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The Liouvillian exceptional points for a quantum Markovian master equation of an oscillator in a generic environment are obtained. They occur at the points when the modified frequency of the oscillator vanishes, whereby the eigenvalues of the Liouvillian become real. In a generic system there are two parameters that modify the oscillator's natural frequency. One of the parameters can be the damping rate. The exceptional point then corresponds to critical damping of the oscillator. This situation is illustrated by the Caldeira--Leggett (CL) equation and the Markovian limit of the Hu--Paz--Zhang (HPZ) equation. The other parameter changes the oscillator's effective mass whereby the exceptional point is reached in the limit of extremely heavy oscillator. This situation is illustrated by a modified form of the Kossakowski--Lindblad (KL) equation. The eigenfunctions coalesce at the exceptional points and break into subspaces labelled by a natural number $N$. In each of the $N$-subspace, there is a $(N+1)$-fold degeneracy and the Liouvillian has a Jordan block structure of order-$(N+1)$. We obtain the explicit form of the generalized eigenvectors for a few Liouvillians. Because of the degeneracies, there is a freedom of choice in the generalized eigenfunctions. This freedom manifests itself as an invariance in the Jordan block structure under a similarity transformation whose form is obtained. We compare the relaxation of the first excited state of an oscillator in the underdamped region, critically damped region which corresponds to the exceptional point, and overdamped region using the generalized eigenvectors of the CL equation., Comment: To appear in Physica A (2023)

We study a linear chain of oscillators with inhomogeneity in their interactions with phonon bath. In a previous work on the Markovian master equation of the system, we investigated a model in which the difference in the site-phonon coupling between adjacent oscillators is the same throughout the chain. Here we look into another model in which the oscillators are coupled to the phonon bath with alternating strength at successive sites. Whereas in the first model all exciton modes are connected, in the second model they are coupled in pairs that are not connected to each other. Owing to this special structure in the coupling, the excitation numbers of different modes can be solved exactly in the steady state. In the first model, the minima of the excitation profile in the site basis occur at the edges of the chain, whereas in the second model the maxima occur at the edges. The energy transfer efficiency in the first model is affected by the source power whereas in the second model the efficiency is independent of it. A distinct feature in the second model is that a sink placed at the middle of the chain is able to distinguish between chains with even and odd number of sites. The energy transfer efficiency in a chain with even number of sites is higher than a chain with odd number of sites. Therefore, it reveals the discrete nature of the chain. In the limit of very long chain when the discreteness of the chain is less evident, the efficiencies approach each other., Comment: To appear in AIP Proceedings (2022)

We consider the reduced dynamics of a molecular chain weakly coupled to a phonon bath. With a small and constant inhomogeneity in the coupling, the excitation relaxation rates are obtained in closed form. They are dominated by transitions between exciton modes lying next to each other in the energy spectrum. The rates are quadratic in the number of sites in a long chain. Consequently, the evolution of site occupation numbers exhibits longer coherence lifetime for short chains only. When external source and sink are added, the rate equations of exciton occupation numbers are similar to those obtained earlier by Fr\"{o}hlich to explain energy storage and energy transfer in biological systems. There is a clear separation of time scale into a faster one pertaining to internal influence of the chain and phonon bath, and a slower one determined by external influence, such as the pumping rate of the source, the absorption rate of the sink and the rate of radiation loss. The energy transfer efficiency at steady state depends strongly on these external parameters, and is robust against a change in the internal parameters, such as temperature and inhomogeneity. Excitations are predicted to concentrate to the lowest energy mode when the source power is sufficiently high. In the site basis, this implies that when sustained by a high power source, a sink positioned at the center of the chain is more efficient in trapping energy than a sink placed at its end. Analytic expressions of energy transfer efficiency are obtained in the high power and low power source limit. Parameters of a photosynthetic system are used as examples to illustrate the results.

