L. I. Plimak

Commuting Heisenberg operators as the quantum response problem: Time-normal averages in the truncated Wigner representation

The applicability of the so-called truncated Wigner approximation (−W) is extended to multitime averages of Heisenberg field operators. This task splits naturally in two. First, what class of multitime averages the −W approximates and, second, how to proceed if the average in question does not belong to this class. To answer the first question, we develop a (in principle, exact) path-integral approach in phase space based on the symmetric (Weyl) ordering of creation and annihilation operators. These techniques calculate a new class of averages which we call time-symmetric. The −W equations emerge as an approximation within these path-integral techniques. We then show that the answer to the second question is associated with response properties of the system. In fact, for two-time averages, Kubo’s renowned formula relating the linear-response function to two-time commutators suffices. The −W is directly generalized to the response properties of the system allowing one to calculate approximate time normally ordered two-time correlation functions with surprising ease. The techniques we develop are demonstrated for the Bose-Hubbard model.

Causal signal transmission by quantum fields. I. Response of the harmonic oscillator

It is shown that response properties of a quantum harmonic oscillator are in essence those of a classical oscillator, and that, paradoxical as it may be, these classical properties underlie all quantum dynamical properties of the system. The results are extended to non-interacting bosonic fields, both neutral and charged.

Causal signal transmission by quantum fields. II. Quantum-statistical response of interacting bosons

Abstract We analyse nonperturbatively signal transmission patterns in Green’s functions of interacting quantum fields. Quantum field theory is reformulated in terms of the nonlinear quantum-statistical response of the field. This formulation applies equally to interacting relativistic fields and nonrelativistic models. Of crucial importance is that all causality properties to be expected of a response formulation indeed hold. Being by construction equivalent to Schwinger’s closed-time-loop formalism, this formulation is also shown to be related naturally to both Kubo’s linear response and Glauber’s macroscopic photodetection theories, being a unification of the two with generalisation to the nonlinear quantum-statistical response problem. In this paper we introduce response formulation of bosons; response reformulation of fermions will be subject of a separate paper.

Signatures of attosecond electronic?nuclear dynamics in the one-photon ionization of molecular hydrogen: analytical model versus ab initio calculations

We present an analytical model based on the time-dependent WKB approximation to reproduce the photoionization spectra of an H2 molecule in the autoionization region. We explore the nondissociative channel, which is the major contribution after one-photon absorption, and we focus on the features arising in the energy differential spectra due to the interference between the direct and the autoionization pathways. These features depend on both the timescale of the electronic decay of the autoionizing state and the time evolution of the vibrational wavepacket created in this state. With full ab initio calculations and with a one-dimensional approach that only takes into account the nuclear wavepacket associated to the few relevant electronic states we compare the ground state, the autoionizing state, and the background continuum electronic states. Finally, we illustrate how these features transform from molecular-like to atomic-like by increasing the mass of the system, thus making the electronic decay time shorter than the nuclear wavepacket motion associated with the resonant state. In other words, autoionization then occurs faster than the molecular dissociation into neutrals.

Three-body bound states in atomic mixtures with resonant p-wave interaction.

We employ the Born-Oppenheimer approximation to find the effective potential in a three-body system consisting of a light particle and two heavy ones when the heavy-light short-range interaction potential has a resonance corresponding to a nonzero orbital angular momentum. In the case of an exact resonance in the p-wave scattering amplitude, the effective potential is attractive and long range; namely, it decreases as the third power of the interatomic distance. Moreover, we show that the range and power of the potential, as well as the number of bound states, are determined by the mass ratio of the particles and the parameters of the heavy-light short-range potential.

Operator ordering and causality beyond the rotating wave approximation

Conventional definition of the time-normal operator ordering (Kelley P. L. and Kleiner W. H., Phys. Rev., 136 (1964) A316) is prone to causality violations (de Haan M., Physica A, 132 (1985) 375; 397). We show that such violations disappear if this definition is amended outside the rotating wave approximation. Nonrelativistic causality of an arbitrary time-normal product turns out to be a property of quantum kinematics, while relativistic causality is demonstrated for a time-normal product of two operators under the most general assumptions about quantum dynamics (commutativity of operators at space-like intervals). This eliminates the key obstacle preventing phase-space techniques of quantum optics from being extended to arbitrary quantum fields including fermions.

Efimov states in atom-molecule collisions

We analyse scattering of a heavy atom off a weakly bound molecule comprising an identical heavy and a light atom in the Born-Oppenheimer approximation. We focus on the situation where the heavy atoms are bosons, which was realized in several experiments. The elastic and inelastic cross sections for the atom-molecular scattering exhibit a series of resonances corresponding to three-body Efimov states. Resonances in elastic collisions are accessible experimentally through thermalization rates, and thus constitute an alternative way of observing Efimov states.

