Non-equilibrium Thermo Field Dynamics Research Articles

4 × 4-matrix transformation for systems with spontaneous symmetry breakdown and self-consistent renormalization in nonequilibrium Thermo Field Dynamics

This study aims to construct a theoretical formulation of a nonequilibrium process for a system of Bose–Einstein condensate associated with a spontaneous symmetry breakdown. For this, Thermo Field Dynamics is used. We then describe the most general forms of a 4 × 4-matrix transformation and an unperturbed Hamiltonian. After calculating the 4 × 4-matrix self-energy and identifying its on-shell, we impose the renormalization condition in which the entire on-shell self-energy should vanish. This condition provides a sufficient number of independent equations to determine all of the parameters in an unperturbed Hamiltonian, among which the quantum transport equations for normal and anomalous number distributions are included.

Transverse susceptibility and resonance absorption of an anti-ferromagnetic spin wave interacting with a phonon reservoir

Abstract A form of the transverse magnetic susceptibility for an anti-ferromagnetic spin system interacting with a phonon reservoir is derived employing the TCLE method (a method in which the admittance of a physical system is directly derived from time-convolutionless equations with external driving terms) in terms of the non-equilibrium thermo-field dynamics (NETFD) in the spin-wave approximation. The region valid for the lowest spin-wave approximation is numerically investigated in detail, and the transverse susceptibility for the anti-ferromagnetic system of one-dimensional infinite spins of magnitude $S = 2$ is numerically studied in that region. It is confirmed that the effects of the memory and initial correlation for the spin system and phonon reservoir, which are represented by the interference terms in the TCLE method, increase the power absorption in the resonance region. In the region valid for the lowest spin-wave approximation, it is shown that the line half-width $\Delta \omega_{\mathrm{RF}}$ of the power absorption in the resonance region for a transversely rotating magnetic field increases as the temperature $T$ rises, and decreases as the uniaxial anisotropy energy $\hbar K$ of the $z$ direction increases or as the wave number $k$ becomes large, and also that the line peak-height $H_{\mathrm{RF}}$ of the power absorption in the resonance region for the transverse magnetic field decreases as $T$ rises and increases as $\hbar K$ increases or as $k$ becomes large. According to analytic considerations, it is anticipated that $\Delta \omega_{\mathrm{RF}}$ and $H_{\mathrm{RF}}$ vary approximately as $\Delta \omega_{\mathrm{RF}} \approx \{A_k + B_k \bar{n} (\omega_{{\mathrm{R}} k} )\} \{ \bar{n}(\omega_{{\mathrm{R}} k}) + \,1\}$ and $H_{\mathrm{RF}} \propto \{A_k + B_k \bar{n}(\omega_{{\mathrm{R}} k})\}^{- 1} \{ \bar{n}(\omega_{{\mathrm{R}} k}) + 1\}^{- 1}$, with $\bar{n}(\omega_{{\mathrm{R}} k}) = \{ \exp(\hbar \omega_{{\mathrm{R}} k}/ (k_{\mathrm{B}} T)) -1 \}^{- 1}$, where $\omega_{{\mathrm{R}} k}$ is the characteristic frequency of the phonon reservoir. Here, $A_k$ is a positive quantity dependent on the Zeeman frequency, the spin magnitude, and the anisotropy energy of the spin system, and $B_k$ is a positive quantity dependent on the anisotropy energy. At low temperatures ($k_{\mathrm{B}} T \ll \hbar \omega_{{\mathrm{R}} k}$), it is anticipated that $\Delta \omega_{\mathrm{RF}}$ and $H_{\mathrm{RF}}$ vary approximately as $\Delta \omega_{\mathrm{RF}} \approx A_k \exp\{((A_k + B_k)/ A_k) \exp(- \hbar \omega_{{\mathrm{R}} k}/ (k_{\mathrm{B}} T))\}$ and $H_{\mathrm{RF}} \propto A_k^{- 1} \exp\{- ((A_k + B_k)/ A_k) \exp(- \hbar \omega_{{\mathrm{R}} k}/ (k_{\mathrm{B}} T))\}$.

Expanding the Range of Hierarchical Equations of Motion by Tensor-Train Implementation.

The non-equilibrium thermo-field dynamics formulation of the hierarchical equations of motion combined with the tensor-train representation of the density matrix is discussed, and a new numerical integration scheme is introduced. The numerical methodology is based on an adaptive low-rank Galerkin reduction scheme and can preserve linear invariants (such as the trace of the density matrix). The method is applied to the study of the charge transfer dynamics in model pentacene molecular aggregates. The combined effect of a discrete set of molecular vibrational modes and a thermal bath is investigated, paying special attention to the coherent-incoherent transition of the charge transport. The new computational framework is shown to be a very promising methodology for the study of the quantum dynamics of complex molecular systems in the condensed phase.

