Coherent State Expansion Research Articles

Phase space representation of sound field excited by a noise source in underwater acoustic waveguide.

The analysis of the field excited in a waveguide by a point noise source is performed using the phase space representation of this field given by the distribution of its amplitude in the depth-angle-time space. The transition from the traditional description of the field amplitude as a function of depth and time to phase space representation is performed using the coherent state expansion developed in quantum mechanics. In this paper, the correlation function of noise signals arriving at different points of the phase plane depth-angle is investigated. Numerical simulation data show that measurements of signal correlations in phase space, performed with the help of a receiving vertical antenna, can be used as input data in solving the problem of source localization and reconstruction of unknown parameters of the sound speed profile. It is shown that in phase space there is an analog of the classical interference pattern observed in the distribution of sound intensity in the distance-frequency plane. The slopes of striations in this interference pattern, as in the conventional one, are given by the Chuprov waveguide invariant.

Numerical variational studies of quantum phase transitions in the sub-Ohmic spin-boson model with multiple polaron ansatz

With extensive variational calculations, dissipative quantum phase transitions in the sub-Ohmic spin-boson model are numerically studied in a dense limit of environmental modes. By employing a generalized trial wave function composed of coherent-state expansions, transition points and critical exponents are accurately determined for various spectral exponents, demonstrating excellent agreement with those obtained by other sophisticated numerical techniques. Besides, the quantum-to-classical correspondence is fully confirmed over the entire sub-Ohmic range, compared with theoretical predictions of the long-range Ising model. Mean-field and non-mean-field critical behaviors are found in the deep and shallow sub-Ohmic regimes, respectively, and distinct physical mechanisms of them are uncovered.

Admissible Vectors and Hilbert Algebras

Admissible vectors for unitary representations of locally compact groups are the basis for group frame and covariant coherent state expansions. Main tools in the study of admissible vectors have been Plancherel and central integral decompositions, of applicability only under certain separability and semifiniteness restrictions. In this work, we present a study of admissible vectors in terms of convolution Hilbert algebras valid for arbitrary unitary representations of general locally compact groups.

Generalized coherent-squeezed-state expansion for the super-radiant phase transition

We develop a systematic variational coherent-squeezed-state expansion for the ground state of the quantum Rabi model, which includes an additional squeezing effect with comparisons to previous coherent-state approach. For finite large ratio between the atomic and field frequency, the essential feature of the ground-state wave function in the super-radiant phase appears, which has a structure of two delocalized wake packets. The single-peaked wave function with one coherent-squeezed state works well even around the critical regime, exhibiting the advantage over the coherent-state method. As the coupling increases to form strong correlations physics in the vicinity of phase transition, we develop an improved wave function with a structure of two Gaussian wave packets, which is a linear superposition of two coherent-squeezed state. The ground-state energy and the average photon number agree well with numerical ones even in the strong-correlated regimes, exhibiting a substantial improvement over the coherent-state expansion. The advantage of the coherent-squeezed-state expansion lies in the inclusion of the second coherent-squeezed state and the additional squeezed deformation of the wave function, providing a useful tool for multi-modes spin-boson coupling systems of greater complexity.

Matched Field Processing in Phase Space

The traditional method of matched field processing is based on comparing the complex amplitudes of the measured and calculated sound fields in an underwater waveguide. Because of the high sensitivity of the wave field to variations in environmental parameters, the use of this approach requires accurate knowledge of the ocean-acoustic environment. In this paper, it is shown that under conditions of uncertain environment, instead of comparing the depth dependencies of complex field amplitudes, it is advisable to compare their field amplitude distributions in the phase plane “grazing angle-depth.” Such distributions are calculated using the coherent state expansion borrowed from quantum mechanics. Due to the absence of multipath in the phase space, the amplitudes of the coherent states are less sensitive to variations in the environmental parameters than the total wave field. This makes it possible to construct the similarity coefficients of measured and calculated fields that almost “do not notice” the differences of the compared fields caused by weak sound-speed variations, and “react” only to differences caused by strong changes in the sound-speed field and/or source position.

