# Non-trivial dynamic regimes of small (nano-scale) quantum systems.

V.A.Benderskii  Institute of Problems of Chemical Physics, RAS
142432 Moscow Region, Chernogolovka, Russia  E. I. Kats  Landau Institute for
Theoretical Physics, RAS,
142432, Chernogolovka, Moscow region, Russia.

###### Abstract

Small (but still containing many, about $10^{2}-10^{4}$, atoms) quantum
systems (traditionally termed nano-systems) are dramatically different from
their macroscopic or genuine microscopic (atomic) cousins. Microscopic
molecular systems (with a few atoms) obey a regular quantum dynamics
(described by time dependent Schrodinger equation), whereas in macroscopic
systems with continuous energy spectra, one can expect, also regular, although
typically relaxation, dynamic behavior. The topic of our paper is in-between
these limits. Nano-scale systems are characterized by small (but finite) mean
inter-level spacing. In such a case with recurrence periods in pico-second
range, Loschmidt echo and double resonance phenomena come into the game.
System behavior becomes non-trivial and manifests a sort of transitions
between regular and chaotic dynamics. We show that such dynamic transitions
occur when the Loschmidt echo time of life exceeds the typical recurrence
cycle period. We illustrate this behavior in the frame work of a few versions
of the exactly solvable quantum problem, proposed long ago by Zwanzig. It is
based on the study of time evolution of the initially prepared vibrational
state coupled to a reservoir with dense spectrum of its vibrational states. In
the simplest version of the Zwanzig model, the reservoir has an equidistant
spectrum, and the system - reservoir coupling matrix elements are independent
of the reservoir states. We generalize the model to include into consideration
the coupling of the initially prepared single state to system phonon
excitations. The coupling results to temperature dependent broadening and
decay of the echo components. Another generalization is to replace a single
level by two states coupled to the Zwanzig reservoir. We anticipate that the
basic ideas inspiring our work can be applied to a large variety of
interesting for the applications nano-systems (e.g., dissipative free
propagation of excitations along molecular chains, or as a model for exchange
reactions).

##  I Introduction

There are materials which do not exhibit unusual properties in the nano-scale
regime. For example simple non-polar hydrocarbon molecules when aggregated are
added merely additively. In other systems, which we are investigating in this
work, the properties may exhibit anomalous values (or behavior of a system is
unusual) in the nano-regime. Common wisdom borrowed from textbooks on quantum
mechanics teaches that a population of somehow initially prepared state of a
macroscopic system, monotonically decreases in time due to energy flow from
this initial state into the states of the reservoir (formed by all continuous
states of the macroscopic system under consideration). In the opposite limit
(a small system with a few degrees of freedom) system behavior is also well
known. The system dynamics is reduced to the recurrence cycles, with their
periods (according to the famous Poincare theorem) determined by the lowest
rationally independent inter-level spacings. As it is often the case, an
intermediate case (relatively large but not macroscopically large quantum
system) is the most difficult one for theoretical analysis. However just such
systems (generically termed as nano-systems) containing about several hundreds
of atoms, become more and more attractive for various applications.

Our intent in this work is to investigate dynamic behavior of these
intermediate nano-size quantum systems. In principle, the full information
about system (both dynamics and statics) is naturally contained in quantum
mechanical solution of the corresponding Schrodinger equation. In practice,
however the quantum solution is unfeasible even for not too large (about
$10^{2}-10^{4}$ atoms) systems. Therefore one has to rely either on heavy ab-
initio numeric, or to look for analytically doable approximations. Luckily for
us, many years ago Robert Zwanzig in a remarkable work (although published not
in a regular and easy accessible journal ZW60 ) proposed a simple (but not
trivial) exactly solvable model of quantum dynamics. Within this model, an
initially prepared single state of a quantum system, which evenly couples to
the dense but discrete spectrum of the reservoir levels. Within the Zwanzig
model, the coupling strength is assumed to be a constant independent of the
reservoir levels. Surprisingly enough that in spite of this evidently
erroneous assumption, Zwanzig approximation correctly identifies the dynamic
regimes and characteristic time scales in the problem. Thus, although the
Zwanzig model is a toy model (in the sense of caricaturing some physical
features), when properly interpreted it can yield quite reasonable values for
a variety of essential (and principally measured) quantities. That such a
simple theory can predict rather complex and subtle features for nano-size
quantum systems is remarkable.

Here we compile a brief self-contained review that we wish had existed when we
first entered the field BF07 . In what follows in the frame work of
generalized Zwanzig approach we study a number of nano-size quantum systems.
Since our work is primary about physics (and only then how it can be modeled
theoretically) it is worth to note first what are physical systems we have in
mind. Our topic is nano-size quantum objects, possessing discrete vibrational
energy spectra. Such objects are carbon nano-tubes, graphene flacks, metallic
clusters, some large organic molecules or their clusters. Typical feature
(relevant for everything what follows in the paper) of such systems is that
their vibrational spectra are discrete and characteristic inter-level spacing
${\bar{\Delta}}$ is on the order of $0.1\,-\,10\,cm^{-1}$. This value of
${\bar{\Delta}}$ is translated into the pico-second time scale for the periods
of the recurrence cycles

$\displaystyle T=\frac{2\pi}{c{\bar{\Delta}}}\simeq
10^{-10}\,-\,10^{-12}\,s\,,$ (1)

where $c$ is the light speed. In turn, experimental studies of the excited
state evolution in large (nano-size) molecular systems UM91  \- LE05  have
established that the processes of vibrational relaxation, energy transfer and
ultra-fast chemical reactions occur in the same pico-second time range as
recurrence cycles. Therefore, the recurrence cycles are essential ingredients
to be included to describe theoretically such features of the quantum dynamics
of nano-systems as irreversibility, chaotic behavior and loss-free distant
energy transfer. These and some other, high resolution experiments (which will
be cited and shortly discussed in our paper) demonstrate that a system initial
vibrational state evolution, as a rule, has a complicated form with irregular
oscillations. The methods of double resonances in the non-linear femto-second
spectroscopy MU95  \- AS04  appear to be especially useful to observe these
irregular oscillations of the initial and final state populations
simultaneously. The phenomenon has been observed for the wide variety of
large-size molecules in liquid and solid solutions and on interfaces FE03  \-
SC02 . Note to the point that irregular oscillations of the population of
initially prepared excited state have been confirmed by the numerous quantum
dynamical calculations LW97  \- FZ06 .

The loss-free excitation energy transfer, observed in linear molecules
consisting of $12-26\,CH_{2}$ fragments BM98 , PN01  also belongs to a similar
class of irregular in time behavior. In such kind of experiments the initial
excitation of the molecular terminal group produces the running wave with
approximately constant speed (about of $2\,ps$ per one fragment). What is even
more surprising that instead of naively expected random distribution of the
excitation energy over the total set of intra-molecular modes, the wave
propagation lasts during several passages along the chain before the uniform
distribution is established. The distant energy transfer occurs between
localized vibrations of fragments separated by about $1\,nm$ distance during
$20\,-\,40\,ps$ whereas the life-time of these vibration excitations
themselves does not exceed $3\,ps$. The matter is that the excitation energy
transfer includes intermediate excitations of delocalized long-lived
vibrations binding the initial and final fragments HT03  \- CW09 .

These data permit to suggest that the irregular evolution is the generic
property of the systems with dense discrete spectra. Therefore, the role of
the reservoir with a discrete spectrum is fundamentally different from that
with a continuous spectrum, typical for macroscopic systems. In the latter
case (with the infinite recurrence period time) the reservoir serves as a sink
for energy flow. Contrary in the case of the discrete spectrum, repetitive
reverse transitions from the reservoir to the initial state and in the
opposite direction, determine the non-trivial and often irregular long-time
dynamics. The synchronization of these reverse transitions results in the
appearance of a multi-component Loschmidt echo phenomenon with a partial
recovery of the initial state population (at the frequency corresponding to
the initial excited state energy), and double resonances (at the frequencies
of the reservoir states). Both effects are responsible for non-monotonic time
evolution. The counterpart of such behavior is the periodic energy
concentration in one of the vibration mode, arising as a result of the time
dependent exchange between reservoir states far from equilibrium.

In a few of previous works of our group BF07 , BG09  \- BK11b  we illustrated
how such complex behavior might appear in the frame work of the simple Zwanzig
model, and how the simplest version of the model can be generalized to relax
some unphysical assumptions of the model. Besides in these works we proposed a
method which makes it possible to solve the dynamical problem analytically
beyond the bare Zwanzig model approximations ( BG09b  \- BK11b ). The main
ingredients of our new method (only schematically and briefly described in the
previous papers) is the representation of the partial amplitudes of recurrence
cycles. Unlike the standard Fourier expansion over eigen-frequencies, this
representation reveals explicitly the time dependent exchange between intra-
molecular states in each recurrence cycle. The fine structure of the Loschmidt
echo arises as a result of the dephasing phenomena associated with the fact
that the exchange of the different reservoir states with the initial state
occurs not at the same instants of time. The synchronization of the reverse
transitions is destroyed when the Loschmidt echo components of the different
recurrence cycles start to overlap. The cycle overlapping determines the
critical recurrence cycle number. Then at larger time (cycle numbers) the
system dynamics is expected to be in the stochastic-like regime.

Since that time we have realized that the quantum dynamics of nano-size
systems is much richer than that predicted for the systems with continuous
spectra, and that non-monotonous in time and irregular dynamics is a robust
and generic feature of almost arbitrary quantum system with
$10^{2}\,-\,10^{4}$ degrees of freedom. Moreover the regular-stochastic
dynamic transition (crossover) yields to a loss of the expected one-to-one
correspondence between system spectrum and its long time dynamics. For all our
examples the spectrum remains deterministic, while long time dynamics
eventually becomes stochastic. This unusual combination is the specific
feature of nano-systems.

Motivated by this new understanding we decided to combine altogether our
previous works to illustrate our method on a number of particular physical
realizations (only partially overlapping with those in the previous works).
The aim to present this review arises from the conviction that unifying our
previous works supplemented by the new applications of the developed
theoretical approach and by new experimental data and observations collected
in very recent year, yield to a new stage of development of this field:
dynamics of nano-systems.

Our manuscript is divided into 7 sections. After this introduction in section
II we summarize shortly the results on quantum dynamics of the bare Zwanzig
model. It is discussed also a possible physical realization most closely
satisfying the model assumptions. The section contains also an extended list
of references to compensate partially its brevity. Then in section III
utilizing developed in our works BG09b  \- BK11b  theoretical approach we
investigate with all details the evolution of the population of the initially
prepared single state of the system. In section IV we analyze various
physically motivated generalizations of the bare Zwanzig model, and how our
analytical method should be modified to describe theoretically these
generalizations. In the same section (subsection IV.3 we show that on similar
footing we can study dynamics not only for an individual (single) nano-system
but as well for the ensemble of somehow distributed nano-systems. We
investigate the reservoir states evolution in section V. We collect some
already discussed in the literature and new applications of our approach to
physically interesting phenomena in nano-systems. Namely, two-level systems
coupled to reservoir (subsection VI.1), propagation of vibrational excitations
in nano-size chains (subsection VI.2). The last section VII summarizes the
main findings of our work, with a discussion of possible physical consequences
and interpretation of the results.

##  II Zwanzig model and its approximate physical realization

Mentioned in the previous section Zwanzig model ZW60  considers a quantum
system where somehow initially prepared a single excited vibrational energy
level is coupled with a constant coupling strength ($C$) to the equidistant
discrete reservoir states. In such a formulation (termed in what follows as
the bare Zwanzig model) the model allows the exact analytic solution. We will
present below briefly the exact solution (for time evolution of the population
for the initial state and reservoir states) found in BF07 , however it is
worth to noting first about a possibility of physical realizations of the bare
Zwanzig model assumptions. As an approximate realization one can have in mind
the well-known scheme of radiation-less transitions UM91 . In this
experimentally realizable scheme the initially excited state is prepared by a
short optical pulse pumping (see Fig.1).

Figure 1: Scheme of radiation-less transitions $\\{R_{n}\\}\leftarrow S$,
accompanying by the optical transition from the ground to initial state
$S\leftarrow S_{0}$. Direct optical transitions $\\{R_{n}\\}\leftarrow S_{0}$
are forbidden.