Operator ordering and causality

Introduction.— Causality in photocounting is a longstanding issue [1–5]. No one really doubts that field radiation and detection is a causal process (cf, e.g., Refs. [5]), but to identify universal causal quantities measured by a macroscopic detector of arbitrary design remains an open question. Such quantities naturally emerge in response formulation of quantum electrodynamics (QED) [6–10]. In this letter we consider causality properties of these quantities and show that their use eliminates all causality issues from the theory. Referring the reader for details to the cited papers, here we only outline the key points. In Glauber’s photodetection theory [11–13], spectral properties of the detected field are determined by the quantum averageA formal implementation of the concepts of mesoscopic electromagnetic interaction and of the propagating wave in quantum electrodynamics beyond the rotating wave approximation is discussed. Used as a guide, these concepts lead to a natural resolution of a long-standing controversy: causality violations in the Glauber–Kelley–Kleiner photodetection theory. The Glauber–Kelley–Kleiner definition of the time-normal operator ordering must be amended without the rotating wave approximation, which eliminates all causality problems.

Causal signal transmission by quantum fields. IV. The causal Wick theorem

In the previous papers of these series [1–3] we investigated response properties of quantised fields. Analyses in those papers were, so to speak, pure kinematics. Starting from this paper we extend our analyses to dynamics. Here we derive the formal tool on which other dynamical results depend, and which we term the causal Wick theorem. The latter is a functional (Hori’s) [4, 5] form of Wick’s theorem [6–8] in the Schwinger-Perel-Keldysh closed-time-loop formalism [9–11], written in terms of the linear response function [12] characteristic of the field in question. The term causal Wick theorem was introduced in the unpublished paper [13]. In the standard Feynman-Dyson technique of quantum field theory [7], Wick’s theorem supplies dynamical information about free fields, expressed by their contractions , or Feynman propagators (we avoid the term causal Green function, because we associate “causal” with “retarded,” as in causal response and causal space-time evolution). In the closed-time-loop formalism, propagators becomes matrices, but the fact that Wick’s theorem is a supplier of dynamical information contained in propagators does not change. As was shown in Refs. [1, 3] (see also [14, 15]), components of the Perel-Keldysh matrix propagator may be expressed by the linear response function (known in quantum field theory as the retarded Green function [7, 8]) characteristic of the field. As a physical quantity, this function emerges in Kubo’s linear response theory [12] for the field in question, see Refs. [1, 3] and sections IVC and IID of this paper. Linear response of the free electromagnetic field was considered by Schwinger [16]. The only information about the field needed for Wick’s theorem is its response. Thus the formal operation of operator reordering which is the subject of Wick’s theorem is inherently connected to how the field propagates in space-time. One may say that propagation of the field is the actual physical content of Wick’s theorem. The causal Wick theorem makes this evident. An immediate word of caution is in place here. In the literature, the term Wick’s theorem is used for two different facts. Wick’s original theorem expresses time-ordered products of free-field operators as linear combinations of normally-ordered products of the same operators. This theorem holds irrespective of the quantum state of the system and ipso facto may be used for systems out of thermal equilibrium. The other meaning of this term is a Gaussian factorisation of Green functions of free fields in thermal equilibrium (underlying, e.g., the Matsubara diagram techniques [17]). Phase-space methods may be applied to quantum dynamics in both cases, cf. Refs. [14] and [18, 19], respectively. This particular paper is part of the effort directed at extending the out-of-equilibrium methods of Ref. [14] to space-time evolution of arbitrary interacting quantum fields, cf. the introduction to our paper [1]. The question of how far the equilibrium phasespace techniques of Refs. [18, 19] may be generalised is outside the scope of this paper (and very much outside the scope of this series of papers). Consequently, here the term Wick’s theorem is used strictly in the sense of a relation between time-ordered and normally-ordered operator products. Had the need arisen to emphasise the said distinction, we shall talk about Wick’s theorem for operators .

Real-time quantum trajectories for classically allowed dynamics in strong laser fields

Both the physical picture of the dynamics of atoms and molecules in intense infrared fields and its theoretical description use the concept of electron trajectories. Here, we address a key question which arises in this context: Are distinctly quantum features of these trajectories, such as the complex-valued coordinates, physically relevant in the classically allowed region of phase space, and what is their origin? First, we argue that solutions of classical equations of motion can account for quantum effects. To this end, we construct an exact solution to the classical Hamilton–Jacobi equation which accounts for dynamics of the wave packet, and show that this solution is physically correct in the limit . Second, we show that imaginary components of classical trajectories are directly linked to the finite size of the initial wave packet in momentum space. This way, if the electronic wave packet produced by optical tunnelling in strong infrared fields is localised both in coordinate and momentum, its motion after tunnelling ipso facto cannot be described with purely classical trajectories – in contrast to popular models in the literature.