Finite temperature quantum dynamics of complex systems: Integrating thermo‐field theories and tensor‐train methods

AbstractThis review provides the fundamental theoretical tools for the development of a complete wave‐function formalism for the study of time‐evolution of chemico‐physical systems at finite temperature. The methodology is based on the non‐equilibrium thermo‐field dynamics (NE‐TFD) representation of quantum mechanics, which is alternative to the commonly used density matrix representation. TFD concepts are extended and integrated with the tensor‐train (TT) numerical tools leading to a novel and powerful theoretical and computational framework for the study of complex quantum dynamical problems. In addition, NE‐TFD techniques are extended to enable the study of dissipative open systems via a new formulation of the hierarchical equations of motion (HEOM) fully integrated with TT methodologies. We demonstrate that the combination of the TFD machinery with computational advantages of TTs results in a powerful theoretical and computational framework for scrutinizing dynamics of complex multidimensional electron‐vibrational systems. We illustrate the validity and the computational advantages of the developed methodologies by applying them to the study of quantum coherence effects in the energy‐transfer processes in antenna systems, to the analysis of fingerprints of vibrational modes in electron‐transfer and charge‐transfer processes in various model and realistic multidimensional molecular systems, as well as to simulation of other fundamental models of physical chemistry.This article is categorized under: Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics

Unification of Thermo Field Kinetic and Hydrodynamics Approaches in the Theory of Dense Quantum–Field Systems

A formulation of nonequilibrium thermo-field dynamics has been performed using the nonequilibrium statistical operator method by D.N. Zubarev. Generalized transfer equations for a consistent description of the kinetics and hydrodynamics of the dense quantum field system with strongly-bound states are derived.

Zubarev’s Nonequilibrium Statistical Operator Method in the Generalized Statistics of Multiparticle Systems

We present a generalization of Zubarev’s nonequilibrium statistical operator method based on the principle of maximum Renyi entropy. In the framework of this approach, we obtain transport equations for the basic set of parameters of the reduced description of nonequilibrium processes in a classical system of interacting particles using Liouville equations with fractional derivatives. For a classical systems of particles in a medium with a fractal structure, we obtain a non-Markovian diffusion equation with fractional spatial derivatives. For a concrete model of the frequency dependence of a memory function, we obtain generalized Kettano-type diffusion equation with the spatial and temporal fractality taken into account. We present a generalization of nonequilibrium thermofield dynamics in Zubarev’s nonequilibrium statistical operator method in the framework of Renyi statistics.

Nonequilibrium Dynamics of the σ-Model Modes on the de Sitter Space

The two-dimensional σ-model with the de Sitter target space is locally canonic in the north pole diamond of the Penrose diagram in the cosmological gauge. The left and right moving modes on the cylindrical base space are entangled among themselves and interact with the de Sitter metric. Firstly, we show that the untangled oscillators can be obtained from the entangled operators by applying a set of Bogoliubov transformations constrained by the requirement that the partial evolution generator be diagonal. Secondly, we determine the nonequilibrium dynamics of the untangled modes in the nonequilibrium thermofield dynamics formalism. The thermal modes are represented as thermal doublet oscillators that satisfy partial evolution equations of Heisenberg-type. From these we compute the local free one-body propagator of an arbitrary mode between two times. Thirdly, we discuss the field representation of the thermal modes. We show that there is a set of thermal doublet fields that satisfy the equal time canonical commutation relations, are solutions to the σ-model equations of motion, and can be decomposed in terms of thermal doublet oscillators. Finally, we construct a local partial evolution functional of Hamilton-like form for the thermal doublet fields.

Relaxation Process of Interacting Two-mode System Influenced by Markovian Thermal Reservoirs

Two different models of a relaxation process are considered for a linearly interacting two-mode system under the influence of independent Markovian thermal reservoirs. One is to describe the relaxation process of bare particles and the other is to describe the one of quasi particles which are derived from bare particles by the Bogoliubov transformation. The difference is that the former does not includes the effect of the inter-mode interaction on the damping operator while the latter does. The equations of motion are solved algebraically by making use of non-equilibrium thermo field dynamics. The relaxation processes in the two models are investigated in detail. The results are applied for investigating a non-ideal beam splitter with photon loss and noise addition.

Postselected quantum systems and their time evolution in non-equilibrium thermo-field dynamics

A postselected quantum system is studied within the framework of non-equilibrium thermo-field dynamics, where an extended Hilbert space is used for describing a quantum system. A two-operator state defined on the extended Hilbert space is introduced to describe a property of a postselected quantum system. By imposing some conditions on the two-operator state, it reduces to the previously introduced quantities such as the two-vector state, the W-operator, the connection state and the 2-time quantum state. The correlation included in the two-operator state is clarified. Furthermore, time evolution of a postselected quantum system influenced by a Markovian environment is investigated. The linear response to an external field is also discussed. The results show the symmetric property of a postselected quantum system, even if time evolution is not unitary.