It is essential that the initial vibrational state $S$ belongs to the so-
called narrow potential energy surface. If the intervals between neighboring
levels on this surface exceed the spectral width of the pumping pulse, only a
single initial state enters into the game. The final states $\\{R_{n}\\}$
belong to the bright low-lying energy surfaces, but the direct optical
transitions to these states are forbidden (e.g., due to symmetry
restrictions). The $SR$ interaction leads to a formation of the mixed
(coupled) states, which are certain superpositions of the initial and
reservoir states. The absorption band related to the transition from the
ground to these mixed states acquires the fine structure with the components
which are determined by the individual mixed states. The envelope of the
absorption band $S\leftarrow S_{0}$ is determined by the width of the $SR$
interaction region. Since the reservoir states are the eigen-states of the
single energy surface, they are ortho-normal states. The Hamiltonian
describing the $SR$ interaction between the initial state (with its eigen-
function $\varphi_{s}$) and the reservoir states (with their eigen-functions
$\\{\varphi_{n}\\}$ has the following form (in the units where $\hbar=1$)

$\displaystyle
H=\varepsilon^{0}_{s}b_{s}^{+}b_{s}+\sum_{n}\varepsilon_{n}^{0}b_{n}^{+}b_{n}+\sum_{n}C_{n}(b_{s}^{+}b_{n}+b_{s}b_{n}^{+})\,,$
(2)

where $b_{s}^{+},b_{n}^{+},b_{s},b_{n}$ are creation and annihilation
operators of the corresponding states

$\displaystyle
b_{s,(n)}\Phi_{0}=0\,,\,b_{s,(n)}^{+}\Phi_{0}=\varphi_{s,(n)}\,,\,b_{s(n)}\varphi_{s(n)}=\Phi_{0}\,,\,b_{s(n)}^{+}\varphi_{s(n)}=0\,,$
(3)

with $\Phi_{0}$ being the ground state wave function. The last of the Eqs (3)
is valid if the resonance region of the $SR$ interaction contains only one
excited state per each vibrational mode. For the bare Zwanzig model
Hamiltonian the coupling constant $C_{n}$ is independent of $n$, i.e.,
$C_{n}\equiv C$. Since the basis set of the reservoir states is ortho-normal,
the Hamiltonian matrix has only one non-zero column and one non-zero line of
the interaction matrix elements (apart from the diagonal elements
corresponding to the bare (non-perturbed by the interaction) eigen-values.
Owing to this simple structure of the Hamiltonian matrix, the secular equation
reads as

$\displaystyle
F(\varepsilon)=\epsilon\sum_{n}\frac{C_{n}^{2}}{\varepsilon-\varepsilon_{n}^{0}}=0\,,$
(4)

where the energy of the initial state is chosen as the zero, and the inter-
level spacing is chosen as the unity of the energy scale. The dipole matrix
element of the $S\leftarrow S_{0}$ transition is determined by the energy
presentation of the Green function (see e.g., UM91  or its textbook version in
MA76 . Within the above assumptions (the single initial state and forbidden
direct $\\{R_{n}\\}\leftarrow S_{0}$ transitions), the dipole moment $\mu$ of
the transition is energy independent, and the absorption band is determined by
the following function $I(\varepsilon)$

$\displaystyle
I(\varepsilon)=-\pi^{-1}|\mu|^{2}Im\langle\varphi_{s}|G(\varepsilon)|\varphi_{s}\rangle\,,$
(5)

where

$\displaystyle G(\varepsilon)=(\varepsilon-H)^{-1}\,.$ (6)

Since the poles of the Green function $G(\varepsilon)$ in the lowest half-
plane are defined by the roots of the secular equation

$\displaystyle lim_{\eta\to
0}\langle\varphi_{s}|G(\varepsilon+i\eta)|\varphi_{s}\rangle=(F(\varepsilon))^{-1}\,,$
(7)

and the function $1/F(\varepsilon)$ in the vicinity of the pole
$\varepsilon=\varepsilon_{n}^{*}$ is proportional to the residue in this pole,
the shape of the absorption band is given by

$\displaystyle\langle\varphi_{s}|G(\varepsilon)|\varphi_{s}\rangle\simeq-i\pi\sum_{n}\left.\left(\frac{dF}{d\varepsilon}\right)^{-1}\right|_{F(\varepsilon)}=0\,.$
(8)

The sum in Eq. (4) is singular in all points
$\varepsilon=\varepsilon_{n}^{0}$, so that one root is located in each
interval $[\varepsilon_{n}^{0}\,,\,\varepsilon_{n+1}^{0}$. Separating the
singular term $(u_{n}^{-1})$ from the bare energy value $\varepsilon_{n}^{0}$,
(i.e., $\varepsilon_{n}=\varepsilon_{n}^{0}+u_{n}$) one can rewrite Eq. (4) as
the expansion over the moments of the mixed reservoir states

$\displaystyle
F(u_{n})=u_{n}+\varepsilon_{n}^{0}-\frac{C_{n}^{2}}{u_{n}}+\sum_{\nu=0}^{\infty}M_{n\nu}u^{\nu}\,.$
(9)

Here

$\displaystyle
M_{n\nu}=\sum_{k}\frac{C_{n+k}^{2}}{\Delta_{n+k}^{\nu+1}}\,;\,\Delta_{n+k}=\varepsilon_{n+k}^{0}-\varepsilon_{n}^{0}\,.$
(10)

In the simplest bare Zwanzig model ZW60 , the spectrum of the unperturbed
reservoir is assumed to be equidistant and interaction matrix elements $C_{n}$
and line widths $\gamma_{n}$ are the same for the all reservoir states, i.e.,

$\displaystyle\varepsilon_{n}^{0}=n\,,\,C_{n}=C\,,\,\gamma_{n}=\gamma\,.$ (11)

Then, the Eq. (4) reads as

$\displaystyle F(\varepsilon)=\varepsilon+i\gamma-\Gamma
cot(\pi(\varepsilon+i\gamma))=0\,,$ (12)

where for the sake of simplicity, the reservoir level $n=0$ is assumed to be
in the resonance with the initial state. Eqs. (5), (8), and (12) show that the
envelop of the absorption band is the Lorentzian and its half-width is equal
to the interaction region width $\Gamma$

$\displaystyle\rho(\varepsilon)=\pi^{-2}\frac{\Gamma}{\varepsilon^{2}+\Gamma^{2}}\sum_{n}\frac{\gamma}{(\varepsilon-n)^{2}+\gamma^{2}}\,.$
(13)

where $\Gamma=\pi C^{2}$. To get the well-resolved spectrum, the half-widths
of the components $\gamma_{n}$ has to be smaller than the characteristic
inter-level spacing

$\displaystyle\gamma_{n}\ll\varepsilon_{n+1}-\varepsilon_{n}\,.$ (14)

##  III Initial state evolution within the bare Zwanzig model.

Textbook quantum mechanical wisdom teaches that an arbitrary wave function is
expanded over the complete basis set of the eigen-functions with time
dependent coefficients

$\displaystyle\Psi(t)=a_{s}(t)\varphi_{s}(q)+\sum_{n}a_{n}(t)\varphi_{n}(q_{n})\,,$
(15)

where $q$ and $q_{n}$ stand for the initial state and reservoir states degrees
of freedom. These coefficients satisfy the Heisenberg equations of motion

$\displaystyle i\frac{\partial a_{s}}{\partial
t}=\epsilon_{s}^{0}a_{s}+\sum_{n}C_{n}a_{n}\,;\,i\frac{\partial
a_{n}}{\partial t}=\varepsilon_{n}^{0}+C_{n}a_{s}\,.$ (16)

Supplementing the Eqs. (16) by the natural initial conditions

$\displaystyle a_{s}(0)=a_{s0}\,,\,a_{n}(0)=a_{n0}\,$ (17)

their solution can be expanded into Fourier series. The terms in this series
are determined by the energy eigen-values of the Hamiltonian matrix, i.e., by
the secular equation (4)

$\displaystyle a_{s}(t)=(2\pi
i)^{-1}\int_{-\infty}^{\infty}(a_{s0}+R(\varepsilon))\exp(-i\varepsilon
t)\frac{d\varepsilon}{dF(\varepsilon)}=\sum_{n}(a_{s0}+R(\varepsilon))\left.\left(\frac{dF}{d\varepsilon}\right)\right|_{F(\varepsilon)=0}\,,$
(18)

where the entering (18) function $R(\varepsilon)$ is

$\displaystyle
R(\varepsilon)=\sum_{n}\frac{C_{n}a_{n0}}{\varepsilon-\varepsilon_{n}^{0}}\,.$
(19)

For the bare Zwanzig model and zero value for the initial amplitudes of all
reservoir states, $a_{n0}=0$, the time-dependent initial state amplitude is
given by

$\displaystyle
a_{s}(t)=2\sum_{n=0}^{n=\infty}\frac{\cos(\varepsilon_{n}t)}{1+\pi^{2}C^{2}(\varepsilon_{n}/C)^{2}}\,.$
(20)

The sum (20) gives the exact solution of the dynamical problem. If the initial
state is strongly coupled to reservoir states ($\Gamma\gg 1$), the sum
contains the large number of terms oscillating with rationally independent
frequencies. It is not an easy task to identify the recurrence cycles from
such a sum of the oscillating terms. To overcome the difficulty we apply the
generalized Poisson summation formula introduced in BG09 , BG09b . This method
makes it possible to express the initial state amplitude $a_{s}(t)$ in terms
of the recurrence cycle partial amplitudes $a_{s}^{(k)}$

$\displaystyle a_{s}(t)=\sum_{k}a_{s}^{(k)}(t-2k\pi)\,$ (21)

with the partial cycle amplitudes expressed as

$\displaystyle
a_{s}^{(k)}(t-2k\pi)=\frac{\Gamma}{\pi}\exp(-2k\pi\gamma)\int_{-\infty}^{\infty}d\varepsilon\frac{(\varepsilon+i(\Gamma-\gamma_{s}))^{k-1}}{(\varepsilon-i(\Gamma+\gamma_{s}))^{k+1}}\exp(i\varepsilon(t-2k\pi))\,,$
(22)

where $\gamma_{s}$ and $\gamma$ are the widths of the initial and reservoir
states respectively. Pure mathematically Eq. (22) replaces a discrete set of
poles situated close to the real axis by a single pole on the imaginary axis,
which characterizes the decay probability of the initial quasi-stationary
state.

We are interested in the initial state population evolution for $t\geq 0$.
Then the sum (21) contains only the finite number of partial amplitudes for
the cycle numbers in the interval $0\leq k\leq[t/2\pi]$ ($[.]$ denotes the
integer part). For the $k=0$ cycle, the integrand the in Eq (22) has two poles
at $\varepsilon=\pm i\Gamma$. Only the positive pole is inside the integration
contour comprising the real axis and half-circle of an infinite (very large)
radius in the upper complex half-plane, where the integrand is zero. For $k<0$
only the negative pole remains. and it is outside the integration contour.
Hence, the cycles with $k<0$ do not contribute to $a_{s}(t)$. For
$k>[t/2\pi]$, the integration contour lies in the lower complex half-plane and
no pole $\varepsilon=i\Gamma$. Therefore $a_{s}^{(k)}$ are nonzero only for
$t\geq 2k\pi$, i.e., they proportional to the Heaviside step functions
$\theta(t-2k\pi)$. In the initial cycle $k=0$, the amplitude, describing the
initial state $S$ population evolution, exhibits exponential decay, with its
decrement including that of the intrinsic (own) energy level $S$ decay and the
decrement due to $S-R$ -transitions

$\displaystyle a_{s}^{(0)}(t)=\exp(-(\Gamma+\gamma_{s})t)\,.$ (23)

The strong coupling condition $\Gamma\gg 1$ ensures exponential decay when the
$SR$ -transitions occur into the large number of the final states of the
reservoir with $n\leq\Gamma$. Then the wave function destructive interference
suppresses coherent oscillations. In the weak coupling limit, resonance
transitions between the initial state $S$ and the reservoir state $n=0$ govern
the dynamic behavior. It is worth to note to the point that the decay rate
predicted by the Fermi Golden Rule approximation is $2\Gamma$. The reason for
the error is obvious: this approximation is valid only if the amplitude is a
slow (in comparison with the rapidly oscillating terms) function of time, what
is evidently not the case for the essential oscillating contributions of the
recurrence cycles in the strong coupling case BK08 . All the more the
subsequent revivals of the initial state amplitude for the cycles $k\geq 1$
can not be reproduced within the Fermi Golden Rule approximation.

Calculating residue in the $(k+1)$-th order pole at
$\varepsilon=i(\Gamma+i\gamma_{s})$, one can express the integral (22) in
terms of the ortho-normalized set of the Laguerre functions (the products of
the Laguerre polynomials and exponential factors) BG09 , BG09b

$\displaystyle a_{s}^{(k)}(t)=\exp(-2k\pi\Gamma
t)\exp(-(\gamma_{s}(t-2k\pi)))\,,$ (24)

where $a_{s0}^{(k)}(t)$ is the partial amplitude at $\gamma=\gamma_{s}=0$

$\displaystyle
a_{s0}^{(k)}(t)=\frac{\tau_{k}}{k}\frac{dL_{k}(\tau_{k})}{d\tau_{k}}\exp(-(\tau_{k}/2))\theta(\tau_{k})=-\frac{\tau_{k}}{k}L^{1}_{k-1}(\tau_{k})\exp(-(\tau_{k}/2))\theta(\tau_{k})\,,$
(25)

where $\tau_{k}$ stands for the local time of the $k$ -th cycle

$\displaystyle\tau_{k}=2\pi(t-2k\pi)C^{2}\,.$ (26)

The expressions (21) - (26) above present the exact solution of the dynamical
problem for the bare Zwanzig model. Note that the initial state evolution can
be described by two equivalent sums (20) and (21). The first sum being the
Fourier transform of the energy spectrum, presents the initial state amplitude
as a superposition of the coherent oscillations, whereas the second sum
depending on the total transition probability into all reservoir states,
manifests explicitly the partial recovery of the initial state amplitude
during each recurrence cycle. This phenomenon is known as Loschmidt echo. From
the expressions (24) - (26) we see the following properties of the Loschmidt
echo: (i) the number of zeros and the number of components of $k$-th partial
amplitude is $k$, (ii) the total width of the state population evolution in
time (which is determined by the width of the oscillation region for the
Laguerre polynomials) is $4k$. We illustrate these properties in Fig. 2, where
we show the results of the direct Matlab solutions of the Eqs. (21) - (26).

