Time-dependent Expectation Values Research Articles

On thermalization in classical scalar field theory

Thermalization of classical fields is investigated in a \phi^4 scalar field theory in 1+1 dimensions, discretized on a lattice. We numerically integrate the classical equations of motion using initial conditions sampled from various nonequilibrium probability distributions. Time-dependent expectation values of observables constructed from the canonical momentum are compared with thermal ones. It is found that a closed system, evolving from one initial condition, thermalizes to high precision in the thermodynamic limit, in a time-averaged sense. For ensembles consisting of many members with the same energy, we find that expectation values become stationary - and equal to the thermal values - in the limit of infinitely many members. Initial ensembles with a nonzero (noncanonical) spread in the energy density or other conserved quantities evolve to noncanonical stationary ensembles. In the case of a narrow spread, asymptotic values of primary observables are only mildly affected. In contrast, fluctuations and connected correlation functions will differ substantially from the canonical values. This raises doubts on the use of a straightforward expansion in terms of 1PI-vertex functions to study thermalization.

Ultrafast Non-Adiabatic Curve Crossing and Energy Transfer in the IBr B0+·Ar van der Waals Complex

Photodissociation of the T-shaped IBr B0+·Ar complex initiated by an ultrafast laser pulse is investigated theoretically by integrating the time-dependent Schrödinger equation for wavepacket motion in two dimensions. The fragmentation dynamics are explored through calculations of time-dependent expectation values, electronic state populations and vibrational state probabilities. The focus of this study is on the competition between electronic predissociation of IBr, induced by non-adiabatic transitions between the bound B0+ and repulsive Y0+ excited states, and vibrational predissociation of Ar from the intact diatomic, mediated by vibrational energy redistribution between intramolecular and van der Waals degrees of freedom. This is facilitated by comparative calculations in which non-adiabatic curve crossing is either included or artificially disabled. It is found that molecular dissociation to form ground-state I + Br at energies below the B0+ asymptotic limit takes place over time scales ranging from 200-500 fs for vibrational levels that are strongly coupled to the Y0+ continuum, to longer than 10 ps for the most metastable levels. Sequential loss of vibrational quanta from IBr B0+ into the van der Waals mode over a 2-10 ps time scale is significantly disturbed by dissociation of the diatomic moiety: instead of time-ordered vibrational cooling via consecutively lower-energy levels, a complex pattern of energy transfer evolves during the lifetime of metastable B0+ vibrational levels until the Ar atom is propelled into the van der Waals continuum. The probabilities of final IBr B0+ vibrational levels are determined, therefore, by the strength of non-adiabatic transfer to the molecular dissociative continuum in addition to the dynamical coupling of intramolecular and van der Waals modes.

Dissipative chaotic quantum maps: Expectation values, correlation functions and the invariant state

I investigate the propagator of the Wigner function for a dissipative chaotic quantum map. I show that a small amount of dissipation reduces the propagator of sufficiently smooth Wigner functions to its classical counterpart, the Frobenius-Perron operator, if \(\). Several consequences arise: the Wigner transform of the invariant density matrix is a smeared out version of the classical strange attractor; time dependent expectation values and correlation functions of observables can be evaluated via hybrid quantum-classical formulae in which the quantum character enters only via the initial Wigner function. If a classical phase-space distribution is chosen for the latter or if the map is iterated sufficiently many times the formulae become entirely classical, and powerful classical trace formulae apply.

Visualizing the collapse and revival of wave packets in the infinite square well using expectation values

We investigate the short-, medium-, and long-term time dependence of wave packets in the infinite square well. In addition to emphasizing the appearance of wave packet revivals, i.e., situations where a spreading wave packet reforms with close to its initial shape and width, we also examine in detail the approach to the collapsed phase where the position-space probability density is almost uniformly spread over the well. We focus on visualizing these phenomena in both position- and momentum-space as well as by following the time-dependent expectation values of and uncertainties in position and momentum. We discuss the time scales for wave packet collapse, using both an autocorrelation function analysis as well as focusing on expectation values, and find two relevant time scales which describe different aspects of the decay phase. In an Appendix, we briefly discuss wave packet revival and collapse in a more general, one-dimensional power-law potential given by V(k)(x)=V0|x/a|k which interpolates between the case of the harmonic oscillator (k=2) and the infinite well (k=∞).