On spatial structure of the wave field in a vertical section of a deep water acoustic waveguide

The spatial structure of the wave field excited by a tonal point source in an underwater acoustic waveguide with random sound speed fluctuations is analyzed. This is done using an approach in which the field is approximated by a sum of components formed by contributions of narrow beams of rays. These components fluctuate less than the total field and, therefore, they are called stable. Each ot them can be isolated from the total field recorded by a long vertical antenna, using the coherent state expansion developed in quantum mechanics. An approximate analytical expression is obtained for the correlation matrix of the superposition of several stable components. The applicability of this equation is confirmed by the results of Monte Carlo simulation. The simulation has confirmed the theoretical prediction that nearly the same stable components can correspond to the sound speed fluctuations with completely different correlation functions.

Frequency-renormalized multipolaron expansion for the quantum Rabi model

We present a frequency-renormalized multipolaron expansion method to explore the ground state of quantum Rabi model (QRM). The main idea is to take polaron as starting point to expand the ground state of QRM. The polarons are deformed and displaced oscillator states with variationally determined frequency-renormalization and displacement parameters. This method is an extension of the previously proposed polaron concept and the coherent state expansion used in the literature, which shows high efficiency in describing the physics of the QRM. The proposed method is expected to be useful for solving other more complicated light-matter interaction models.

Anatomy of quantum critical wave functions in dissipative impurity problems

Quantum phase transitions reflect singular changes taking place in a many-body ground state, however, computing and analyzing large-scale critical wave functions constitutes a formidable challenge. New physical insights into the sub-Ohmic spin-boson model are provided by the coherent state expansion (CSE), which represents the wave function by a linear combination of classically displaced configurations. We find that the distribution of low-energy displacements displays an emergent symmetry in the absence of spontaneous symmetry breaking, while experiencing strong fluctuations of the order parameter near the quantum critical point. Quantum criticality provides two strong fingerprints in critical low-energy modes: an algebraic decay of the average displacement and a constant universal average squeezing amplitude. These observations, confirmed by extensive variational matrix product states (VMPS) simulations and field theory arguments, offer precious clues into the microscopics of critical many-body states in quantum impurity models.

Universal spatial correlations in the anisotropic Kondo screening cloud: Analytical insights and numerically exact results from a coherent state expansion

We analyze the spatial correlations in the spin density of an electron gas in the vicinity of a Kondo impurity. Our analysis extends to the spin-anisotropic regime, which was not investigated in the literature. We use an original and numerically exact method, based on a systematic coherent-state expansion of the ground state of the underlying spin-boson Hamiltonian, which we apply to the computation of observables that are specific to the fermionic Kondo model. We also present an important technical improvement to the method, that obviates the need to discretize modes of the Fermi sea, and allows one to tackle the problem in the thermodynamic limit. One can thus obtain excellent spatial resolution over arbitrary length scales, for a relatively low computational cost, a feature that gives the method an advantage over popular techniques such as NRG and DMRG. We find that the anisotropic Kondo model shows rich universal scaling behavior in the spatial structure of the entanglement cloud. First, SU(2) spin-symmetry is dynamically restored in a finite domain in parameter space in vicinity of the isotropic line, as expected from poor man's scaling. We are also able to obtain in closed analytical form a set of different, yet universal, scaling curves for strong exchange asymmetry, which are parametrized by the longitudinal exchange coupling. Deep inside the cloud, i.e. for distances smaller than the Kondo length, the correlation between the electron spin density and the impurity spin oscillates between ferromagnetic and antiferromagnetic values at the scale of the Fermi wavelength, an effect that is drastically enhanced at strongly anisotropic couplings. Our results also provide further numerical checks and alternative analytical approximations for the recently computed Kondo overlaps [PRL 114, 080601 (2015)].

Generalized multipolaron expansion for the spin-boson model: Environmental entanglement and the biased two-state system

We develop a systematic variational coherent-state expansion for the many-body ground state of the spin-boson model, in which a quantum two-level system is coupled to a continuum of harmonic oscillators. Energetic constraints at the heart of this technique are rationalized in terms of polarons (displacements of the bath states in agreement with classical expectations) and antipolarons (counterdisplacements due to quantum tunneling effects). We present a comprehensive study of the ground-state two-level system population and coherence as a function of tunneling amplitude, dissipation strength, and bias (akin to asymmetry of the double-well potential defining the two-state system). The entanglement among the different environmental modes is investigated by looking at spectroscopic signatures of the bipartite entanglement entropy between a given environmental mode and all the other modes. We observe a drastic change in behavior of this entropy for increasing dissipation, indicative of the entangled nature of the environmental states. In addition, the entropy spreads over a large energy range at strong dissipation, a testimony to the wide entanglement window characterizing the underlying Kondo state. Finally, comparisons to accurate numerical renormalization-group calculations and to the exact Bethe ansatz solution of the model demonstrate the rapid convergence of our variationally optimized multipolaron expansion, suggesting that it should also be a useful tool for dissipative models of greater complexity, as relevant for numerous systems of interest in quantum physics and chemistry.