Figure 2: The initial state evolution for the bare Zwanzig model. $C^{2}=1$,
and $\gamma=0$ The upper panel shows the initial cycles with the regular
dynamics. The lower panel demonstrates chaotic-like long time dynamics in the
case of the overlapping recurrence cycles.

One essential physical remark is in order here. The fact is that since the
recurrence cycle period increases proportionally to density of the reservoir
states. Eventually it becomes longer than the time interval available for
measurements. It happens when the density of states exceeds the corresponding
particular experiment-dependent limit. In this case only the exponential decay
in the cycle $k=0$ is observed. This limit, (depending on the measurement
resolution), distinguishes the systems with continuous and discrete spectra.
As we emphasize in the Introduction section, nano-particles posses typically
the discrete vibrational spectra. For such a system during the cycle $k=0$ the
initial state population is completely transferred to the reservoir states. At
the beginning of the cycle $k=1$, the reverse transitions partially restore
the initial state amplitude, $a_{s}(t)$. However at the end of the first cycle
the initial state population becomes to be depleted again. Since the $SR$
exchange of the populations leads to the additional phase shifts, the
amplitudes of the reservoir states at the beginning of second cycle $k=2$
consists of two components with different phases, and their exchange with the
initial state produces the two-component echo signal. The first its component
is related to amplitude that comes from the cycle $k=1$, while the second
component appears due to the $SR$ exchange in the cycle $k=2$. Similarly, the
partial amplitude of the $k$ -th cycle split into $k$ components with
different phases coming from the preceding cycles. Since the total width of
the echo signals grows with increasing of the cycle number, the components of
the neighboring cycles begin to overlap when the cycle width becomes equal to
the cycle period, $4\pi\Gamma$. This critical cycle number is

$\displaystyle k_{c}=\pi^{2}C^{2}\,.$ (27)

For $k<k_{c}$ the cycles do not overlap and the Loschmidt echo signal has the
well-resolved components. For $k>k_{c}$ the echo signal structure becomes more
and more complex due to mixing of the components of the different cycles. The
specific features of the above described evolution, can be formulated and
analyzed in terms of the Langevin-like dynamic equations. Indeed, eliminating
the amplitudes of the reservoir states, the Heisenberg equations of motion
(16) can be written as those for the harmonic oscillator with dissipative
(friction) and reactive forces

$\displaystyle{\ddot{a}_{s0}}+C^{2}\int_{0}^{t}{\dot{a}_{s0}}(t^{\prime})G(t-t^{\prime})dt^{\prime}=-C^{2}G(t)\,;\,G(t)=2\pi\sum_{k=-\infty}^{\infty}\delta(t-2k\pi)-1\,.$
(28)

The Eqs. (28) show that the transitions to the resonance level $n=0$ lead to
the coherent oscillations of the initial state population with the frequency
which is determined by the $SR$ interaction matrix element. The transitions to
the non-resonant levels $n\neq 0$ produces the friction and reactive forces.
In the bare Zwanzig model, the reactive force has the form of the periodic
$\delta$ -pulses with the recurrence cycle periodicity. It is instructive to
note the difference in the dynamic behavior between the Zwanzig model and the
extensively studied models for the kicked periodically systems (see e.g., the
monographs ZA85 , TA89 , and reviews works FH96 , IZ90 , and the literature
cited therein). The matter is that Within the Zwanzig model the friction force
(the integral term in the Eq. (28) is non-local and it depends on the initial
state amplitude. Besides the reactive force has a fine structure and it
contains the infinite number of the periodic terms.

The average value of the initial state population per cycle can be expressed
as

$\displaystyle\langle\left(a_{s0}^{(k)}\right)^{2}\rangle_{k}=\frac{\pi}{2kk_{c}}\int_{0}^{4k_{c}}|a_{s0}^{(k)}(\tau_{k})|^{2}d\tau_{k}\,.$
(29)

The integral (29) is $k$-independent and in the both limits $k<k_{c}$ and
$k>k_{c}$ it is approximately $1/\Gamma$. The constant average value for cycle
amplitudes in the limit $k\gg k_{c}$ signals that the mixing in the strongly
overlapped cycles is a stationary random process. Owing to the cycle overlap
the dynamics includes the large number of combination frequencies. Their
interference results in the stochastic behavior of the time dependent
amplitudes. The stationary behavior is confirmed also by the constant number
of the extremal points per cycle ($\simeq 2k_{c}$) and is $k$ -independent
decay of the average per cycle correlation functions. These properties are
typical for a dynamical system with mixing. Such kind of the paradoxical
behavior (irregular, quasi-chaotic dynamics of the deterministic systems)
admits a statistical description. The characteristics of the bare Zwanzig
model, found above, show that this model can also be treated as a
deterministic dynamical system whose long-time evolution is a random-like.
Mixing of zeros of the state population amplitudes, and decrease in the
intervals between these zeroes make the initial state amplitude very sensitive
to small variations of the time. Therefore an arbitrary small coarse graining
destroys the deterministic dynamics. Upon such coarsening, the system loses
both the reversibility in time (i.e., the initial state wave function cannot
be determined from the long-time evolution of the coarsening amplitudes) and
one-to-one correspondence between the system dynamics and its spectrum. In
order to retain the deterministic character of the long-time evolution, it is
necessary to measure more and more rapidly oscillating amplitudes within more
and more narrow time intervals, which is impossible in practice for the finite
measurement accuracy. One can say that the chaos appears due to a coupling of
the dynamical system to a measurement device (c.p., with ZU82  or GR93 ).

##  IV Generalizations of the Zwanzig model

The first step to relax the rather severe restrictions of the bare Zwanzig
model, is to deform somehow the reservoir equidistant spectrum. To illustrate
how it works in the following subsection IV.1 we study only two special types
of the spectrum deformations, and then, armed by this knowledge in subsection
IV.2 we investigate more general spectral deformations, enabling to describe
almost arbitrary reservoir spectra. It is worth to note that if the order of
the reservoir levels does not change upon deformations
$(\varepsilon_{n+1}^{0}-\varepsilon_{n}^{0}>0)$ (i.e., for any $n$, there are
no level permutations), the spectrum remains regular. If it is not the case,
mixing of the levels occurs. One should distinguish this spectral mixing,
producing chaotic behavior in quantum mechanic systems, and also known dynamic
mixing phenomena leading to chaotic classical dynamics SI77  \- HA90 .

###  IV.1 Simple deformations of the equidistant reservoir spectrum

To illustrate how it works in this subsection we study only two special types
of the spectrum deformations. Namely,

  * •

Homogeneous spectral deformations, producing from the bare Zwanzig model
reservoir, that with non-equidistant but still regular spectrum

$\displaystyle\varepsilon_{n}^{2}=n^{2}(1+a^{2}n^{2})^{\pm
1};\,C_{n}^{2}=C^{2}(1+b^{2}n^{2})^{\pm 1}\,,$ (30)

where the sign $\pm$ correspond to stretching (sign $+$) or compression (sign
$-$) of the spectrum, and constants $a$ and $b$ are assumed to be small
$a\,,\,b\,\ll 1$.

  * •

Splitting of the spectrum into $K$ displaces sub-lattices with the same
periods for each of the sub-lattice

$\displaystyle\varepsilon_{n0}=\pm nK\,;\,\varepsilon_{nk}=\pm(nK+k)+x_{k}\,,$
(31)

where $k=0\,,\,1\,,\,K-1$ and $x_{1}<x_{2}...<x_{K-1}<1/2$.

Spectral mixing occurs when the reservoir spectrum is split into sub-lattices
with rationally independent periods, i.e.,

$\displaystyle\varepsilon_{n0}=\pm
nK\,;\,\varepsilon_{nk}=\pm(nK+k)(1+\delta_{k})\,,$ (32)

where $k=0\,,\,1\,,\,K-1$ and $\delta_{1}<\delta_{2}...<\delta_{K-1}<1/2$. The
deformation (32) gives to the resonances between the levels from the different
sub-lattices at the values of $nk$ and $n^{\prime}k^{\prime}$ leading to small
denominators in the expansion (9). As a result, the tails of the envelope of
the absorption band are deformed in the far spectral region for the reservoir
levels with $n\simeq 1/\delta_{k}$ (although the absorption band central
region is almost unchanged under this spectral deformation).

In order to study the absorption band shape for an arbitrary reservoir
spectrum, it is convenient to utilize the formal scaling relation transforming
the arbitrary spectrum into the equidistant one. One more advantage of this
scaling transformation that it allows us to discuss the spectral and dynamical
effects of deformations on the same footing. For the both types of the non-
equidistant spectra listed above the secular equation has the form

$\displaystyle F(\lambda)=P(\lambda)(Q(\lambda)-cot(\pi\lambda))=0\,,$ (33)

where the auxiliary variable $\lambda$ is introduced in order to map the sub-
sequence of the intervals $[\varepsilon_{n}^{0}\,\varepsilon_{n+1}^{0}]$ for
the reservoir spectrum under consideration into the sub-sequences $[n\,n+1]$
for the bare Zwanzig model reservoir levels. To perform the mapping we
introduce two functions $g(\lambda)$ and $h(\lambda)$ which determine the
deformation of the spectrum

$\displaystyle\varepsilon_{n}^{0}=ng(\lambda)\delta(\lambda-n)\,;\,C_{n}^{2}=C^{2}h(\lambda)\delta(\lambda-n)\,.$
(34)

The summation in the secular equation (4) for the deformed spectrum gives

$\displaystyle P(\lambda)=\Gamma\frac{h(\lambda)}{g(\lambda)}\,.$ (35)

The existence of small or even vanishing inter-level spacings in the
generalized secular equation (33) is related to the characteristic function
$Q(\lambda)$, which in own turn depends on $\Gamma$ and the deformation
parameters entering the Eqs. (30) - (32). For mixing-free deformations the
moments $M_{\nu n}$ (10) are monotonic functions of the deformation
parameters. However when the deformation parameter $\delta$ entering in the
Eq. (32) grows upon certain critical value. at least one interval approaches
to zero. These very small intervals correspond to the maxima of the $M_{\nu
n}$ moments. The number of small intervals increases upon increasing of the
mixing parameter, $\delta$. Moreover unlike the homogeneous deformations (30)
and (31) which decrease the number of mixed states, the mixing deformation
(32) leads to a broadening of the absorption band. The critical value of the
deformation parameter $\delta$ corresponds to the appearance of one
permutation of order of the intervals within the interaction region, $\Gamma$.
If the number of mixing sub-lattices is three ($K=3$), the critical value is
$\delta_{c}\simeq(1/3\Gamma)$. Our numerical computations suggest the
following relation between the critical deformation parameter and $\Gamma$,
namely

$\displaystyle\delta_{c}\Gamma\simeq 0.20\,-\,0.35\,.$ (36)

Eq. (36) is valid for two-parametric deformations including superposition of
homogeneous deformations with mixing. When the deformation parameters grow,
the isolated resonances start to overlap and initially regular spectra look as
random-like ones. The results of our numeric computations, presented in Fig. 3
illustrate this phenomenon (crossover from the regular to quasi-random
spectrum upon increasing the deformation parameter).

Figure 3: Crossover from the regular to quasi-random spectrum of the mixed
$RS$ states upon the mixing deformation. $K=3$,
$\delta_{1}=0\,;\,0.1\,;\,0.4\,,\,0.7$, $\delta_{2}=2\delta_{1}$ (from the top
to the bottom). $C^{2}=2\gamma=0.1$.

It is worth to note that this figure shows that the duration of the transition
from the beginning of overlapping of the recurrence cycles to deep mixing
behavior is several times longer than the overlapping time. The wide crossover
region, (”weak chaos”) is due to the multicomponent structure of the Loschmidt
echo, where oscillations with different frequencies and phases are involved.
The influence of the reservoir spectrum on the duration of the transition
region is manifested in the ”lag effect” discussed in Section V.