The approach to thermal equilibrium in quantized chaotic systems

We consider many-body quantum systems that exhibit quantum chaos, in the sense that the observables of interest act on energy eigenstates like banded random matrices. We study the time-dependent expectation values of these observables, assuming that the system is in a definite (but arbitrary) pure quantum state. We induce a probability distribution for the expectation values by treating the zero of time as a uniformly distributed random variable. We show explicitly that if an observable has a nonequilibrium expectation value at some particular moment, then it is overwhelmingly likely to move towards equilibrium, both forwards and backwards in time. For deviations from equilibrium that are not much larger than a typical quantum or thermal fluctuation, we find that the time dependence of the move towards equilibrium is given by the Kubo correlation function, in agreement with Onsager's postulate. These results are independent of the details of the system's quantum state.

Dynamic matrix ansatz for integrable reaction-diffusion processes

We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work \cite{Stin95a,Stin95b} we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath with Kawasaki dynamics. We solve the functional equations arising from the algebraic description and show non-perturbatively on the level of all finite-order correlation functions that the coupling to the infinite-temperature heat bath does not change the late-time behaviour of the zero-temperature process. The associated quantum chain is a non-hermitian anisotropic Heisenberg chain related to the seven-vertex model.

Displaced and squeezed number states

From a background of displaced (coherent) and squeezed states, previous (often forgotten) work on displaced and squeezed number states is reviewed. We derive the most general of such states. The time-dependent expectation values, uncertainties, wave-functions, and probability densities are obtained. We comment on the possibility of experimentally observing these states.

Solution of quantum master equations in terms of a non-Hermitian Hamiltonian.

It is known that a quantum master equation that describes dissipative relaxation (such as spontaneous decay) can be separated in a part that describes spontaneous jumps and a jump-free part. The jump-free part of the evolution is determined by a non-Hermitian Hamiltonian H, which acts on state vectors, rather than on density operators. We identify a class of systems for which the complete evolution is entirely determined by the jump-free evolution, which is governed by H. Although this class is restricted, it contains a sizable number of systems of physical significance. Among these are two-, three-, and four-state atoms. The method is used to find explicit expressions of time-dependent expectation values in terms of the complex eigenvalues of H. As a first application we evaluate exactly the photon statistics in the driven \ensuremath{\Lambda} system. As a second example of the method, we treat the configuration of a V system driven by two standing waves. This system constitutes an effective beam splitter.

Quantum chaos of atoms in a resonator driven by an external resonant field.

A system of N two-level atoms in a resonator is considered interacting with a resonant eigenmode field and with an external coherent field, with a frequency slightly different from the frequency of the atomic transition. A model Hamiltonian is constructed for describing the slow quantum dynamics of the system, and a set of closed c-number equations for time-dependent quantum expectation values is derived in the boson and spin coherent states. In the region of parameters for developed chaos in the semiclassical limit (when the radiation field is considered classically) we show that the semiclassical approximation is violated by quantum effects at the time scale ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$\ensuremath{\sim}lnN. We also show that quantum correlations grow exponentially during the time interval 0${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$. An additional averaging over the initial density matrix is considered. The connection of our model with the Dicke model [Phys. Rev. 93, 99 (1956)] is demonstrated, and the possiblity of measuring the quantum chaos effects considered in this paper is discussed.

