validity are discussed in [73] and also in [72, 74].
For non-Markovian interactions, contrary to the Markov case, correlations verify
a different set of equations than quantum mean values. In addition, these equations
contain the memory kernel associated with the environment (the environment cor-
relation function).
In the next section we shall focus on the dynamics of two-time correlation func-
tions of system observables. While higher order correlations can also be evaluated,
we find that to illustrate the theory it is enough to consider only two-time correla-
tions. We will use the system of equations derived in [34, 75] in the weak coupling
limit.