###  IV.2 Arbitrary reservoir spectra.

To analyze more general deformations of the reservoir spectra, let us re-write
the expression for the partial recurrence cycle amplitudes (25) in the form
valid for the deformed reservoir spectrum. For the secular equation (33), the
partial amplitudes can be derived using as above the generalized Poisson
summation (which works even for the complex-valued $\lambda$) we get

$\displaystyle
a_{s}^{(k)}(\tau)=\pi^{-1}\int_{-\infty}^{\infty}\exp(i(P(\lambda)Q(\lambda)t-2k\pi\lambda))\frac{d\varepsilon}{d\lambda}\frac{1}{P(\lambda)(1+Q^{2}(\lambda))}\left(\frac{Q(\lambda)+i}{Q(\lambda)-i}\right)^{k}d\lambda\,.$
(37)

For the bare Zwanzig model, $P(\lambda)=1$, $Q(\lambda)=\lambda$, and
$\lambda=\varepsilon/\Gamma$. The main new feature (in comparison to its
Zwanzig model counterpart (22)) of the expressions (37) for the partial cycle
amplitudes, derived from the generalized secular equation (33), is that the
recurrence cycle period in Eq. (37) depends on time and, hence, local time of
$k$-th cycle depends on $\varepsilon/\lambda$

$\displaystyle\tau_{k}(\lambda)=\frac{\varepsilon}{\lambda}t-2k\pi\,.$ (38)

The deformations change the number and positions of the poles in the integrand
in Eq. (37). Therefore the integration contour should be also changed to be
adapted to the spectrum deformations. If the deformation has a form

$\displaystyle Q(\lambda)=\lambda+\eta\phi\,,$ (39)

where $\eta\ll 1$, and $\phi(\lambda)$ is assumed to be a single power
function

$\displaystyle\phi(\lambda)=\lambda^{2s+1}\,,\,s\geq 1\,.$ (40)

For such a deformation in the integrand in Eq. (37) for the partial cycle
amplitudes there appear new $2s$ poles (in the addition to a slightly
displaced the pole $\lambda=i$ of the bare Zwanzig model). Correspondingly, in
such a case, the initial state evolution is described by superposition of
$(2s+1)$ exponential terms. For the sake of simplicity let us consider the
case $s=1$. Then the 3 poles are

$\displaystyle\lambda_{1}=i\left(1+\eta+\frac{3}{2}\eta^{2}+....\right)\,;\lambda_{2,3}=\pm
i\left(\eta^{-1/2}\mp(1/2)+...\right)\,.$ (41)

At the critical value of the deformation parameter, $\eta_{c}$ the poles
$\lambda_{2}$ and $\lambda_{3}$ become complex conjugated and are merged
together forming a single second order pole. If $\eta\ll\eta_{c}$, then
$|\lambda_{1}|\ll|\lambda_{2,3}|$, and the contribution of the new poles is
small. As a result, we find in the agreement with Eq. (29), that the initial
state amplitude decays almost exponentially with its decrement close to
$\Gamma$. When the parameter, $\eta$, grows the decrement constant increases.
The initial state population exhibits the damped oscillations when the complex
conjugated poles appear. In this case, for the initial cycle $k=0$ its time
evolution becomes non-exponential. The level population decay can be
characterized by the following time-dependent effective decrement

$\displaystyle d_{s}(t)=-\frac{d(log(a_{s}^{(0)}))}{dt}\,.$ (42)

The described above spectrum deformations create also the poles on the real
axis, which result in oscillations of the partial amplitudes in the negative
local time region, $t<2k\pi$. This a bit counter-intuitive backward evolution
(which, as it should be, disappears at $\eta=0$ is shown in the Fig.4

Figure 4: Backward evolution of the initial state amplitude for the cycle
$k=1$. The reservoir posses the increasing inter-level spacings according to
the Exp.(30). $C^{2}=1$. The entering Eq. (30) parameters: $b=0$,
$a=0\,;\,0.15\,;\,0.25\,;\,0.35$ (curves 1-4, respectively).

From the all above examples we conclude that the crossover from regular to
quasi-chaotic behavior is strongly dependent on the spectrum of the $SR$ mixed
states. The number of the cycles with regular dynamics increases when the
spectrum approaches to the equidistant one. In the case of the special
spectrum deformation (31), the function $\phi(\lambda)$ can be represented as
a trigonometric function of the angle $2\pi\varepsilon/(2s+1)$. The period of
this trigonometric function is smaller than the basic period in the secular
equation (33). This energy dependent angle in the deformation function
$\phi(\lambda)$ leads to the satellites (weak components) in each of the
partial cycle amplitude (apart from the basic Loshmidt echo characteristic for
the bare Zwanzig model). In time domain the satellites are shifted by $2\pi
m/(2s+1)$ (where $m=1,.....,s$ in each of the recurrence cycle. The
intensities of the new components increase with increasing of the deformation
parameter $\delta$ in the Eq. (31). Alike the basic echo, the satellites are
split into components, which start to overlap upon increasing of the cycle
number. As a result, the critical cycle number decreases due to the mixing. As
it is shown in the Fig. 5, the similar change in the evolution behavior
results from the mixing deformations (32). For $\delta=0$ (panel 5 a) the
critical cycle number is given by the Eq. (27). For $\delta=0.019$ and
$\delta=0.049$ (panels 5b and 5c) the satellite intensities are smaller than
the basic components. The spectral triplets (which can be clearly visible in
Fig 3) are accompanied by the new triplets in the fine structure of the
absorption band. The decreasing of the critical cycle number, $k_{c}(\delta)$,
corresponds to transfer of the satellites population for a given cycle into
the next cycle. The mixing of the basic components and satellites produce the
Loshmidt echo signal, consisting of many combined components with looking
irregular intensity distributions. In a contrast with the bare Zwanzig model,
the number of the echo signal components becomes several times greater than
the cycle number. For $\delta=0.079$ (panels 5d) the intensities of the basic
components and satellites are already comparable, and the initial triplet
structure is destroyed already in a few initial cycles, $k\leq 3$. Our numeric
results presented in the Fig 5 demonstrate the change of the time evolution
behavior under combined action of the dynamical (at $\delta=0$) and spectral
mixing. If these processes are assumed to be mutually independent, the
effective critical cycle number ${\tilde{k}}_{c}$ (for the reservoir
consisting of three sub-lattices, $K=3$) is given by the following natural
interpolation relation (confirmed by our numeric computations)

$\displaystyle{{\tilde{k}}_{c}}^{-1}\simeq k_{c}^{-1}+3\delta\,.$ (43)

The numerical calculations indicate that the Eq. (43) is satisfactory when
$0\leq\delta\leq 0.15$ and $1\leq C^{2}\leq 4$. Note that the very possibility
of such statistical description of the systems with spectral mixing is based
on the ergodic hypothesis, which suggests that a single dynamical system
visits all available states in its phase space, during the time evolution
process (see, for example, KM10 , IZ90 , KB98 . A comment concerning the
interpolation formulas (43) is needed here. The fact is that for realistic
parameters of nano-tubes or fullerens (or some other nano-particles) their
dynamic evolution (essential for physical applications) depends mainly on the
$SR$ transitions into the limited number of states within the interaction
region window. In this case, the spectral mixing affects the evolution
dynamics only if the mixing occurs just in the interaction region (and all
remaining system states in the phase space are almost irrelevant). For the
considered above types of the spectral deformations this condition is
fulfilled if $3\delta k_{c}>1$ (c.p., with our interpolation formulas (43)).
Although the cycle period is energy dependent in the non-equidistant spectra,
the regular evolution maintains in the sufficiently wide range of the
deformation parameters. The recurrence cycles are destroyed only if the strong
spectral mixing holds in the interaction region.

To conclude this subsection we would like to emphasize that the model spectral
deformation with $K=3$ sub-lattices we mainly considered in this subsection,
is not merely a toy model to illustrate time evolution for the non-equidistant
reservoir spectra. For example, vibrational spectra of carbon nano-tubes KB98
, fullerene molecules SP02 , AC11 , or forming biological membranes lipids
IY03  typically contain indeed three vibrational energy levels progressions
with nearly constant (but different in each of the progression) inter-level
spacings (in many lipids, see e.g., IY03 , these intervals are about $20\,cm$,
and the corresponding period of the recurrence cycle is in pico-second time
window).

Figure 5: Time evolution of the initial state coupled to reservoir with
spectral mixing. $C^{2}=1$, $K=3$, $\delta=0\,;\,0.019\,;\,0.049\,;\,0.079$;
$a=0\,,\,0.15\,,\,0.25\,,\,0.5$ (from the top to the bottom).

###  IV.3 Ensemble of nano-particles at finite temperatures.

Usually in almost any realistic system there is not a single nano-particle but
a composite consisting of many weakly coupled not-identical nano-particles.
Therefore applying the Zwanzig model approach in such a case, one has to deal
with an ensemble of the reservoirs with somehow distributed energy levels and
their widths. The natural physical mechanism leading to the ensemble of the
reservoirs is related to the collective vibrational excitations (phonons)
coupled to the individual nano-particle displacements in the composite system.
For instance the line width introduced ad hoc by hands in the section II
results just from the relative displacements of the individual nano-particles.
These displacements can be described in terms of the interaction between
intra-molecular vibrations and phonons. Owing to this interaction the levels
of the Hamiltonian (2) are displaced and acquired the finite width depending
on the phonon quantum numbers. The phonon spectrum is limited by the Debye
frequency characterizing the vibrations of the nearest neighbor particles. The
Debye frequency $\Omega_{D}$ is typically by several times smaller than the
intra-molecular frequencies in nano-systems under consideration (with their
characteristic scale frequency $\Omega_{0}$ scale frequency, and comparable
with the thermal energy $k_{B}T$ at room temperatures

$\displaystyle\Omega_{D}\simeq k_{B}T\ll\Omega_{0}\,.$ (44)

Phonons yield to transitions between mixed internal vibration - phonon states

$\displaystyle\langle n^{\prime}q^{\prime}|\leftarrow|nq\rangle\,,$ (45)

where $n$ stands for the internal vibration energy levels, and $q$ is phonon
wave vector. Introducing phonon creation and annihilation operators
$b_{q}^{+}\,,b_{q}$, the interaction of the arbitrary selected state $(n\,q)$
with the vibration-phonon reservoir is described by the Hamiltonian

$\displaystyle
H_{nq}=\varepsilon_{nq}^{0}b_{n}^{+}b_{n}b_{q}^{+}b_{q}+\sum_{n^{\prime}q^{\prime}}\varepsilon_{n^{\prime}q^{\prime}}b^{+}_{n^{\prime}}b_{n^{\prime}}b_{q^{\prime}}^{+}b_{q^{\prime}}+\sum_{n^{\prime}q^{\prime}}V_{nq\,;\,n^{\prime}q^{\prime}}(b_{n}^{+}b_{n^{\prime}}+b_{n}b_{n^{\prime}}^{+})(b_{q}^{+}b_{q^{\prime}}+b_{q}b_{q}^{+})\,.$
(46)

To retain the well-defined absorption band, the phonon induced shifts and
broadening should be smaller than the characteristic inter-level spacings in
the intra-molecular spectrum. The weak vibration-phonon coupling condition
reads

$\displaystyle|V_{nq;n^{\prime}q^{\prime}}|\ll|\varepsilon_{n}^{0}-\varepsilon_{n^{\prime}}^{0}|\,.$
(47)

The weak vibration-phonon coupling is often realized in molecular crystals. In
the opposite case of strong vibration-phonon coupling, the fine structure of
the absorption band is smeared out. This limit can be treated theoretically
within the conventional theory of radiation-less transitions in systems with
continuous spectra. When the condition of (47) is fulfilled, the integral over
the continuous spectrum can be represented as the sum of the corresponding
integrals over all individual intervals with one simple pole within each
interval

$\displaystyle\sum_{n^{\prime}q^{\prime}}\to\sum_{n^{\prime}}\int
d\varepsilon_{q}\rho(n^{\prime},\varepsilon_{q})\,,$ (48)

where $\rho(\varepsilon)$ is phonon spectral density. Then we can derive the
secular equation

$\displaystyle
F_{nq}=\varepsilon-\varepsilon_{n}^{0}-\int\frac{V_{nq}^{2}(u)}{u-\varepsilon_{nq}^{0}}\rho(u)du=0\,,$
(49)

where we denote

$\displaystyle
V_{nq}(u)=V_{nq;n^{\prime}q^{\prime}}\delta(u-\varepsilon_{n^{\prime}q^{\prime}}^{0})\,.$
(50)

Following Van Hove general formalism for the perturbation theory in systems
with continuous spectra HO55 , and using exact relation

$\displaystyle lim_{\eta\to
0}\int\Phi(x)(x+i\eta)^{-1}dx=-i\pi\Phi(0)+P\int\Phi(x)x^{-1}dx\,$ (51)

($P$ means the principal part of the integral over the single interval), we
find that the pole $\varepsilon_{nq}^{0}$ in the lower complex half-plane
determines the level width, while the sum of the principal parts leads to the
displacements of the intra-molecular energy levels

$\displaystyle\varepsilon_{nq}=\varepsilon_{n}^{0}+D_{nq}-i\gamma_{nq}\,;\,D_{nq}=\sum_{n^{\prime}}P\int\frac{V_{nq}^{2}(u)}{u-\varepsilon_{n^{\prime}q}^{0}}\rho(u)du\,;\,\gamma_{nq}=\pi
V_{nqnq}\,.$ (52)

Eq. (52) defines complex eigen-values of the eigen-states $|nq\rangle$. Due to
vibration-phonon interaction a single intra-molecular level splits into the
band with half-width $\gamma_{nq}$. According to the weak coupling condition
(47) the bands are not overlapped, and the vibration-phonon states are
approximately orthogonal. With found from (52) values for the level
displacements and broadening, the effective intra-molecular Hamiltonian matrix
(with small imaginary parts of the diagonal elements resulted from vibration
phonon coupling) can be written for a given value of the phonon wave number
$q$ as