Highly vibrationally excited HCN/HNC: Eigenvalues, wave functions, and stimulated emission pumping spectra

We report the large scale calculation of accurate vibrational eigenstates of X̃ state HCN/HNC up to ∼37 000 cm−1. The discrete variable representation-distributed Gaussian basis (DVR-DGB) approach is used. In the final major computational step, large dense Hamiltonian matrices (≥11 000 rows or columns) must be diagonalized. The use of new software for out-of-core eigenproblems is described in some detail. Representative wave functions of high lying vibrational eigenstates near the isomerization barrier and at energies up to 39 313 cm−1 are presented. Theoretical stimulated emission pumping (SEP) spectra for several initial (Ã) vibrational states are presented along with calculations of the quantum survival probability for these spectra. The spectra range from being dominated by localized eigenstates to being dominated by delocalized eigenstates, depending on the geometry of each initial state. In addition, time-dependent expectation values of the coordinates of the evolving wave packet are plotted.

Observation of longitudinal three-spin order for measuring dipole-dipole cross correlation

Renewed interest in cross relaxation and nuclear Overhauser effects (I) has been stimulated by the advent of two-dimensional exchange spectroscopy (NOESY) (2, 3). This technique can be used to measure the migration of longitudinal magnetization between different sites in a molecule resulting from cross relaxation. The longitudinal relaxation behavior of systems with many spins is frequently treated using generalized Solomon equations (4) in which the dynamics of the eigenstate populations are expressed solely in terms of (time-dependent) expectation values of single-spin angular momentum operators I&, i.e., (I&) = Tr{ cr(t)Zk=} . It has long been recognized, however, that relaxation phenomena cannot be treated accurately without taking into account the fact that the fluctuations of different dipolar interactions are correlated (5), and Fagerness et al. (6) have demonstrated that such dipole-dipole cross correlation may lead to partial conversion of Zeeman order (I& into longitudinal three-spin order (41&.1,,). Such cross-correlation effects are not apparent from the traditional Solomon equations, and hence it is often assumed, although seldom explicitly stated, that these effects can be neglected without severe consequences regarding the accuracy of the structural data derived from NOE studies. Recently, however, Bull (7) has shown in a theoretical paper that the time dependence of crossand diagonal-peak intensities in NOESY spectra can be significantly altered by cross-correlation effects, which give rise to additional pathways for the migration of Zeeman order, e.g., (L) --, (~uL) + (L). Cross-correlation effects can be best observed by monitoring the growth of longitudinal three-spin order (41&lmz). However, the use of 90” pulses in the normal two-dimensional NOESY experiment precludes this direct observation. Three-spin order can only be observed if two conditions are fulfilled simultaneously: the scalar couplings between at least two of the three relevant spins must be resolved, and the flip angle of the RF pulse used to monitor the state of the system must differ from 90” (8-10). If these conditions are met, longitudinal three-spin order leads to a doubly antiphase contribution to the multiplets: for a doublet of doublets in a system ofthree spin-j nuclei this contribution has the form + I:1 :1 :+ I. This antiphase component can, however, be observed separately from the intense background of Zeeman order (which gives rise to in-phase multiplets) by triple-quantum filtration techniques. Bull (7) and Jaccard et al. (1 I) have recently suggested this approach, which is analo-

Equation of motion method for the time-dependent Newns-Anderson model

The time-dependent Newns-Anderson model including the intra-atomic Coulomb interaction is investigated by means of the Heisenberg equation of motion. It is shown that a hierarchy of equations can be derived for the time-dependent expectation values of charged states for atoms moving near metal surfaces. A truncation procedure is used to reduce these equations to a closed form. A simple case of time-dependence for parameters in the model is examined numerically.