The eigenvalues of the Liouvillians of Markovian master equation for a harmonic oscillator have a generic form. The Liouvillians considered are quadratic in the position coordinates or creation and annihilation operators, as well as having positive renormalized frequencies. We prove this by showing that a generic Liouvillian of this form can be similarly related to the Liouvillian of the Kossakowski--Lindblad equation, whose eigenvalues are already known. The left and right eigenfunctions of the generic Liouvillian also form a complete and biorthogonal set. Examples of similarly related right eigenfunctions are given., Comment: To appear in Physica A

Through a set of generators that preserves the hermiticity and trace of density matrices, we analyze the damping of harmonic oscillator in open quantum systems into four modes, distinguished by their specific effects on the covariance matrix of position and momentum of the oscillator. The damping modes could either cause exponential decay to the initial covariance matrix or shift its components. They have to act together properly in actual dynamics to ensure that the generalized uncertainty relation is satisfied. We use a few quantum master equations to illustrate the results., Comment: Accepted by Physica A

We study general transformation on the density matrix of two-level system that keeps the expectation value of observable invariant. We introduce a set of generators that yields hermiticity and trace preserving general transformation which casts the transformation into simple form. The general transformation is in general not factorized and not completely positive. Consequently, either the parameter of transformation or the density matrix it acts on needs to be restricted. It can transform the system in the forward and backward direction with regard to its parameter, not as a semigroup in the time translation symmetry of dynamical maps. The general transformation can rotate the Bloch vector circularly or hyperbolically, dilate it or translate it. We apply the general transformation to study the general symmetry of amplitude damping and phase damping in two-level system. We generalize the generators to higher level systems., Comment: Accepted by European Physical Journal B

We obtain the solutions of the generic bilinear master equation for a quantum oscillator with constant coefficients in the Gaussian form. The well-behavedness and positive semidefiniteness of the stationary states could be characterized by a three-dimensional Minkowski vector. By requiring the stationary states to satisfy a factorized condition, we obtain a generic class of master equations that includes the well-known ones and their generalizations, some of which are completely positive. A further subset of the master equations with the Gibbs states as stationary states is also obtained. For master equations with not completely positive generators, an analysis on the stationary states suggests conditions on the coefficients of the master equations that generate positive evolution for a given initial state., Comment: Accepted by Physica A

We study the most general continuous transformation on the generators of bilinear master equations of a quantum oscillator. We find that transformation operators that preserve the hermiticity of density operators and conserve the probability of reduced dynamics should be adjoint-symmetric, and they are not limited to the pure product of unitary operators in the bra and ket space but could be a mixture of them. We need to include the more general transformation operators to explore the full symmetry of generic reduced dynamics. We discuss how the operators are related to those considered in previous works, and illustrate how they leave the reduced dynamics form invariant, or map one into the other. The positive semidefinite requirement on the density operator can be imposed to give a valid range of transformation parameters., Comment: Typos in Eq.(14) are corrected

We study a series of $N$ oscillators each coupled to its nearest neighbours, and linearly to a phonon field through the oscillator's number operator. We show that the Hamiltonian of a pair of adjacent oscillators, or a dimer, within the series of oscillators can be transformed to a form in which they are collectively coupled to the phonon field as a composite unit. In the weak coupling and rotating-wave approximation, the system behaves effectively like the trilinear boson model in the one excitation subspace of the dimer subsystem. The reduced dynamics of the one excitation subspace of the dimer subsystem coupled weakly to a phonon bath is similar to that of a two-level system, with a metastable state against the vacuum. The decay constant of the subsystem is proportional to the dephasing rate of the individual oscillator in a phonon bath, attenuated by a factor that depends on site asymmetry, intersite coupling and the resonance frequency between the transformed oscillator modes, or excitons. As a result of the collective effect, the excitation relaxation lifetime is prolonged over the dephasing lifetime of an individual oscillator coupled to the same bath., Comment: Misprints in Eqs. (21), (22) and (27) are corrected

We study the reduced dynamics of a pair of non-degenerate oscillators coupled collectively to a thermal bath. The model is related to the trilinear boson model where the idler mode is promoted to a field. Due to nonlinear coupling, the Markovian master equation for the pair of oscillators admits non-Gaussian equilibrium states, where the modes distribute according to the Bose-Einstein statistics. These states are metastable before the nonlinear coupling is taken over by linear coupling between the individual oscillators and the field. The Gibbs state for the individual modes lies in the subspace with infinite occupation quantum number. We present the time evolution of a few states to illustrate the behaviors of the system., Comment: Misprints are corrected. Final version