$\displaystyle
H=\left(\begin{array}[]{ccccccc}\varepsilon_{2q}-i\gamma_{2q}&0&0&C_{2s}&0&0&....\\\
0&\varepsilon_{1q}-i\gamma_{1q}&0&C_{1s}&0&0&...\\\
0&0&\varepsilon_{0q}-i\gamma_{0q}&C_{0s}&0&0&....\\\
C_{2s}&C_{1s}&C_{0s}&\varepsilon_{sq}-i\gamma_{sq}&C_{-1s}&C_{-2s}&....\\\
0&0&0&C_{-1s}&\varepsilon_{-1q}-i\gamma_{-1q}&0&0\\\
0&0&0&C_{-1s}&0&\varepsilon_{-2q}-i\gamma_{-2q}&0\\\
....&....&....&....&....&....&...\\\ \end{array}\right)\,,$ (60)

where

$\displaystyle\varepsilon_{sq}=\Delta_{0}+D_{sq}\,;\,\varepsilon_{nq}=\varepsilon_{n}^{0}-D_{nq}\,,$
(61)

and in the weak coupling limit

$\displaystyle\gamma_{nq}\ll|\varepsilon_{nq}-\varepsilon_{(n+1)q}|\,.$ (62)

If characteristic optical excitation time is shorter than inverse Debye
frequency and longer than period of intra-molecular vibrations, the initial
state (prepared by the vertical optical pumping as it is schematically shown
in the figure 1) is in the frozen phonon configuration and includes the set of
phonon states determined by their equilibrium distribution in the ground state
$\rho_{g}(\varepsilon_{q},T)$. Then, according to the random phase principle,
this distribution is random and can be characterized by the mean value and
dispersion. Therefore, in order to describe the ensemble of nano-particles the
Hamiltonian matrix (60) should be averaged over this distribution. In the weak
coupling limit (62) we can introduce independently energy mean values and
level widths of intra-molecular states

$\displaystyle{\bar{\varepsilon}_{n}(T)}=\int\varepsilon_{nq}(\varepsilon_{q})\rho_{g}(\varepsilon_{q},T)d\varepsilon_{q}\,;\,{\bar{\gamma}_{n}}(T)=\int\gamma_{nq}(\varepsilon_{q})\rho_{g}(\varepsilon_{q},T)d\varepsilon_{q}\,,$
(63)

and the corresponding dispersions

$\displaystyle{\bar{\delta}}_{n}^{2}(T)=\langle(\varepsilon_{nq}-{\bar{\varepsilon}}_{n})^{2}\rangle_{\rho}\,;\,\gamma_{nq}\ll|\varepsilon_{nq}-\varepsilon_{(n+1)q}|\,.$
(64)

From the other hand the fluctuation-dissipation theorem (see, for example the
classical textbook LL77 , or its more advanced version in YS94 ) states that
the temperature dependent dispersion is given by

$\displaystyle{\bar{\delta}_{n}}^{2}(T)={\bar{\delta}_{n}^{2}(0)}\coth\frac{\varepsilon_{q}}{k_{B}T}\,.$
(65)

where ${\bar{\delta}_{n}(0)}$ is the width of the ground state distribution at
$T=0$. Eqs. (63) - (65) suggests the following steps to describe an ensemble
of nano-particles at finite temperatures: (i) to replace real-valued intra-
molecular eigen-frequencies of the individual nano-particle by the complex-
valued counterparts for the ensemble of nano-particles, taking into account
the weak vibration-phonon coupling; (ii) to average over the set of the ground
state configurations ( i.e. over random distribution with temperature
dependent widths). The latter procedure to replace the mean values (63) by

$\displaystyle{\tilde{\varepsilon}_{n}}={\bar{\varepsilon}_{n}}-i{\bar{\gamma}_{n}}+f_{n}\,,$
(66)

where $f_{n}$ is a random variable with the zero mean value and the dispersion
${\bar{\delta}_{n}}^{2}$. This Eq. (66) assumes that there are two mechanisms
for the line broadening. Homogeneous broadening results from the transitions
(45), whereas the non-homogeneous broadening arises from the temperature
dependent distribution of phonon configurations in the ground state. As a
result, the fine structure of the absorption band is not changed upon non-
homogeneous broadening if

$\displaystyle{\bar{\delta}_{n}}\ll|{\bar{\varepsilon}_{n}}-{\bar{\varepsilon}_{n+1}}|\,.$
(67)

It is instructive to compare the absorption band structures for a single
particle and their ensemble. For a single particle the absorption band has the
Lorentzian shape, while due to non-homogenous broadening (occurring as a
result of the ensemble averaging) it acquires the Gaussian shape. If the
values ${\bar{\gamma}_{n}}$ and ${\bar{\delta}_{n}}$ are comparable the fine
structure components have a rather complicated shape consisting of the
Gaussian central part and Lorentzian tails. Let us stress that in the ensemble
of nano-particles decay of the reservoir states (due to vibron-phonon
interactions) suppresses the Loschmidt echo signal. Indeed as we said already
above, to study the initially prepared state evolution in the ensemble of
nano-particles, one has to average the initial state amplitude over the random
distribution of the phonon configurations. This procedure leads to the
broadening of the Loschmidt echo components. Then the number of observable
recurrence cycles decreases. Only a finite number of the reservoir states is
essential for the time evolution of the initial state, and this number is on
the order of the $SR$ interaction region width, $\Gamma$. In fact for the
Zwanzig model with finite number of the reservoir states the time evolution is
almost the same as that for the infinite reservoir systems BG09 .

Note to conclude this subsection that similar results can be obtained within
the random matrix formalism widely used in nuclear ME68  \- PW07  and
molecular DF00  spectroscopy.

##  V Reservoir states evolution

In the macroscopic systems the continuous reservoir states play a passive
role, merely as a sink for the population of the initially prepared state. The
reason is evident, there are no reverse $R\to S$ transitions since the
recurrence cycles periods are infinitely long. By contrast in nano-systems
possessing reservoirs with discrete dense spectra, the reservoir enters
actively into the system dynamics. The states of the reservoir with a dense
discrete spectrum are periodically occupied in the direct $S\to R$ transitions
and then depleted in the reverse $R\to S$ transitions. Synchronicity of the
reverse transitions leads to the double resonance phenomenon at the reservoir
eigen-frequencies. These double resonances hold (and can be observed) even
when the Loshmidt echo components are overlapped, therefore the double
resonances provide a convenient and powerful method for experimental studies
of the ultra-fast energy redistribution processes HO07  \- AS04 .

According to the Heisenberg equations of motion (16)

$\displaystyle
a_{n}(t)=-iC_{n}\int_{0}^{t}\exp(in(t-t^{\prime})a_{s}(t^{\prime})dt^{\prime}\,$
(68)

or in terms of the partial cycle amplitudes

$\displaystyle a_{n}(t)=\sum_{k=0}^{[t/2\pi]}a_{n}^{(k)}\,.$ (69)

In the bare Zwanzig model, these amplitudes have the form BG09b

$\displaystyle
a_{n}^{(k)}=i\frac{C}{2k\Gamma}\exp(-i\beta_{n}\tau_{k})\int_{0}^{\tau_{k}}xL_{k-1}^{1}(x)\exp(-s_{n}x)dx\,,$
(70)

where

$\displaystyle\beta_{n}=\frac{n}{2\Gamma}\,;\,s_{n}=\frac{1}{2}\left(1-i\frac{n}{\Gamma}\right)\,.$
(71)

Each of the amplitude in the expression (70) consists of the periodic
(harmonic) and non-periodic components. At the end of each cycle, the non-
periodic component is exponentially small while the periodic component,
${\tilde{a}}_{n}^{(k)}(t)$ has the constant magnitude

$\displaystyle{\tilde{a}}_{n}^{(k)}=C\frac{2\Gamma}{\Gamma^{2}+n^{2}}\exp(-i(nt-k(\pi+\varphi_{n})))\,,$
(72)

where the phase shift $\varphi_{n}$

$\displaystyle\varphi_{n}=tan^{-1}\frac{n}{\Gamma}\,.$ (73)

The amplitude of the population in the $n$-th state of the reservoir is the
sum of the partial amplitudes (72) for all previous cycles

$\displaystyle
a_{n}(2(k+1)\pi)\simeq\sum_{k^{\prime}=0}^{k}{\tilde{a}}_{n}^{(k^{\prime})}(4(k-k^{\prime}+1)k_{c})=iC(-1)^{k+1}\frac{(\Gamma+in)^{k}}{(\Gamma-
in)^{k+1}}\,,k<k_{c}\,.$ (74)

Then from (74) we conclude that the magnitudes of the reservoir state
amplitudes at the end of the non-overlapped cycles are $k$-independent. Their
form is given by the Lorentzian with the half-width $\Gamma$

$\displaystyle|a_{n}(2k\pi)|^{2}=\frac{\Gamma}{\pi(\Gamma^{2}+n^{2})}\,.$ (75)

The phase shift $\varphi_{n}$ (73) by its meaning is the additional (with
respect to standard dynamic phase ) geometric phase shift which results from
each population exchange event. Since the number of exchange events is equal
to the cycle number, the total shift is given by

$\displaystyle arg(a_{n}(2k\pi))=k\pi+(2k+1)\varphi_{n}\,.$ (76)

The periodic (harmonic) components describe the motion of a point in the phase
space along a circle with the radius, $R_{n}={\sqrt{\Gamma^{2}+n^{2}}}$, which
is independent of the cycle number. The long sequence of the rotations by the
angles (75) incommensurable with $2\pi$, yields to a quasi-chaotic dynamic
behavior. In the limit $k\to\infty$ all possible rotation angles densely cover
the circle. Based on our numeric results, the resonance time can be estimated
as

$\displaystyle\tau_{kn}^{*}\simeq\tau_{k0}^{*}\frac{\Gamma}{n}tan^{-1}\frac{n}{\Gamma}\,,$
(77)

where $\tau_{0k}^{*}\simeq 4k-0.74k^{1/3}$ corresponds to the resonance line
for $n=0$ state in the $k$-th cycle. The Eq. (77) is the condition for
synchronic depopulation of the reservoir states with different numbers. The
width of the distribution in the resonance times is proportional to the ratio,
$(n/2\Gamma)^{2}$, which is small for the states contributing to the Loschmidt
echo signal. It is worth to emphasize the difference between the shapes of the
echo and double resonance lines. The width of the echo signal increases upon
perturbations of the equidistant reservoir spectrum spectrum, whereas the
double resonances occur irrespective of this property of the reservoir
spectra. Thus the double resonances can be detected more easily than the
Loschmidt echo signal which is smeared out due to large inhomogeneous
broadening effects.

The amplitude of the resonance state, $a_{0}(t)$, always remains imaginary
(similarly to the two-level system case). The additional phase shift is zero,
$\varphi_{0}=0$, and the overturn (phase shift $\pi$) occurs along the
imaginary axis between the points $\pm iC\Gamma^{-1}$. In each cycle the
partial amplitudes are changed by $\pm 2iC\Gamma^{-1}$. The double resonance
time, $\tau_{0k}^{*}$, is a characteristic time for a point in the phase space
to pass through the center of the circle, when $a_{0}(\tau_{0k}^{*})=0$. The
global maximum of the double resonance signal is determined by the vanishing
population for the maximal number of the reservoir states. The half-width of
the double resonance line in the non-overlapping cycles increases slowly with
increasing the cycle number. Our numeric results suggest the following law

$\displaystyle\Delta\tau_{1/2}\simeq(32k)^{1/3}\,;\,2\leq k\leq k_{c}-1\,.$
(78)

Owing to the overturns, the amplitude of the state $n=0$ has different signs
in the odd and even cycles. In the odd cycles the population flow is directed
from the reservoir toward the initial state, and the population of the
resonance state decreases. By contrast, the population in the even cycles
increases and becomes higher than the equilibrium population given by the
Lorentzian distribution (74). The population increases because the states with
$n\simeq\Gamma$ more rapidly (in comparison with $n=0$ state) transfer the
population into the initial state. The transitions from these $n\simeq\Gamma$
states produce the flow from the initial to the resonance state and lead to
its super-equilibrium population. Therefore, the double resonance transitions
$n\to s\to n^{\prime}$ redistribute the population between the reservoir
states. The double resonance transitions lead also to satellites in the double
resonance spectra. At the end of the cycle $k=0$ the entire population of the
initial state is transferred to the reservoir states ($a_{s}^{(0)}=0$) with
the distribution (74). In the first cycle, the reservoir states populated in
the zeroth cycle return the population back to the initial state with the
initial rate proportional to $C^{2}$. Since the resonance time increases with
increasing of the reservoir level number $n$, different reservoir states are
depleted in the different times. However, the reservoir states are not
depleted completely during the cycle, and therefore the amplitude of the
initial state is not totally recovered.

