Reduced density matrices and phase-space distributions in thermofield dynamics.

Thermofield dynamics (TFD) arose out of efforts to define operators at finite temperatures in the theory of superconductivity.1 Since then it has developed into a powerful formalism for dealing with quantum field theories at finite temperatures2-4 and has found numerous applications in non-equilibrium phenomena5 and in quantum optics.6-15 The central idea in TFD is to represent a density operator as a vector in an extended Hilbert space. The same idea, in fact, also forms the basis of another formalism known as the Liouville space representation.16-19 The difference between the two formalisms is as follows. Consider a density operator ρ on a Hilbert space ℌ. Any operator on ℌ and, in particular, the density operator ρ can be expanded in terms of the operators |M > constitutes a complete orthonormal basis in ℌ. In TFD the operators |M > . In Liouville space representation,16-19 on the other hand, |M >< N| are regarded as the basis vectors in the Hilbert space ℌ of linear operators on ℌ. In what follows, we shall confine ourselves to the TFD representation and consider its application to quantum optics. In particular, we shall highlight its usefulness for exact or approximate solution of master equations encountered in quantum optics.

In many problems in quantum optics, damping plays an important role. These include, for example, the decay of an atom in an excited state to a lower state and the decay of the radiation field inside a cavity with partially transparent mirrors. In general, damping of a system is described by its interaction with a reservoir with a large number of degrees of freedom. We are interested, however, in the evolution of the variables associated with the system only. This requires us to obtain the equations of motion for the system of interest only after tracing over the reservoir variables. There are several different approaches to deal with this problem.

The Bogoliubov transformation in thermofield dynamics, an operator formalism for the finite-temperature quantum-field theory, is generalized to describe a field in arbitrary confined regions of space and time. Starting with the scalar field, the approach is extended to the electromagnetic field and the energy-momentum tensor is written via the Bogoliubov transformation. In this context, the Casimir effect is calculated for zero and non-zero temperature, and therefore it can be considered as a vacuum condensation effect of the electromagnetic field. This aspect opens an interesting perspective for using this procedure as an effective scheme for calculations in the studies of confined fields, including the interacting fields.

An algorithm to compute vibronic spectra of harmonic surfaces including Dushinsky rotation and Hertzberg–Teller terms is described. The method, inspired by thermo field dynamics, maps the thermal density matrix onto the vacuum state and uses the time-dependent coupled cluster ansatz to propagate it in time. In the Franck–Condon approximation where the dipole matrix elements are taken to be constants, this reduces to the auto correlation function of the new vacuum. In the Hertzberg–Teller approximation, the full time evolution operator is needed. This too is governed by a closed set of equations. The theoretical development is presented along with an application to anthracene.

Random phase approximation (RPA) electron correlation methods have gained in popularity in the recent years. A number of RPA correlation energy variants emerged in the Kohn–Sham DFT framework and in the theory of strongly orthogonal geminals. Foundations of most RPA approaches trace back to an exact expression for two‐electron reduced density matrix written in terms of one‐electron density matrix and dynamic one‐electron response functions, originally presented in the seminal paper of McLachlan and Ball (Rev. Mod. Phys. 1964, 36, 844). The aim of this article is to give a pedagogical introduction to possible approaches for describing electron correlation energy based on the McLachlan and Ball formula. The focus of the presentation is to formulate electron‐interaction energy expressions as functions of reduced density matrices. On one hand, it provides a common umbrella for RPA approximations proposed for uncorrelated (Hartree–Fock or Kohn–Sham) as well as partially correlated (strongly orthogonal geminals) references. On the other hand, such presentation may stimulate new developments in density matrix functional theory.

With the use of the density equation [Phys. Rev. A 14, 41 (1976)], second-order density matrices of atoms and molecules are calculated directly without any use of the wave function. The decoupling of the third- and fourth-order density matrices is done using the diagrammatic procedure of the Green's-function method, including terms up to second order in electron correlation. In addition to the results given in a previous communication [Phys. Rev. Lett. 76, 1039 (1996)], we give more detailed formulations, discussions, and additional results for ${\mathrm{H}}_{2}\mathrm{O},$ ${\mathrm{NH}}_{3},$ ${\mathrm{CH}}_{4},$ HF, ${\mathrm{N}}_{2},$ CO, and acetylene using double-\ensuremath{\zeta} basis sets, and for ${\mathrm{CH}}_{3}\mathrm{OH},$ ${\mathrm{CH}}_{3}{\mathrm{NH}}_{2},$ and ${\mathrm{C}}_{2}{\mathrm{H}}_{6}$ (staggered and eclipsed) using minimal basis sets. The present density-equation method gave energies as accurate as, and reduced density matrices (RDM's) more accurate than the single and double excitation configuration interaction method. The present method seems to give better quality results as the system becomes large. The convergence was fairly good and the calculated second-order RDM's almost satisfied some necessary conditions of the $N$ representability, the so-called $P,$ $Q,$ and $G$ conditions, while the first-order RDM's were exactly $N$ representable. The variety of the molecules calculated and the quality of the calculated results show that the density-equation method can be a promising alternative to the wave-function approach in quantum mechanics.