We consider the eigenvalue problem of a kinetic collision operator for a quantum Brownian particle interacting with a one-dimensional chain. The quantum nature of the system gives rise to a difference operator. For the one-dimensional case, the momentum space separates into infinite sets of disjoint subspaces dynamically independent of one another. The eigenvalue problem of the collision operator is solved with the continued fraction method. The spectrum is non-negative, possesses an accumulation point and exhibits a band structure. We also construct the eigenvectors of the collision operator and establish their completeness and orthogonality relations in each momentum subspaces., Comment: 12 pages, 4 figures

The quantum Markovian master equation of the reduced dynamics of a harmonic oscillator coupled to a thermal reservoir is shown to possess thermal symmetry. This symmetry is revealed by a Bogoliubov transformation that can be represented by a hyperbolic rotation acting on the Liouville space of the reduced dynamics. The Liouville space is obtained as an extension of the Hilbert space through the introduction of tilde variables used in the thermofield dynamics formalism. The angle of rotation depends on the temperature of the reservoir, as well as the value of Planck's constant. This symmetry relates the thermal states of the system at any two temperatures. This includes absolute zero, at which purely quantum effects are revealed. The Caldeira-Leggett equation and the classical Fokker-Planck equation also possess thermal symmetry. We compare the thermal symmetry obtained from the Bogoliubov transformation in related fields and discuss the effects of the symmetry on the shape of a Gaussian wave packet., Comment: Eqs.(64a), (65a)-(68) are corrected

We derive an exact Markovian kinetic equation for an oscillator linearly coupled to a heat bath, describing quantum Brownian motion. Our work is based on the subdynamics formulation developed by Prigogine and collaborators. The space of distribution functions is decomposed into independent subspaces that remain invariant under Liouville dynamics. For integrable systems in Poincar\'e's sense the invariant subspaces follow the dynamics of uncoupled, renormalized particles. In contrast for non-integrable systems, the invariant subspaces follow a dynamics with broken-time symmetry, involving generalized functions. This result indicates that irreversibility and stochasticity are exact properties of dynamics in generalized function spaces. We comment on the relation between our Markovian kinetic equation and the Hu-Paz-Zhang equation., Comment: A few typos in the published version are corrected

Unstable state furnishes a semigroup irreducible representation of the Poincar\'e group. The state vector is represented by a superposition of energy eigenkets. As a consequence of this superposition, the state vector can be transformed into the rest frame through {\it a} Lorentz transformation only when the eigenkets are labeled by velocity variable, but not momentum variable. We also clarify the meaning of the velocity variable in the state vector with respect to the velocity derived from kinematical consideration of the scattering process., Comment: Major revision

The definition of mass and width of relativistic resonances and in particular of the $Z$-boson is discussed. For this we use the theory based on time asymmetric boundary conditions given by Hardy class spaces ${\mathbf \Phi}_-$ and ${\mathbf \Phi}_+$ for prepared in-states and detected out-states respectively, rather than time symmetric Hilbert space theory. This Hardy class boundary condition is a mathematically rigorous form of the singular Lippmann-Schwinger equation. In addition to the rigorous definition of the Lippmann-Schwinger kets $|[j,{\mathsf s}]^{\pm}>$ as functionals on the spaces ${\mathbf \Phi}_{\mp}$, one obtains Gamow kets $|[j,{\mathsf s}_R]^- >$ with complex centre-of-mass energy value ${\mathsf s}_R=(M_R-i\Gamma_R/2)^2$. The Gamow kets have an exponential time evolution given by $\exp{(-iM_Rt-\Gamma_Rt/2)}$ which suggests that $(M_R,\Gamma_R)$ is the right definition of the mass and width of a resonance. This is different from the two definitions of the $Z$-boson mass and width used in the Particle Data Table and leads to a numerical value of $M_R=(91.1626\pm 0.0031) {\rm GeV}$ from the $Z$-boson lineshape data., Comment: 21 pages revtex file