Hydrogen transfer in double minimum potential: Kinetic properties derived from quantum dynamics

The interaction of hydrogen transfer in a double minimum potential with a condensed phase environment is studied. For a symmetric double minimum system, the tunneling motion in the vibrational ground state is retarded efficiently by fluctuation as well as by rearrangement of the lattice consisting of harmonic oscillators. Environmental displacements with large inertia cause dynamic asymmetry by failing to cooperate with the transfer motion and favor a thermally activated process even at low temperatures. To describe such processes, an effective Hamiltonian is derived, which consists of a leading term referring to a one-dimensional transfer motion along an asymmetric potential profile and of a random perturbation term linear in the transfer coordinate. The power spectral density is derived for the perturbation given as a superposition of the time-dependent quantum mechanical expectation values of the vibrational displacements in the environment. A master equation treatment is proposed to describe the kinetic properties and is applied to a model for benzoic acid dimers in the crystalline state. The model reproduces the full temperature dependence of the observed NMR T1 data for (C6H5COOH)2 and (C6H5COOD)2 with plausible parameters and relates the temperature-dependent apparent activation energy to the energy level scheme of the transfer motion.

On the initial conditions of time-dependent mean-field equations of evolution: (II). Zero temperature

We analyze the problem so far untouched of determining the initial mean-field wavefunction in the context of zero-temperature mean-field descriptions of time-dependent expectation values and quantum fluctuations of nuclear observables. The nucleus, at zero temperature, is taken to be in a low-lying excited many-body eigenstate and is approximated by the corresponding RPA wavefunction as a continuous superposition of coherent states (i.e. Slater determinants). A generating function G A ( λ) for time-dependent expectation values and quantum fluctuations is constructed within the formalism of functional integration. By applying the saddle-point method to the functional action of G A ( λ) and then taking its λ-derivatives, we recover the well-known TDHF theory and propose a simple determination of the initial Slater determinant for an appropriate mean-field description of time-dependent expectation values. The analog mean-field description of quadratic-quantum fluctuations proceeds similarly and in addition includes the contribution of the uncorrelated TDHF-RPA phonons coupled to collective excitations of the initial (static) mean-field configuration. When the collective TDHF-RPA excitations are solely taken into account, we obtain an improved version of the Balian-Veneroni dispersion formula by showing how to determine the initial mean-field wavefunction. By first taking the λ-derivatives of G A ( λ) before applying the saddle-point method, the initial mean-field wavefunction is found to be non-linearly coupled to the mean-field dynamics themselves. In return, and in contrast to the first quantization scheme, these both depend non-trivially upon the observable A ̂ being measured so that approximations must be proposed to simplify the resulting mean-field equations.

Expectation values and fluctuations in a time-dependent mean-field approach

Time-dependent expectation values of many-body nuclear observables and their quantum fluctuations are considered within the framework of functional integration. We propose two quantization procedures through saddle-point approximations to a functional integral representation of the real-time generating function. The first method, referred to as Derivation After Variation, leads to the well-known TDHF approximation obtained also in Alhassid, Muller, and Koonin, Phys. Rev. C 23 (1981), 487, with a different functional integration method. The resulting saddle-point approximation of the time-dependent quantum fluctuations is found to include the contribution of the collective phonons of the corresponding time-dependent R.P.A. in addition to the bare contribution of the TDHF mean-field. Direct comparison is made with the time-dependent mean-field approach of Balian and Veneroni, Phys. Lett. B 136 (1984), 301, obtained by application of the variational principle proposed in Balian and Veneroni, Phys. Rev. Lett. 47 (1981), 1353, 1765(E). The second method, referred to as Variation After Derivation, reproduces in a much simpler formulation the results of Troudet and Koonin, Phys. Rev. C 28 (1983), 1465, where the mean-field is found to be dependent upon the observable being measured. We propose this mean-field theory as a possible alternative to the previous mean-field approaches for the evaluation of many-body measurements that are known to be poorly described within the bare TDHF approximation, such as widths of mass distributions of final fragments in heavy-ion collisions. A comparative illustration of these various methods is proposed through the numerical evaluation of the time-dependent expectation values of a one-body observable for a simple shell-model hamiltonian.

Strong damping and low-temperature anomalies for the harmonic oscillator.

We consider Ullersma's model of a damped harmonic oscillator interacting with a heat bath. We show that for large damping the oscillator does not undergo Brownian motion at any temperature. For high temperatures non-Markovian effects show up in initial slips only. For low temperatures we find, as further anomalies, power-law decay of time-dependent expectation values as well as a strong squeezing of the displacement and momentum uncertainties; again these anomalies are most strongly pronounced for strong damping.