Stabilizing spin coherence through environmental entanglement in strongly dissipative quantum systems

The authors propose an entangled variational coherent-state expansion for the ground state of the Ohmic spin-boson model, and observe a surprisingly dramatic influence of environmental entanglement on qubit coherence.

Properties of ground state and anomalous quantum fluctuations in one-dimensional polaron—soliton systems—the effects of electron-two-phonon interaction and non-adiabatic quantum correlations

Based on the Holstein model Hamiltonian of one-dimensional molecular crystals, by making use of the expansion approach of the correlated squeezed-coherent states of phonon instead of the two-phonon coherent state expansion scheme, the properties of the ground state and the anomalous quantum fluctuations are investigated in a strongly coupled electron—phonon system with special consideration of the electron-two-phonon interaction. The effective renormalization (α̃i) of the displacement of the squeezed phonons with the effect of the squeezed-coherent states of phonon and both the electron-displaced phonon and the polaron-squeezed phonon correlations have been combined to obtain the anomalous quantum fluctuations for the corrections of the coherent state. Due to these non-adiabatic correlations, the effective displacement parameter α̃i is larger than the ordinary parameter αi(0). In comparison with the electron-one-phonon interaction (g) corrected as α̃ig, we have found the electron-two-phonon interaction (g1) corrected as α̃i2g1 is enhanced significantly. For this reason, the ground state energy (E0(2)) contributed by the electron-two-phonon interaction is more negative than the single-phonon case (E0(1)) and the soliton solution is more stable. At the same time, the effects of the electron-two-phonon interaction greatly increase the polaron energy and the quantum fluctuations. Furthermore, in a deeper level, we have considered the effect of the polaron-squeezed phonon correlation (f-correlation). Since this correlation parameter f > 1, this effect will strengthen the electron-one and two-phonon interactions by fα̃ig and f2α̃i2g1, respectively. The final results show that the ground state energy and the polaron energy will appear more negative further and the quantum fluctuations will gain further improvement.

Generalization of the coherent-state path integrals and systematic derivation of semiclassical propagators

The coherent path integral is generalized such that the identity operator represented in a complete (actually overcomplete) set of the coherent states with the ``time-variable'' exponents are inserted between two consecutive short-time propagators. Since such a complete set of any given exponent can constitute the identity operator, the exponent may be varied from time to time without loss of generality as long as it is set common to all the Gaussians. However, a finite truncation of the coherent state expansion should result in different values of the propagator depending on the choice of the exponents. Furthermore, approximation methodology to treat with the exact propagator can also depend on this choice, and thereby many different semiclassical propagators may emerge from these combinations. Indeed, we show that the well-known semiclassical propagators such as those of Van Vleck, Herman-Kluk, Heller's thawed Gaussian, and many others can be derived in a systematic manner, which enables one to comprehend these semiclassical propagators from a unified point of view. We are particularly interested in our generalized form of the Herman-Kluk propagator, since the relative accuracy of this propagator has been well established by Kay, and since, nevertheless, its derivation was not necessarily clear. Thus our generalized Herman-Kluk propagator replaces the classical Hamiltonian with a Gaussian averaged quantum Hamiltonian, generating non-Newtonian trajectories. We perform a numerical test to assess the quality of such a family of generalized Herman-Kluk propagators and find that the original Herman-Kluk gives an accurate result. The reason why this has come about is also discussed.

Dynamics of Jaynes–Cummings Model in the Absence ofRotating-Wave Approximation

The Jaynes–Cummings model (JCM) is studied in the absence of therotating-wave approximation (RWA) by a coherent-state expansion technique.In comparison with the previous paper in which the coherent-state expansionwas performed only to the third order, we carry out in this paper a completeexpansion to demonstrate exactly the dynamics of the JCM without the RWA.Our study gives a systematic method to solve the non-RWA problem, whichwould be useful in various physical systems, e.g., in a system with anultracold trapped ion experiencing the running waves of lasers.