Numerically we find that in the cycle $k=1$ the maximum of population
amplitude $max|a_{n}|^{2}\simeq 0.46$ is at $\tau_{s1}^{*}=2$. The states
$n>\Gamma$ are depleted at $\tau_{n1}^{*}<\tau_{s1}^{*}$, when the population
of the initial state is still increasing, and the states $n<\Gamma$ are
depleted when this population decreases. At the end of the cycle, the
population is distributed over the reservoir states, as it is dictated by the
Eq. (74). The similar behavior of the population redistribution occurs in the
subsequent cycles. We illustrated this behavior for the cycle $k=2$ in Fig. 6.

Figure 6: Redistribution of the initial state population over the mixed
reservoir states, during the second recurrence cycle, $k=2$. The numbers of
the reservoir states are shown on the curves.

One can find also that apart from the main double resonance line caused by the
overturn, at $k\geq 2$ there arise $k-1$ weak lines (satellites) which are
associated with the oscillations of the initial state amplitude. The number,
intensity and width of the satellites increase upon increase of the cycle
number. Nevertheless this change in the satellite spectrum does not affect the
main double resonance, which intensity and width depend only weakly on the
cycle number. As it is shown in Fig. 6, the satellites leave the main
resonance behind in the states $n<\Gamma$, and they occur after the main
resonance line in the states $n>\Gamma$. In the states $n\simeq\Gamma$ there
are the both satellites, advanced and retarded ones.

The initial state plays the role of the intermediate energy level in the
redistribution process between the reservoir states. The sign of the
population flow changes $k$ times during the cycle depending both on the
amplitudes of the population and phases. As a result, the dynamics of the
redistribution is not described by the detail balance conditions. The fine
structure of the Loschmidt echo signal is associated with the deviations of
the reservoir state populations from the equilibrium distribution (74). In
terms of the phase shift the condition for the Loschmidt echo mixing $k>k_{c}$
means that the phase shifts for all reservoir states $n\geq 1$ are larger than
$2\pi$ when $k>k_{c}$. Mixing affects also the shape of the double resonance
signal. It occurs the double resonance line width given by Eq (78) becomes
comparable with the state under consideration vibration period, $2\pi/n$. The
critical cycle number corresponding to this condition drops rapidly when the
state number increases

$\displaystyle k_{n}\simeq\frac{1}{4}\left(\frac{k_{c}}{n}\right)^{3}\,.$ (79)

The criterion (79) defines also crossover from the regular to quasi-chaotic
dynamics in the $n$-th reservoir state evolution. When $k>k_{n}$ , the
intensities of the main lines and satellites are comparable. Our numerical
computations also support quasi-random dynamics in the cycles $k>k_{n}$. The
computed trajectories in the phase space $a_{n}$ and $da_{n}/dt$ look as
strongly entangled and fill the entire phase space available for a given
reservoir state $n$.

##  VI Other applications of the Zwanzig model

The purpose of this section is to illustrate how the bare Zwanzig model (or
its generalization) approach can be applied to describe quantum dynamics of
various experimentally studied (and therefore interesting for theorists) nano-
systems.

###  VI.1 Two-level systems coupled to reservoirs

The main difficulty for experimental detection of the double resonances on the
reservoir state frequencies is related to the fact that these frequencies are
very close to the pumping frequency of the initial state. The required
spectral resolution should be of the order of the characteristic inter-level
spacing, and it is not yet available. However the double resonances can be
observed between two spectral well-separated intra-molecular states (see KM10
, or KN07  \- RU09 ). For such a case the initial (prepared by the optical
pumping) and final (observed) states form a two-level system (TLS). Thus our
aim in this subsection is to generalize described in the previous sections
Zwanzig model approach to study quantum dynamics of the TLS nano-systems. Note
to the point that the TLS interacting with intra-molecular and phonon
reservoirs is a convenient minimal model to investigate ultra-fast molecular
processes SS95  \- MN08 .

The dynamical problem for the TLS coupled to the reservoir with continuous
spectrum has been investigated already in details (see, e.g., review paper
LC87 , monograph BM94 , or our publication BK11 ). The specific feature of
this model is the transition from coherent oscillations between the initial
and final state to exponential decay of the initial state population when the
transition matrix element between the TLS states becomes smaller than the
matrix element for the TLS-reservoir coupling. In this subsection, we
investigate the problem for the case of the intra-molecular reservoir with the
equidistant spectrum (the bare Zwanzig model reservoirs). The time dependent
wave function of the TLS coupled to the reservoirs is described by a column
with 2 terms which are the wave functions of the initial and final
configurations coupled to the reservoirs

$\displaystyle\Psi(t)=\left(\begin{array}[]{c}\Psi_{L}(t)\\\
\Psi_{R}(t)\end{array}\right)\,.$ (82)

In own turns the functions $\Psi_{L}$ and $\Psi_{R}$ are superpositions of the
initial (and final) TLS states ($\Phi_{L}$ and $\Phi_{R}$ respectively)
coupled to the reservoir eigen-functions, $\Phi_{nL}$ and $\Phi_{nR}$, with
time dependent amplitudes

$\displaystyle\Psi_{L,R}(t)=a_{L,R}(t)\Phi_{L,R}+\sum_{n}a_{nL,nR}(t)\Phi_{nL,nR}\,.$
(83)

The Hamiltonian, describing the TLS system coupled to the corresponding
reservoirs, can be written in the following compact form

$\displaystyle H=\left(\begin{array}[]{cc}H_{L}&0\\\
0&H_{R}\end{array}\right)+\Delta{\hat{\sigma}_{x}}\,,$ (86)

where $H_{L}$ and $H_{R}$ are the Zwanzig Hamiltonian (2) for the $L$ and $R$
configurations, ${\hat{\sigma}_{x}}$ is the Pauli matrix, and $2\Delta$ is a
level splitting in the isolated TLS (i.e., uncoupled to the reservoirs). The
Hamiltonian matrix (86) contains two non-zero columns and two non-zero lines
of the coupling matrix elements for the both configurations and the diagonal
matrix elements of the unperturbed TLS and reservoir states. Within the bare
Zwanzig model, the both reservoir spectra (renormalized by the vibration-
phonon interaction, described in Sec. IV) (subsection IV.3) are equidistant,
and the coupling constants and decay rates are assumed to be the same for all
reservoir states

$\displaystyle
E_{nL}+D_{n}=E_{nR}+D_{n}=n\,;\,C_{n}=C\,;\,\gamma_{L}=\gamma_{R}=\gamma_{0}\,;\,\gamma_{nR}=\gamma_{nL}=\gamma\,.$
(87)

The secular equation is

$\displaystyle
F(\varepsilon)=\left(\varepsilon+i\gamma_{0}-\sum_{n}C_{n}^{2}(E_{n}+D_{n}-\varepsilon+i\gamma_{n})^{-1}\right)^{2}-\Delta^{2}=0\,.$
(88)

This equation (88) can be split into two equations for odd and even states.
Performing explicitly the summation in the Eq. (88) we find these equations

$\displaystyle F^{\pm}(\varepsilon)=\left(\varepsilon+i\gamma_{0}\pm\Delta-\pi
C^{2}\cot(\pi\varepsilon+i\pi\gamma\right)=0\,.$ (89)

Similarly we find the Heisenberg equations of motion, generalizing those (16)
for the bare Zwanzig model

$\displaystyle i\frac{\partial a_{L}}{\partial t}=\Delta
a_{R}+\sum_{n}C_{n}a_{nL}\,;\,i\frac{\partial a_{R}}{\partial t}=\Delta
a_{L}+\sum_{n}C_{n}a_{nR}\,;\,$ (90) $\displaystyle i\frac{\partial
a_{nL}}{\partial t}=E_{n}a_{nL}+C_{n}a_{L}\,;\,i\frac{\partial
a_{nR}}{\partial t}=E_{n}a_{nR}+C_{n}a_{R}\,.$

Solutions of these equations can be easily found, and can be represented in
the following form

$\displaystyle
a_{L}(t)=\sum_{\varepsilon_{n}^{+}}\frac{\cos(\varepsilon_{n}^{+}t)}{1+\pi\Gamma+C^{-2}(\varepsilon_{n}^{+}-\Delta)^{2}}\,;\,a_{R}(t)=i\sum_{\varepsilon_{n}^{-}}\frac{\sin(\varepsilon_{n}^{-}t)}{1+\pi\Gamma+C^{-2}(\varepsilon_{n}^{-}-\Delta)^{2}}\,.$
(91)

Transformation of (91) to the partial amplitude representation yields

$\displaystyle
a_{L}(t)=\frac{1}{2}\sum_{k=-\infty}^{\infty}\left(a_{k}^{(+)}(t)+a_{k}^{(-)}(t)\right)\,;\,a_{R}(t)=\frac{i}{2}\sum_{k=-\infty}^{\infty}\left(a_{k}^{(+)}(t)-a_{k}^{(-)}(t)\right)\,;\,$
(92) $\displaystyle
a_{k}^{(\pm)}(t)=C^{2}\int_{-\infty}^{+\infty}d\varepsilon\exp(i\varepsilon(t-2k\pi))\frac{(\varepsilon\pm\Delta+i\Gamma)^{k-1}}{\varepsilon\pm\Delta-i\Gamma)^{k-1}}\,.$
(93)

Calculating the contour integral we end up with the explicit expressions for
the partial cycle amplitudes

$\displaystyle
a_{L}^{(0)}=\exp(-(\Gamma+\gamma_{0})t)\cos(\Delta\,t)\,;\,a_{L}^{(k)}(\tau_{k})=a_{s0}^{(k)}(\tau_{k})\exp(-2k\pi\gamma)\cos\left(\frac{\Delta}{2\Gamma}\tau_{k}\right)\exp\left(-\frac{\gamma_{0}}{2\Gamma}\tau_{k}\right)\,,$
(94)

and

$\displaystyle
a_{L}^{(0)}=\exp(-(\Gamma+\gamma_{0})t)\sin(\Delta\,t)\,;\,a_{R}^{(k)}(\tau_{k})=ia_{s0}^{(k)}(\tau_{k})\exp(-2k\pi\gamma)\sin\left(\frac{\Delta}{2\Gamma}\tau_{k}\right)\exp\left(-\frac{\gamma_{0}}{2\Gamma}\tau_{k}\right)\,,$
(95)

where in the both cases $k\geq 1$, and $a_{s0}^{(k)}$ and $\tau_{k}$ are
defined by the Eqs. (37) - (38). The above equations (94), (95) include three
mutually independent factors related to the transitions between the TLS
states, between these states and the reservoirs, phonons with and without
phonons are taking into the consideration. The factorization of these
phenomena is the main advantage of the partial amplitude representation.

With these formal results, on the TSL coupled to the equidistant spectrum
reservoirs, in hands we are in the position to proceed further, and to analyze
the details of the quantum dynamics, and corresponding physical consequences.
At beginning of each cycle $k\geq 1$ the TLS states are depleted
($a_{L}^{(k)}(0)=a_{R}^{(k)}(0)=0$). The $LR$ transitions occur in the local
time $\tau_{k}$ and depend on the total population of the both TLS states. As
it concerns to the reservoir states they are populated in the agreement with
Eq. (75) independently of the populations of the individual TLS states.
Reverse transitions from the reservoir states restore the TLS state
population. As a result of this fact, the transitions between the TLS states
are switched on again. Obviously, this effect is absent if the TLS were
coupled to a reservoir with continuous spectrum. The time dependencies
$a_{L}(t)$ and $a_{R}(t)$ given by the Eqs. (92) - (95) show that the regular
dynamics region exists independently of the value of the TLS splitting,
$\Delta$. When $\Delta\ll\Gamma$, the main part of the initial population of
the unperturbed left state $L_{0}$ is transferred to the reservoir states
$\L_{n}$, within the Lorentzian contour with the half-width $\Gamma$. The rest
of the population participates in the $L-R$ exchange. The effective (time
dependent) rate constant for the $L_{0}$ -state decay in the cycle $k=0$ is

$\displaystyle
k_{L}=-\frac{d}{dt}\ln(a_{L}^{(0)})=\Gamma+\Delta\tan(\Delta\,t)\,.$ (96)

Consequently the amplitude of the final TLS state, $R_{0}$, has a maximum at
$t_{m}=\Delta^{-1}\tan(\Delta/\Gamma)$, so that the rate constant for the
$L-R$ transitions in the vicinity of $t_{m}$ is given by

$\displaystyle k_{L-R}\simeq\frac{\Delta^{2}}{\Gamma}\,.$ (97)

In fact this Eq. (97) defines the rate for the dissipative tunneling. Similar
characteristics are widely used in the theory of the systems strongly coupled
with the thermal bath in the continuous spectrum case (see e.g., LC87 , and
CL83  \- YU96 ). Note that the rate constant (97) is independent of the
density of the reservoir states.