Non-equilibrium thermo-field dynamics for anti-ferromagnetic spin system

The non-equilibrium thermo-field dynamics for an anti-ferromagnetic spin system interacting with a phonon reservoir is proposed for the case of a non-bilinear unperturbed Hamiltonian, which includes not only a bilinear part but also a non-bilinear part, in the spin–wave approximation. The two kinds of quasi-particle operators are introduced, and their forms are derived for the semi-free boson fields. It is shown that the two quasi-particles decay exponentially with the frequencies and life-times which are different from each other. It is also shown that each quasi-particle changes to the other tilde quasi-particle through the spin–phonon interaction. The spin–spin correlation functions and longitudinal magnetization for the anti-ferromagnetic spin system under an external static magnetic field are derived in the forms convenient for the perturbation expansions. The expectation values of the spin–wave energy and longitudinal magnetization and the spin–spin correlation functions are investigated numerically for an anti-ferromagnetic system of one-dimensional infinite spins interacting with a damped phonon-reservoir, in the region valid for the lowest spin–wave approximation and the narrowing-limit approximation in which the relaxation times of the spin system are much larger than the correlation time of the phonon reservoir. The two-point Green’s function of the semi-free spin–wave for the anti-ferromagnetic spin system is derived by introducing the thermal quartet notation, and it is given in a form of 4×4 matrix.

Noisy parametric amplifiers in non-equilibrium thermo field dynamics

Single-mode and two-mode parametric amplifiers under the influence of Markovian environments are studied by means of non-equilibrium thermo field dynamics. In the presence of both parametric coupling and system–environment interaction, the dissipative Heisenberg equations of motion are solved for the optical modes of interest. By making use of the solutions, it is examined whether the noisy parametric amplifiers can exhibit the non-classical properties. Furthermore, it is shown that the two-mode parametric amplifier is equivalent to the two single-mode parametric amplifiers with subsequent beam splitting, even if they are influenced by the environments.

Application of Non-Equilibrium Thermo Field Dynamics to quantum teleportation under the environment

Quantum teleportation for continuous variables is treated by Non-Equilibrium Thermo Field Dynamics (NETFD), a canonical operator formalism for dissipative quantum systems, in order to study the effect of imperfect quantum entanglement on quantum communication. We used an entangled state constructed by two squeezed states. The entangled state is imperfect due to two reasons, i.e., one is the finiteness of the squeezing parameter r and the other comes from the process that the squeezed states are created under the dissipative interaction with the environment. We derive the expressions for one-shot fidelity (OSF), probability density function (PDF) associated with OSF and (averaged) fidelity by making full use of the algebraic manipulation of operator algebra within NETFD. We found that OSF and PDF are given by Gaussian forms with its peak at the original information α to be teleported, and that for r≫1 the variances of these quantities blow up to infinity for κ/χ≤1, while they approach to finite values for κ/χ>1. Here, χ represents the intensity of a degenerate parametric process, and κ the relaxation rate due to the interaction with the environment. The blow-up of the variances for OSF and PDF guarantees higher security against eavesdropping. With the blow-up of the variances, the height of PDF reduces to small because of the normalization of probability, while the height of OSF approaches to 1 indicating a higher performance of the quantum teleportation. We also found that in the limit κ/χ≫1 the variances of both OSF and PDF for any value of r (>0) reduce to 1 which is the same value as the case r=0, i.e., no entanglement.

From classical mechanics with doubled degrees of freedom to quantum field theory for nonconservative systems

The $2 \times 2$-matrix structure of Green's functions is a common feature for the real-time formalisms of quantum field theory under thermal situations, such as the closed time path formalism and Thermo Field Dynamics (TFD). It has been believed to originate from quantum nature. Recently, Galley has proposed the Hamilton's principle with initial data for nonconservative classical systems, doubling each degree of freedom [Phys. Rev. Lett. 110, 174301 (2013)]. We show that the Galley's Hamilton formalism can be extended to quantum field and that the resulting theory is naturally identical with nonequilibrium TFD.

From superoperator formalism to nonequilibrium Thermo Field Dynamics

Emphasizing that the specification of the representation space or the quasiparticle picture is essential in nonequilibrium quantum field system, we have constructed the unique unperturbed representation of the interaction picture in the superoperator formalism. To achieve it, we put the three basic requirements (the existence of the quasiparticle picture at each instant of time, the macroscopic causality and the relaxation to equilibrium). From the resultant representation follows the formulation of nonequilibrium Thermo Field Dynamics (TFD). The two parameters, the number distribution and excitation energy, characterizing the representation, are to be determined by the renormalization condition. While we point out that the diagonalization condition by Chu and Umezawa is inconsistent with the equilibrium theory, we propose a new renormalization condition as a generalization of the on-shell renormalization on the self-energy which derives the quantum transport equation and determines the renormalized excitation energy.