Matching-pursuit/split-operator-Fourier-transform simulations of excited-state nonadiabatic quantum dynamics in pyrazine

A simple approach for numerically exact simulations of nonadiabatic quantum dynamics in multidimensional systems is introduced and applied to the description of the photoabsorption spectroscopy of pyrazine. The propagation scheme generalizes the recently developed matching-pursuit/split-operator-Fourier-transform (MP/SOFT) method [Y. Wu and V. S. Batista, J. Chem. Phys. 121, 1676 (2004)] to simulations of nonadiabatic quantum dynamics. The time-evolution operator is applied, as defined by the Trotter expansion to second order accuracy, in dynamically adaptive coherent-state expansions. These representations are obtained by combining the matching-pursuit algorithm with a gradient-based optimization method. The accuracy and efficiency of the resulting computational approach are demonstrated in calculations of time-dependent survival amplitudes and photoabsorption cross sections, using a model Hamiltonian that allows for direct comparisons with benchmark calculations. Simulations in full-dimensional potential energy surfaces involve the propagation of a 24-dimensional wave packet to describe the S(1)S(2) interconversion of pyrazine after S(0)-->S(2) photoexcitation. The reported results show that the generalized MP/SOFT method is a practical and accurate approach to model nonadiabatic reaction dynamics in polyatomic systems.

Matching-pursuit split-operator Fourier-transform simulations of excited-state intramolecular proton transfer in 2-(2′-hydroxyphenyl)-oxazole

The excited-state intramolecular proton-transfer dynamics associated with the keto-enolic tautomerization reaction in 2-(2(')-hydroxyphenyl)-oxazole is simulated according to a numerically exact quantum-dynamics propagation method and a full-dimensional excited-state potential energy surface, based on an ab initio reaction surface Hamiltonian. The reported simulations involve the propagation of 35-dimensional wave packets according to the recently developed matching-pursuit/split-operator-Fourier-transform (MP/SOFT) method by Wu and Batista. The underlying propagation scheme recursively applies the time-evolution operator as defined by the Trotter expansion to second order accuracy in dynamically adaptive coherent-state expansions. Computations of time-dependent survival amplitudes, photoabsorption cross sections, and time-dependent reactant(product) populations are compared to the corresponding calculations based on semiclassical approaches, including the Herman-Kluk semiclassical initial value representation method. The reported results demonstrate the capabilities of the MP/SOFT method as a valuble computational tool to study ultrafast reaction dynamics in polyatomic systems as well as to validate semiclassical simulations of complex (nonintegrable) quantum dynamics in multidimensional model systems.

Nonlinear canonical transformations in classical and quantum mechanics

p-mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how nonlinear canonical transformations affect p-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of p-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.

Canonical Bose gas simulations with stochastic gauges.

A technique to simulate the grand canonical ensembles of interacting Bose gases is presented. Results are generated for many temperatures by averaging over energy-weighted stochastic paths, each corresponding to a solution of coupled Gross-Pitaevskii equations with phase noise. The stochastic gauge method used relies on an off-diagonal coherent-state expansion, thus taking into account all quantum correlations. As an example, the second-order spatial correlation function and momentum distribution for an interacting 1D Bose gas are calculated.

Analytical Solutions of the Two-Site Holstein Model by the Coherent State Expansion

The polaron problem presents a highly nonlinear dynamical system in which the charge and lattice deformations are intricately coupled together. The coherent state expansion method was applied to the two-site Holstein model. Analytical solutions are found for the first, the second and the third order approximations. Comparison is made among the exact numerical results, based upon solving a recursive relation for the expansion coefficients, and the analytical solutions up to the third order approximation for various e-ph coupling strengths and electron hopping parameters.

Exact solutions for the two-site Holstein model

The two-site Holstein model is studied by the analytical method of the coherent-state expansion. We obtain the recursive relationship among the expansion coefficients, which leads to the exact solutions of the eigenstate properties. The exact ground-state results in the whole coupling strength region and from antiadiabatic to adiabatic limits are in perfect agreement with that available by the exact numerical diagonalization technique. Comparison with the best perturbation results shows the success and shortcoming of the perturbative method. The coherent-state expansion method is a promising approach in studying electron-phonon $(e\ensuremath{-}\mathrm{ph})$ coupling systems.