Time-dependent variational principle for the expectation value of an observable: Mean-field applications: Roger Balian, Service de Physique Théorique, CEA, Saclay, 91191 Gif sur Yvette, France, and Marcel Vénéroni, Division de Physique Théorique, Institut de Physique Nucléaire, 91406 Orsay, France

The problem of the determination of the initial conditions in the mean-field description of time-dependent expectation values and quantum fluctuations of nuclear observables is considered within the framework of functional integration. We propose two quantization procedures by applying the saddle-point method to a temperature-dependent and time-dependent generating function. In the first method, referred to as derivation after variation, the initial condition of the TDHF equation of evolution is given by the temperature-dependent Hartree-Fock approximation of the initial many-body state of the system. In addition, the saddle-point approximation of the quadratic quantum fluctuations includes the contribution of the corresponding RPA which consists of correlated exchanges of HF-RPA phonons and TDHF-RPA phonons and improves the dispersion formula recently proposed by Balian and Veneroni. In the second method, referred to as variation after derivation, the determination of the mean-field initial condition is self-consistently coupled to the mean-field dynamics which approximates the quantum many-body propagation. In contrast to the previous method, the mean-field initial condition and the mean-field dynamics are found to be dependent upon the observable being measured.

On the initial conditions of time-dependent mean-field equations of evolution: (I) finite temperature

The problem of the determination of the initial conditions in the mean-field description of time-dependent expectation values and quantum fluctuations of nuclear observables is considered within the framework of functional integration. We propose two quantization procedures by applying the saddle-point method to a temperature-dependent and time-dependent generating function. In the first method, referred to as derivation after variation, the initial condition of the TDHF equation of evolution is given by the temperature-dependent Hartree-Fock approximation of the initial many-body state of the system. In addition, the saddle-point approximation of the quadratic quantum fluctuations includes the contribution of the corresponding RPA which consists of correlated exchanges of HF-RPA phonons and TDHF-RPA phonons and improves the dispersion formula recently proposed by Balian and Veneroni. In the second method, referred to as variation after derivation, the determination of the mean-field initial condition is self-consistently coupled to the mean-field dynamics which approximates the quantum many-body propagation. In contrast to the previous method, the mean-field initial condition and the mean-field dynamics are found to be dependent upon the observable being measured.

Incoherently driven Dicke model

The exact steady-state solution for the reduced atomic density operator for a model system of N identical two-level atoms confined to a single site (the Dicke model) and driven by a totally incoherent broad-band (chaotic) field is derived through the equivalent Fokker-Planck equation of the system. Steady-state values of the atomic observables, higher moments and atomic fluctuations are calculated for an arbitrary N and an arbitrary field strength. It is shown that when the chaotic field is a thermal (black-body) field the atomic system is driven into a steady relative occupation number that obeys the Boltzmann distribution law only for N=1. (The same result for N>1 atoms can be reached only if the cooperative interactions between the atoms are ignored.) In the thermodynamic limit N to infinity there is no critical behaviour in contrast to the resonant coherently driven case. It is also shown that semiclassical (direct) factorisation of the equations of motion does not lead to the correct results for the N to infinity limit of the exact quantum case. Within the exact theory numerical results for finite N (<or=100) are presented for the time-dependent expectation values of the atomic operators, the intensity and the intensity autocorrelation function. Superradiant pulses are emitted by N initially inverted atoms but as the incident intensity is increased this superradiant behaviour is obscured. The steady-state fluorescence spectrum is a single Lorentzian of width proportional to N in the weak-field case. In the strong-field case the spectrum is insensitive to the number of atoms and the result is similar to the one-atom situation. Photon bunching and antibunching effects are investigated for the weak- and strong-field limits. When the chaotic driving field is simply a single-mode thermal field the approach to atomic saturation is delayed in comparison with the case of the broad-band chaotic field.