NONEQUILIBRIUM THERMO FIELD DYNAMICS FOR RELATIVISTIC COMPLEX SCALAR AND DIRAC FIELDS

Relativistic quantum field theories for complex scalar and Dirac fields are investigated in nonequilibrium thermo field dynamics. The thermal vacuum is defined by the Bogoliubov transformed creation and annihilation operators. Two independent Bogoliubov parameters are introduced for a charged field. Its difference naturally induces the chemical potential. Time-dependent thermal Bogoliubov transformation generates the thermal counterterms. We fix the terms by the self-consistency renormalization condition. Evaluating the thermal self-energy under the self-consistency renormalization condition, we derive the quantum Boltzmann equations for the relativistic fields.

Nonequilibrium dynamics of strings in time-dependent plane wave backgrounds

We formulate and study the nonequilibrium dynamics of strings near the singularity of the time-dependent plane wave background in the framework of the Nonequilibrium Thermo Field Dynamics (NETFD). In particular, we construct the Hilbert space of the thermal string oscillators at nonequilibrium and generalize the NETFD to describe the coordinates of the center of mass of the thermal string. The equations of motion of the thermal fields and the Hamiltonian are derived. Due to the time-dependence of the oscillator frequencies, a counterterm is present in the Hamiltonian. This counterterm determines the correlation functions in a perturbative fashion. We compute the two point correlation function of the thermal string at zero order in the power expansion.

Canonical Quantization for a Relativistic Neutral Scalar Field in Non-Equilibrium Thermo Field Dynamics

A relativistic neutral scalar field is investigated in non-equilibrium thermo field dynamics. The canonical quantization is applied to the fields out of equilibrium. Because the thermal Bogoliubov transformation becomes time-dependent, the equations of motion for the ordinary unperturbed creation and annihilation operators are modified. This forces us to introduce a thermal counter term in the interaction Hamiltonian which generates additional radiative corrections. Imposing the self-consistency renormalization condition on the total radiative corrections, we obtain the quantum Boltzmann equation for the relativistic scalar field.

Boltzmann Equation for Relativistic Neutral Scalar Field in Non-Equilibrium Thermo Field Dynamics

A relativistic neutral scalar field is investigated on the basis of the Schwinger-Dyson equation in the non-equilibrium thermo field dynamics. A time evolution equation for a distribution function is obtained from a diagonalization condition for the Schwinger-Dyson equation. An explicit expression of the time evolution equation is calculated in the $\lambda\phi^4$ interaction model at the 2-loop level. The Boltzmann equation is derived for the relativistic scalar field. We set a simple initial condition and numerically solve the Boltzmann equation and show the time evolution of the distribution function and the relaxation time.

Unifying treatment of nonequilibrium and unstable dynamics of cold bosonic atom system with time-dependent order parameter in Thermo Field Dynamics

The coupled equations which describe the temporal evolution of the Bose–Einstein condensed system are derived in the framework of nonequilibrium Thermo Field Dynamics. The key element is that they are not the naive assemblages of assumed equations, but are the self-consistent ones derived by appropriate renormalization conditions. While the order parameter is time-dependent, an explicit quasiparticle picture is constructed by a time-dependent expansion. Our formulation is valid even for the system with a unstable condensate, and describes the condensate decay caused by the Landau instability as well as by the dynamical one.

Non-Equilibrium Thermo-Field Dynamics for a Fourth-Order Hamiltonian

The non-equilibrium thermo-field dynamics proposed by Arimitsu and Umezawa are generalized to the case of a fourth-order unperturbed Hamiltonian which includes not only a second-order (quadratic) part but also a fourth-order part. Fujita’s analysis for effects of the initial particle correlation of a quantum gas is proved generally in terms of TFD. The forms of the quasi-particle operators for a semi-free boson field are derived. It is shown that the energies and life-times of the quasi-particles depend on the adiabatic boson-reservoir interaction which leads to the fourth-order part of the unperturbed Hamiltonian. The form of the two-point Green’s function for the semi-free boson field is evaluated. A form of the admittance for a boson system interacting with its heat reservoir, which includes effects of the initial correlation and memory, is derived using the TCLE method formulated in terms of the generalized non-equilibrium thermo-field dynamics. A calculation method of the higher-order parts of the admittance in powers of the boson-boson interaction is given. Furthermore, a calculation method of the perturbation expansions of the two-point Green’s function for the boson system is given. Subject Index: 130