The ratio of the amplitudes of the unperturbed $R_{0}$ and $L_{0}$ states the
time $t_{m}$ is

$\displaystyle\frac{a_{L}^{(0)}(t_{m})}{a_{R}^{(0)}(t_{m})}\simeq\frac{\Delta}{\Gamma}\,.$
(98)

Owing to the $L-R$ transitions and transitions from the TLS to the reservoir
states the populations of $L_{0}$ and $R_{0}$ becomes equal in the limit of
$k\gg 1$. The total population of the both the states is independent of the
TLS splitting, $\Delta$, and is determined by the rate of the intrinsic decay
of the reservoir states. The average per cycle TLS population is

$\displaystyle\langle|a_{L}^{(k)}|^{2}+|a_{L}^{(k)}|^{2}\rangle_{k}=\Gamma^{-1}\exp(-4k\pi\gamma)\,.$
(99)

Eq. (101) indicates that the reservoir states are populated during the main
part of the recurrence cycle period, so that their decay with the rate
constant $\gamma$ determines the population of the TLS. When
$\gamma=\gamma_{0}=0$, and $\Delta/\Gamma\ll 1$, the average per cycle
populations of the $L_{0}$ and $R_{0}$ states are described by the approximate
relation

$\displaystyle\langle|a_{L,R}|^{2}\rangle_{k}\simeq(2\Gamma)^{-1}\left(1\pm(1+\alpha^{2})^{-1}J_{0}(4k\alpha(1+\alpha^{2})^{-1/3})\right)\,,$
(100)

where $\alpha=\Delta/\Gamma$, and $J_{0}$ is the zero order Bessel function.
Eq. (102) shows that the average per cycle populations exhibits the damped
oscillations as a function of the cycle number $k$. For $k\alpha\gg 1$ the
populations of the $L_{0}$ and $R_{0}$ states become the same $\simeq
1/2\Gamma$.

Our numeric results, presented in the Fig. 7, are in the agreement with
described above features of the TLS time evolution. The above described
oscillations suggest that the TLS states are dressed (i.e., mixed with the
reservoir states). The transition probability between these dressed states
$L_{0}+\\{L_{n}\\}$ and $R_{0}+\\{R_{n}\\}$ is about $\Gamma$ times smaller
than that for the unperturbed TLS. The damping coefficient of these slow
oscillations results from the exchange of the dressed states with the far
placed reservoir states $n\gg\Gamma$. For $\Delta>2\Gamma$ the oscillations
disappear and the equal populations of the TLS states is established already
in the cycle $k=0$.

Figure 7: Variation of the average per cycle populations of $L_{0}$ and
$R_{0}$, states and the entire population of the TLS in the recurrence cycles.
$C^{2}=1$, $\Delta=0.4\,,\,1.2\,,\,2.7\,,\,7.4$ from the top to the bottom.

###  VI.2 Propagating vibrational excitations in nano-size chains

All previous parts of our work were about systems which are nano-sized in
three orthogonal directions. However there are several reasons behind a
general interest to the systems which have nano-scales in only two or one
directions. Most effects usually manifest themselves when at least one of the
three system dimensions falls in the nano-sized regime. In such a regime a
number of intrinsic system properties change, some qualitatively, others -
quantitatively. We illustrate this conception investigating nano-sized chains,
interesting for fundamental physics and applications. Namely in this
subsection we consider the finite length chain of the $(2N+1)$ coupled
molecular or atomic sites. All sites, except the central one at $n=0$, termed
impurity site, are identical. Each site can be only in the ground and in the
first excited states. All highly excited states are neglected. This
approximation is justified if all these states have sufficiently large
excitation energy, and therefore are far from the low energy band formed by
the coupled first excited states. The Hamiltonian of this model is

$\displaystyle
H=E_{0}Q_{0}^{+}Q_{0}+\Omega_{0}\sum_{n}Q_{n}^{+}Q_{n}+\sum_{\pm}C(Q_{0}^{+}Q_{\pm
1}+Q_{0}Q^{+}_{\pm 1})+J\sum_{n\neq 0,n\neq\pm
1}(Q_{n}^{+}Q_{n-1}+Q_{n}Q_{n-1}^{+})\,,$ (101)

where $Q_{0}^{+}\,,Q_{0}$ are creation and annihilation operators of the
impurity site $n=0$, and $Q_{n}^{+}\,,\,Q_{n}$ are similar operators for all
other sites, $n\neq 0$. $E_{0}$ is the energy of the impurity excited state,
$\Omega_{0}$ is the energy in the center of the vibrational band, $C$ and $J$
are the matrix elements for the corresponding nearest neighbor couplings (all
the same for the uniform chain). The Hamiltonian (101) is written in the site
(space) representation. It can be transformed into the quantum propagating
wave function representation, keeping separately one impurity state and the
standing waves corresponding to two ($n<0$ and $n>0$) uniform half-chains. In
this representation the band spectrum has the form

$\displaystyle E_{j}=\Omega_{0}+2J\cos\frac{\pi j}{N+1}\,;\,j=1,2,....N\,.$
(102)

Eq. (102) represents the unperturbed spectrum of the half-chains which form
the set of reservoir states. In contrast to the spectrum used in the bare
Zwanzig model approximation, this spectrum becomes denser and denser upon
approaching to the band boundaries, while in the middle of the band, it is
almost equidistant with the characteristic inter-level spacing of the order
$\Delta_{0}\simeq 2\pi J/(N+1)$. Therefore, the recurrence cycle period $T$ is
proportional to the chain length, $T\propto 2(N+1)/J$. In the propagating wave
function representation, the coupling matrix elements $C_{j}$ are proportional
to the amplitudes of the sites $n=\pm 1$ in the quantum $j$-th state

$\displaystyle C_{j}=C(N+1)^{-1/2}\sin\frac{\pi j}{N+1}\,.$ (103)

Eq. (103) indicates that the values of these coupling matrix elements are
approximately constant near the center of the band ($j\simeq N/2$) and
approaches to zero near the band boundaries. Alike the Zwanzig model discussed
above, in the nano-scale chains, the dynamics of the vibrational excitation is
governed by the parameter $\Gamma\propto C_{N/2}/\Delta_{0}$. This parameter
determines the number of reservoir states which essential in the impurity-
chain interactions. For $0\leq C\leq 1$ the impurity level lies within the
band (102). In what follows for the sake of simplicity, we chose the zero
energy level as $E_{0}$, and rescale all energies in $J$ units (i.e.,
$E=0\,,J=1$). Then the secular equation for the Hamiltonian (101) reads

$\displaystyle
F(\varepsilon)=(\varepsilon+E)D_{N}^{2}(\varepsilon)-2C^{2}D_{N}(\varepsilon)D_{N-1}(\varepsilon)=0\,,$
(104)

where $D_{N}$ is $C$ independent three-diagonal determinant, which is well
known in the Jacobi matrix theory (see e.g., BE53 , OL74 ), which is

$\displaystyle D_{N}(\varepsilon)=\frac{\sin(N+1)k}{\sin k}\,,$ (105)

where $\varepsilon=2\cos k$. Eq (104)) is reduced to two equations for the
anti-symmetric and symmetric eigen-states of the chain with a single impurity
site in the middle

$\displaystyle
D_{N}=0\,,\,(\varepsilon+E)D_{N}(\varepsilon)-2C^{2}D_{N-1}(\varepsilon)=0\,.$
(106)

For $C=0$ the levels of the uncoupled impurity-reservoir are doubly
degenerated. For $E=0$ and odd $N$ values, the impurity level coincides with
the reservoir level $\varepsilon^{0}_{(N+1)/2}$. All essential physical
ingredients of the excitation propagation along the nano-scale chain, can be
illustrated in this particular case. Changing the variables in the secular
equation (104)

$\displaystyle k=\frac{\pi}{2}-\alpha\lambda\,;\,\alpha=\frac{\pi}{N+1}\,$
(107)

the secular equation (104) can be rewritten in the same form as the Eq. (33))
for the generalized Zwanzig model

$\displaystyle F(\lambda)=f_{2}(\lambda)(\lambda-\Gamma
f_{1}(\alpha\lambda)\cot(\pi\lambda))=0\,,$ (108)

where $f_{1}$ and $f_{2}$ are the following corrective functions to the bare
Zwanzig model

$\displaystyle f_{1}(x)=x\cot x\,;\,f_{2}(x)=\frac{\sin x}{x}\,.$ (109)

Thus, the dynamical problem for the chain with a single impurity site becomes
equivalent to the discussed above problem for the initially excited state (it
is the impurity state in the chain problem) coupled to the reservoir with the
non-equidistant spectrum and energy-dependent coupling matrix elements.
Assuming the following initial conditions for the recurrence cycle amplitudes
$a_{0}(0)=1$, and $a_{k}(0)=0$ we find the the Fourier series for the impurity
state amplitude

$\displaystyle
a_{0}(t)=2\sum_{k=0}^{[N/2]}U_{k}(\lambda_{k})\cos(2t\sin(\alpha\lambda_{k}))\,;\,U_{k}(\lambda)=\frac{f_{1}^{2}(\lambda)}{(1+\pi\Gamma)f_{1}^{2}(\lambda)+(\alpha^{2}+\pi/\Gamma)\lambda^{2}}\,,$
(110)

where

$\displaystyle\Gamma=\frac{C^{2}(N+1)}{\pi(1-C^{2})}\,.$ (111)

Replacing also time $t$ by the recurrence cycle time $\tau_{k}$

$\displaystyle\tau_{k}=\frac{\sin(\alpha\lambda}{\alpha\lambda}t-2k\pi\,$
(112)

we end up with the exact solution (110) describing impurity site excitation
propagation along the nano-size chain. It reduces to Eq. (20) when the
function $f_{1}(\alpha\gamma)\simeq 1$. For $C^{2}<1/2$ the leading
contribution in the expansion (110) results from the states of the almost
equidistant spectrum (i.e., near the band center, where $k\leq\Gamma\ll N/2$).
Since the states located near the band boundaries do not participate in the
impurity-reservoir exchange, the evolution in this case resembles closely that
for the bare Zwanzig model. According with the Eq. (111), the coupling
constant $\Gamma$ grows with increasing $C^{2}$. Owing to the increase in
$\Gamma$, the region of the regular dynamics increases and achieves the
maximum at $C^{2}=1$, where the spectrum is nearly equidistant.

For $0\leq C^{2}\leq 1/2$, the sum (110) can be transformed using the Poisson
summation formula

$\displaystyle
a_{0}^{(k)}(\tau_{k})=\frac{1}{\pi(1-C^{2})}\int_{-\infty}^{\infty}\frac{\exp(i\lambda\tau_{k})}{\lambda^{2}+\Gamma^{2}f_{1}^{2}(\lambda)}\left(\frac{\lambda+i\Gamma
f_{1}(\lambda)}{\lambda-i\Gamma
f_{1}(\lambda)}\right)^{k}f_{1}^{2}(\lambda)d\lambda\,.$ (113)

The contribution of the compressed dense states located near the band
boundaries increases when $C^{2}>1/2$.

In Fig. 8 we present our numeric results to illustrate how the dynamics of the
vibrational excitation propagation along the nano-chain depends on the
coupling constant $C$. For $C^{2}<1/4$ dynamic behavior is similar to that for
the bare Zwanzig model (Sec. III). The number of the components and the total
width of the Loschmidt echo signal increases proportionally to the cycle
number (panel 8a). The critical cycle number for the regular-chaotic crossover
is

$\displaystyle
k_{c}\simeq(N+1)\frac{C^{2}}{1-C^{2}}a_{0}(t)\simeq\sum_{l}(-1)^{l}((1-C^{2})(J_{2l(N+1)}(2t)+J_{2l(N+1)+2}(2t)+J_{2l(N+1)}(2t))\,.$
(114)

However for $C^{2}\geq 1/2$ dynamics is qualitatively different from that
predicted by the bare Zwanzig model. Already for the small cycle numbers
$k\geq 1$ (panel 8b) the impurity state population oscillates (instead of the
exponential decay in the bare Zwanzig model). The oscillation amplitudes
increase with $k$, whereas the intensity of the main component slightly
decreases. Dynamics remains regular up to the cycle where the oscillating
components become comparable with the main peak (panels 8c and 8d). Because
the zeros of the Bessel functions of different orders are rationally
independent, mixing of the partial amplitudes yields the irregular damped
oscillations in $a_{0}(t)$. For $C^{2}\simeq 0.4$, the critical number
achieves the maximal value, and for larger values of $C^{2}$, $k_{c}$
decreases. Therefore, the threshold for the quasi-stochastic irregular
behavior increases in the interval $0\leq C^{2}\leq 1/4$, and it decreases for
$1/2\leq C^{2}<1$. These two regimes correspond to the distinct mechanisms of
randomization due to the mixing of the echo components. Namely: (i) cycle
overlapping, considered in section III and (ii) considered in this subsection
enhancement of the irregular oscillation amplitudes at $k>k_{c}$. A more
abrupt transition from regular (coherent) dynamics to stochastic one occurs
after the first passage of excitation through an impurity site.

Figure 8: Dynamics of the population at the excited impurity cite located in
the middle of the finite $N=49$ size chain.
$C^{2}=0.1\,,\,0.25\,,\,0.5\,,\,0.95$ from the top to the bottom. The critical
cycle number is marked by the a star (for $C^{2}=0.5$ and $k_{c}=23$).