A simple perturbation theory has been developed for reduced density matrices which are known to be N representable as functions of some set of independent parameters. Formulas are presented for single and double perturbations through second order. Extension to higher order would be possible, but involved. The results do not depend explicitly on the order of the density matrix, and when the N-particle density matrix is used, the formulas can be reduced to those of conventional Rayleigh–Schrödinger theory. Details are given for the special cases of spin-unrestricted or extended SCF density matrices and CI or optimized multiconfiguration density matrices which are known to be N representable.

Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.

A functional form for electrons scattering off localized potentials with boundary conditions imposed through the Wigner distribution function is formulated in a manner appropriate for the study of tunneling in nanoscale junctions. Variation of the functional with respect to the electronic density matrix leads to stationary or critical points respecting open-system boundary conditions. Examples are presented for single-electron and many- (noninteracting) electron scattering and, in both instances, the usual outcome of a one-electron density matrix comprised of scattering wave functions results. The examples highlight how the degrees of freedom associated with imposing scattering boundary conditions can be constrained and removed from a variational determination of the electronic wave functions on a scattering region. The form for the scattering functional is motivated in terms of the maximum entropy principle.

Quantum field theory and coalgebraic logic in theoretical computer science

Although the origin of quantum optics can be traced back to the beginning of the 20th century, when the fundamental ideas about the quantum nature of the interaction between light and matter were put forward, the splendid blossoming of this part of physics began half a century later, after the invention of masers and lasers. It is remarkable that after another half a century the tree of quantum optics is not only very strong and spreading, but all its branches continue to grow, showing new beautiful blossoms and giving very useful fruits.

We derive and implement symmetry-projected Hartree-Fock-Bogoliubov (HFB) equations and apply them to the molecular electronic structure problem. All symmetries (particle number, spin, spatial, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. We show that the resulting method yields a comprehensive black-box treatment of static correlations with effective one-electron (mean-field) computational cost. The ensuing wave function is of multireference character and permeates the entire Hilbert space of the problem. The energy expression is different from regular HFB theory but remains a functional of an independent quasiparticle density matrix. All reduced density matrices are expressible as an integration of transition density matrices over a gauge grid. We present several proof-of-principle examples demonstrating the compelling power of projected quasiparticle theory for quantum chemistry.

A new Wigner phase-space distribution function is introduced for thermal quasiparticles currently in use in thermo field dynamics. The properties of this phase-space distribution are studied and compared with those of the corresponding zero-temperature distribution. Generalized uncertainty relations are implemented in order to study the explicit spreading in phase space as a function of temperature and quantum number of the thermal excitation, thereby permitting the investigation of quantum and thermal fluctuations.

Within finite temperature field theory, we show that truncated non-perturbative self-consistent Dyson resummation schemes can be renormalized with local counter-terms defined at the vacuum level. The requirements are that the underlying theory is renormalizable and that the self-consistent scheme follows Baym''s $\Phi$-derivable concept. The scheme generates both, the renormalized self-consistent equations of motion and the closed equations for the infinite set of counter terms. At the same time the corresponding 2PI-generating functional and the thermodynamical potential can be renormalized, in consistency with the equations of motion. This guarantees the standard $\Phi$-derivable properties like thermodynamic consistency and exact conservation laws also for the renormalized approximation schemes to hold. The proof uses the techniques of BPHZ-renormalization to cope with the explicit and the hidden overlapping vacuum divergences.

We study the quark Wigner distributions of the pion meson which are the phase-space distributions containing the most general one-parton information.Using the pion wavefunctions deduced from a light-cone quark model, we calculate the Wigner distributions of the unpolarized, longitudinally polarized and transversely polarized quarks inside the pion meson via the Fock-state overlap representation, respectively.We present the numerical results of the transverse Wigner distributions in which the longitudinal momentum fraction is integrated out. The mixed Wigner distributions as functions of $b_y$ and $k_x$ are also presented.As a intermediate step, we provide the analytic results for the generalized transverse momentum dependent distributions of the pion meson in the same model.