2
1 Introduction: From Bound States to the Continuum
sufficient legitimacy to resonance phenomena such as the unstable atomic nuclei or
decaying particles.
The aforementioned difficulties with Hilbert space formulation of quantum
mechanics have been resolved by extending the Hilbert space to a rigged Hilbert
space [2–9] within which the resonance wave functions find a natural place.
Resonance wave functions are given by Gamow states, also called Siegert states
[10], which are the eigenvectors of the Hamiltonian with a complex eigenvalue,
and their time evolution follows the exponential decay law. The Gamow states can
describe both sharp peaks in the cross section and decay of resonances or unstable
particles, unifying these two facets of the resonance phenomenon.
Open quantum systems, whose properties are affected by the environment of
scattering states, are intensely studied in nuclear physics, atomic and molecular
physics, nanoscience, quantum optics, etc. In spite of their specific features, they
have generic properties, which are common to all weakly bound or unbound systems. Specific experimental conditions of their studies make them complementary.
For example, nuclei and atoms can be prepared experimentally in a well-defined
state; however, one can hardly tune their individual properties. On the other hand, in
artificial open quantum system, such as quantum dots or atomic clusters, to achieve
identical experimental conditions is virtually impossible, but these systems can be
easily tunable by varying an external parameter.
What is being identified as a quantum environment of the system depends
on the considered physics problem. For example, the quantum environments in
quantum cosmology [11], quantum biology [12], or quantum information science
[13] differ one from another and from the environment which is relevant in
nuclear physics [14–16]. Consequently, many different theoretical and numerical
approaches have been proposed for the description of those different open quantum
systems. The standard approach is based on approximations for the exact master
equation. Tracking an exact evolution of the combined system-plus-environment is
usually impossible and also unwanted, as it involves the large amount of redundant
information which can be traced neither experimentally nor theoretically. Hence,
it is natural to develop reduced descriptions where the dynamics of the system
is considered explicitly, whereas the dynamics of the environment is described
implicitly. From this postulate follows the attempt to describe the system evolution
in terms of the reduced density obtained by taking partial trace over the exact
density of a combined system-plus-environment; hence, the evolution of combined
system-plus-environment is unitary. Depending on the nature of coupling between
system and environment, one obtains either Markovian or non-Markovian equations
of motion which describe the evolution of the open quantum system.
The main interest in studies using reduced density matrices is the energy
transfer to environment (the quantum dissipation) and the loss of coherence of
considered state(s) (the quantum decoherence). These are not quantities of interest
in the nuclear physics which deals with the well-defined quantum states and where
precise experiments provide detailed information about structure of these states
and their decays. Hence, the main emphasis in nuclear case is on the unitarity at
the opening of quantum system, i.e., at the transition from well-bound states (the
1 Introduction: From Bound States to the Continuum
sufficient legitimacy to resonance phenomena such as the unstable atomic nuclei or
decaying particles.
The aforementioned difficulties with Hilbert space formulation of quantum
mechanics have been resolved by extending the Hilbert space to a rigged Hilbert
space [2–9] within which the resonance wave functions find a natural place.
Resonance wave functions are given by Gamow states, also called Siegert states
[10], which are the eigenvectors of the Hamiltonian with a complex eigenvalue,
and their time evolution follows the exponential decay law. The Gamow states can
describe both sharp peaks in the cross section and decay of resonances or unstable
particles, unifying these two facets of the resonance phenomenon.
Open quantum systems, whose properties are affected by the environment of
scattering states, are intensely studied in nuclear physics, atomic and molecular
physics, nanoscience, quantum optics, etc. In spite of their specific features, they
have generic properties, which are common to all weakly bound or unbound systems. Specific experimental conditions of their studies make them complementary.
For example, nuclei and atoms can be prepared experimentally in a well-defined
state; however, one can hardly tune their individual properties. On the other hand, in
artificial open quantum system, such as quantum dots or atomic clusters, to achieve
identical experimental conditions is virtually impossible, but these systems can be
easily tunable by varying an external parameter.
What is being identified as a quantum environment of the system depends
on the considered physics problem. For example, the quantum environments in
quantum cosmology [11], quantum biology [12], or quantum information science
[13] differ one from another and from the environment which is relevant in
nuclear physics [14–16]. Consequently, many different theoretical and numerical
approaches have been proposed for the description of those different open quantum
systems. The standard approach is based on approximations for the exact master
equation. Tracking an exact evolution of the combined system-plus-environment is
usually impossible and also unwanted, as it involves the large amount of redundant
information which can be traced neither experimentally nor theoretically. Hence,
it is natural to develop reduced descriptions where the dynamics of the system
is considered explicitly, whereas the dynamics of the environment is described
implicitly. From this postulate follows the attempt to describe the system evolution
in terms of the reduced density obtained by taking partial trace over the exact
density of a combined system-plus-environment; hence, the evolution of combined
system-plus-environment is unitary. Depending on the nature of coupling between
system and environment, one obtains either Markovian or non-Markovian equations
of motion which describe the evolution of the open quantum system.
The main interest in studies using reduced density matrices is the energy
transfer to environment (the quantum dissipation) and the loss of coherence of
considered state(s) (the quantum decoherence). These are not quantities of interest
in the nuclear physics which deals with the well-defined quantum states and where
precise experiments provide detailed information about structure of these states
and their decays. Hence, the main emphasis in nuclear case is on the unitarity at
the opening of quantum system, i.e., at the transition from well-bound states (the