Our results for the symmetric nano-chain with a single impurity site in the
center at $n=0$, can be easily generalized for a non-symmetric case, when the
impurity site is located at $n\neq 0$. For such a case the time dependent
amplitudes of the chain sites can be also expanded over the Bessel functions

$\displaystyle
b_{n}(t)=\frac{1}{1-C^{2}}\sum_{m=0}^{\infty}J_{2m}(2t)S_{nm}(\Lambda)\,,$
(115)

where

$\displaystyle
S_{nm}(\Lambda)=i\frac{C\Gamma}{\pi}\sum_{k=0}^{[N/2]}U_{k}(\lambda_{k})\cos(2m\alpha\lambda_{k})\frac{\sin((N+1-n)\alpha\lambda_{k})}{\sin((N+1)\alpha\lambda_{k})}\,.$
(116)

The coefficients $S_{nm}$ are non-zero only in the narrow (on the order of
$\Gamma$ and periodically repeated time intervals. Therefore, $b_{n}(t)$
similar to $a_{0}(t)$ is expanded over the partial amplitudes of recurrence
cycles

$\displaystyle b_{n}(t)=\sum_{k}b_{n}^{(k)}(t-2(N+1)k)\,,$ (117)

where the $k$-th partial amplitude is expressed in terms of the Bessel
functions with their orders in the interval $[k(N+1)+n\,,(k+1)(N+1)-n]$. Since
the partial amplitudes depend on the combination $t-n$ , this interval
determines also the arrival $t_{nk}^{(1)}$ and departure $t_{nk}^{(2)}$ times
for the located at the $n$-th site, excited impurity

$\displaystyle k(N+1)+n=t_{nk}^{(1)}\leq t\leq t_{nk}^{(2)}=(k+1)(N+1)-n\,,$
(118)

where the interval, $t_{nk}^{(2)}-t_{nk}^{(1)}$ decreases upon increasing the
impurity site $n$ number. As a result, the excitation propagates along the
chain with almost constant speed (see Fig. 9). For the small coupling constant
in the cycle $k=0$ there are two separate echo signals at times $t_{nk}^{(1)}$
and $t_{nk}^{(2)}$ when the excitation wave passes through the $n$-th site.

Figure 9: Impurity site excitation propagated along the chain. $C^{2}=0.5$,
$N=49$. The cite numbers are indicated in the panels.

The excitation spreads over several sites and the spreading width is almost
$n$-independent. For larger $C^{2}$ values, the excitation propagates with the
same (as for the weak coupling) speed but the intensity of the signal
considerably decreases. Ultimately, for $C^{2}\geq 1/2$ the width of the wave
packet becomes of the order of the half-cycle time. In this limit the wave
packet acquires the considerable width, the propagation is not ballistic
anymore. The sites, $n\geq N/2$, are excited irregularly, whereas the impurity
site dynamics remains regular. This phenomenon means a sort of self-
synchronization of the forward and backward (after reflection from the chain
ends) waves when they are passing trough the impurity site. This process
resembles the synchronicity of the reverse reservoir-initial state transitions
we investigated in Sec. V

It is worth to note that the main findings in this subsections are in the
agreement with experimental data WC07 , CW09 , manifesting that (i) in the
uniform molecular nano-scale chain the vibrational excitation of the impurity
site propagates almost ballistically for a long (on molecular scale) time and
distances; (ii) the number of the recurrence cycles with almost regular
behavior achieves its maximal value at the intermediate coupling constant
$C^{2}\simeq 0.4$.

##  VII Conclusion

Understanding all limitations of our simplified exactly solvable models, we
nevertheless hope that the results collected in this review capture the
essential peculiarities in nano-system dynamics. Namely, the dense discrete
spectra characteristic for nano-particles and large-size molecules are
responsible for appearance of recurrence cycles. By contrast to macroscopic
systems with the continuous spectra, where the initial state population
decreases monotonically in time, the Loschmidt echo and double resonances
arise in each recurrence cycle. The revivals in the time evolution makes it
possible the emergence of new mechanisms of vibration stimulated processes in
the single nano-particles and their ensembles as well. The regular behavior in
the initial recurrence cycles transformed into chaotic-like long-time
evolution is an inherent characteristic of nano-world. The combination of the
deterministic spectra with chaotic-like long time dynamics has no analogs in
the quantum dynamics of the macroscopic dissipative systems. Owing to specific
mechanisms of energy transfer and redistribution within the manifold of intra-
molecular modes, the loss-free and distant energy transfer becomes possible.
The unique property of nano-systems is the possibility for periodical
accumulation of the energy on the selected vibrational modes. The new fields
for experimental studies and applications of this unusual effect are opened.

###### Acknowledgements.

We would like to thank the referee for carefully reading the manuscript and
for his or her recommendations.

## References

  * (1) R.Zwanzig, Lectures in Theor. Phys., 3, 106 (1960).
  * (2) V.A.Benderskii, L.A.Falkovsky, E.I.Kats, JETP Lett., 86, 221 (2007).
  * (3) T.User, W.H.Miller, Phys. Repts., 199, 7 (1991).
  * (4) S.Takahashi (Ed), Time-resolved Vibrational Spectroscopy, Springer, Berlin, (1992).
  * (5) A.H.Zewail, Femtochemistry: Ultrafast Dynamics of Chemical Bonds, World Scientific, Singapore (1994).
  * (6) M.D.Feyer, Ultrafast Infrared and Raman Spectroscopy, Marcel Dekker, New York (2001).
  * (7) A.H.Zewail, Phil.Trans.Roy.Soc., A, 364, 315 (2005).
  * (8) D.M.Leitner, Adv. Chem.Phys., B, 130, 205 (2005).
  * (9) S.Mukamel, Principles of Nonlinear Spectroscopy, Oxford Univ.Press., New York (1995).
  * (10) R.M.Hochstrasser, PNAS, 104, 14190 (2007).
  * (11) J.Bredenbeck, A.Chosh, M.Smith, M.Bohn, J.Am.Chem.Soc., 130, 2152 (2008).
  * (12) J.B.Ashbury, T.Steinel, C.Stromberg, S.A.Orcell, C.P.Skinner, M.D.Feyer, J.Phys.Chem. A., 108, 1107 (2004).
  * (13) C.J.Fesko, J.D.Eaves, J.J.Loparo, A.Tokmakoff, P.L.Geissler, Science, 301, 1698 (2003).
  * (14) M.Khalil, N.Demirdoven, A.Tokmakoff, J. Chem. Phys., 121, 362 (2004).
  * (15) A.V.Benderskii, K.B.Eisenthal, J. Phys. Chem A., 106, 7482 (2002).
  * (16) M.L.Cowan, B.D.Bruner, N.Huse, J.R.Dwyer, E.T.J.Nibbering, T.Elsaesser, R.J.D.Miller, Nature, 434, 199 (2005).
  * (17) F.Gai, K.C.Hanson, J.C.McDonald, P.A.Anfinrud, Science, 279, 1886 (1998).
  * (18) S.Sportlein, H.Cartens, H.Satzger, C.Renner, R.Beherendt, L.Moroder, P.Tavan, W.Zinth, J.Wachveitl, PNAS, 99, 7998 (2002).
  * (19) D.M.Leitner, P.G.Wolynes, J.Phys.Chem. A, 101, 541 (1997).
  * (20) M.Ben-Nun, F.Molnar, H.Lu, J.C.Phillips, T.J.Martinez, K.Schulten, Farad. Discuss., 110, 447 (1998).
  * (21) G.K.Paramonov, H.Naundorf, O.Kuhn, Eur.Phys.J., D, 14, 205 (2001).
  * (22) S.Hayashi, E.Tajkhorshid, K.Schulten, Biophys. J., 85, 1440 (2003).
  * (23) H.Fujisaki, Y.Zhang, J.E.Straub, J.Chem.Phys., 124, 144910 (2006).
  * (24) Z.Wang, J.A.Carter, A.Laguchev, Y.K.Koh, N-H.Seong, D.C.Cahil, D.D.Dlott, Science, 317, 787 (2007).
  * (25) J.A.Carter, Z.Wang, D.D.Dlott, Acc. Chem. Res., 42, 1343 (2009).
  * (26) V.A.Benderskii, L.N.Gak, E.I.Kats, JETP, 108, 159 (2009).
  * (27) V.A.Benderskii, L.N.Gak, E.I.Kats, JETP, 109, 505 (2009).
  * (28) V.A.Benderskii, E.I.Kats, JETP Lett , 88, 338 (2008).
  * (29) V.A.Benderskii, E.I.Kats, Eur. Phys. J. D, 54, 597 (2009).
  * (30) V.A.Benderskii, E.I.Kats, JETP Lett., 92, 370 (2010).
  * (31) V.A.Benderskii, E.I.Kats, JETP, 113, 562 (2011).
  * (32) V.A.Benderskii, E.I.Kats, JETP Lett., 94. 459 (2011).
  * (33) R.D.Mattuck, A Guide to Feynmann Diagrams in Many-Body Problem, 2-nd Ed. McGraw-Hill, New York (1976).
  * (34) G.M.Zaslavsky, Chaos in Dynamical Systems, Harwood, New York (1985).
  * (35) M.Tabor, Chaos and Integrability in Nonlinear Dynamics, J.Wiley, New York (1989).
  * (36) M.E.Flatte, M.Holthaus, Ann. Phys., 245, 113 (1996).
  * (37) F.M.Izrailev, Phys.Rep., 196, 299 (1990).
  * (38) W.H.Zurek, Phys.Rev. D, 26, 1862 (1982).
  * (39) P.Grigolini, Quantum Mechanical Irreversibility, World Scientific, Singapore (1993).
  * (40) Ya.G.Sinai, Introducrion to Ergodic Theory, Princeton Univ. Press., Guilford (1977).
  * (41) M.V.Berry, Chaotic Behavior in Deterministic Systems, G.Cassati (Ed), Plenum Press, New York (1985).
  * (42) F.Haake, Quantum Signatures of Chaos, Springer, Berlin (1990).
  * (43) C.Keating, B.A.McClure, J.J.Rack, I.V.Rubtsov, J.Chem.Phys., 133, 144513 (2010).
  * (44) H.Kuzmany, B.Burger, A.Thess, R.E.Smalley, Carbon, 36, 709 (1998).
  * (45) V.Schettino, H.Prigolini, G.Cardini, J.Phys. Chem.. A, 106, 1815 (2002).
  * (46) S.Adhikari, R.Chowdhury, Phys. Lett. A, 375, 2166 (2011).
  * (47) T.Ishioka, W.Yan, H.L.Strauss, R.G.Snyder, Spectrochem. Acta A, 59, 671 (2003).
  * (48) L. Van Hove, Physica, 21, 517 (1955).
  * (49) L.D.Landau, E.M.Lifshitz, Statistical Physics, part 1, Pergamon, Oxford (1977).
  * (50) L.H.Yu, C.P.Sun, Phys.Rev. A., 49, 592 (1994).
  * (51) M.L.Mehta, Random Matrices, Academic Press, New York (1968).
  * (52) O.Bohigas, S.Tomsovic, D.Ulmo, Phys.Rep., 223, 43 (1993).
  * (53) T.Papenbrock, H.A.Weidenmuller, Rev.Mod.Phys., 79, 997 (2007).
  * (54) W.Dietz, S.F.Fischer, J.Chem.Phys., 113, 2741 (2000).
  * (55) D.V.Kurochkin, S.R.G.Naraharisetty, I.V.Rubtsov, PNAS, 104, 14209 (2007).
  * (56) S.G.Naraharisetty, V.M.Kysanenko, I.V.Rubtsov, J.Chem.Phys., 128, 104502 (2008).
  * (57) I.V.Rubtsov, Acc.Chem.Res., 42, 1385 (2009).
  * (58) Z.Smedarchina, W.Siebrand, M.Z.Zgierski, J.Chem.Phys., 103, 5326 (1995).
  * (59) V.A.Benderskii, A.V.Vetoshkin, L. von Laue, H.P.Trommsdorff, Chem.Phys., 219, 119 (1997).
  * (60) V.A.Benderskii, V.A.Vetoshkin, H.P.Trommsdorff, Chem.Phys., 234, 173 (1998); ibid: 244, 273 (1999), and 262, 393 (2000).
  * (61) G.V.Milnikov, H.Nakamura, Phys. Chem. Chem. Phys., 10, 1374 (2008).
  * (62) A.J.Leggett, S.Chakravarty, A.T.Dorsey, M.P.A.Fisher, A.Garg, M.Zwerger, Rev.Mod.Phys., 59, 1 (1987).
  * (63) V.A.Benderskii, D.E.Makarov, C.A.Wight, Chemical Dynamics at Low Temperatures, J.Wiley, New York (1994).
  * (64) A.O.Caldeira, A.J.Leggett, Ann. Phys., 149, 374 (1983).
  * (65) A.I.Larkin, Yu.N.Ovchinnikov, JETP, 58, 876 (1983); ibid, 59, 420 (1984), and 60, 1060 (1984).
  * (66) B.I.Ivlev, Yu.N.Ovchinnikov, JETP, 66, 378 (1987).
  * (67) L.H.Yu, Phys.Rev. A, 54, 3779 (1996).
  * (68) H.Bateman, A.Erdelyi, Higher Transcendental Functions, McGraw Hill, New York (1953).
  * (69) F.W.J.Olver, Asymptotics and Special Functions, Academic Press, New York (1974).