Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Progress of Theoretical and Experimental Physics

Approximate formulas for expectation values using coherent states

Alexandre Lira Foggiatto, R. M. Angelo, A. D. Ribeiro

Published Aug 29, 2017• 3 citations
Publisher

Abstract

For time-independent Hamiltonians, the dynamics of quantum expectation values of observables in coherent states |$\bar A_T$| can be easily represented as an integral formula involving forward and backward propagators |$K_\pm$|⁠. In the semiclassical regime, an approximate formula |$\bar{A}_T^{\rm sc}$| can be constructed via the replacement of |$K_\pm$| by their semiclassical versions, followed by a consistent integration procedure. Alternatively, one can keep the original propagators and rewrite the integral formula for |$\bar A_T$| as a truncated series expansion, thus introducing a new approximate formula |$\bar A_T^{\rm se}$|⁠. Yet a third approximation |$\bar A_T^{\rm cl}$| can be derived by use of a classical statistical approach based on the Liouville equation and Gaussian probability distributions. In the present paper, we develop these three approximate formulas for expectation values, apply them to simple systems, and evaluate their accuracy.

DOI
10.1093/ptep/ptx129

Related papers

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

Shin-ichi Koda, Kazuo Takatsuka

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.

Physical Review A•Mar 21, 2011•6 citations

Semiclassical evaluation of expectation values

K. M. Mittal, O. Giraud, Denis Ullmo

Semiclassical mechanics allows for a description of quantum systems which preserves their phase information, and thus interference effects, while using only the system's classical dynamics as an input. In particular one of the strengths of a semiclassical description is to present a coherent picture which (to negligible higher-order ℏ corrections) is independent of the particular canonical coordinates used. However, this coherence relies heavily on the use of the stationary phase approximation. It turns out, however, that in some important cases, a brutal application of stationary phase approximation washes out all interference, and thus quantum, effects. In this paper, we address this issue in detail in one of its simplest instantiations, namely the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. The efficiency of our approach is based on the fact that we treat analytically all the integrals that can be performed within the stationary phase approximation, implying that the remaining integrals are simple integrals, in the sense that the integrand has no significant variations on the quantum scale (and thus they are very easy to perform numerically). This to be contrasted with other approaches such as the ones based on initial value representation, popular in chemical and molecular physics, which avoid a root search for the classical dynamics, but at the cost of performing numerically integrals whose evaluation requires a sampling on the quantum scale, and which are therefore not well designed to address the deep semiclassical regime. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.

arXiv:1911.10226•Physical review. E•Oct 12, 2020•2 citations

Semiclassical approximations in phase space with coherent states

Michel Baranger, Marcus A. M. de Aguiar, F Keck +2 more

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

arXiv:quant-ph/0105153•Journal of Physics A Mathematical and General•Sep 3, 2001•176 citations

Semiclassical coherent-states propagator

Alejandro M. F. Rivas

In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories; it only involves real ones. For that purpose, we used the symplectically invariant semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After that, for the transformation to CS representation SPA is avoided; instead a quadratic expansion of the complex exponent is used. This procedure also allows us to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point without the need of any root search typical of van Vleck--Gutzwiller-based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a realistic chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.

arXiv:1207.0012•Physical Review A•Jul 8, 2013•2 citations

Semiclassical coherent-state propagator via path integrals with intermediate states of variable width

Fernando Parisio, Marcus A. M. de Aguiar

We derive a semiclassical approximation for the coherent state propagator $〈{z}^{\ensuremath{''}}|{e}^{\ensuremath{-}iHt/\ensuremath{\Elzxh}}|{z}^{\ensuremath{'}}〉$ using a path integral formulation in which the intermediate coherent states can have arbitrary widths. Our semiclassical formula involves complex trajectories of the smoothed Hamiltonian $\mathcal{H}(q,p,b)=〈z|\ifmmode \hat{H}\else \^{H}\fi{}|z〉$ where b, the width of the coherent state $|z〉,$ is a free function that can be chosen conveniently. The generality of this formalism enables us to derive a semiclassical approximation which contains, as particular cases, other similar approximations known in the literature, providing a natural link between them. We present numerical results showing that the semiclassical propagation can be very sensitive to the choice of b and we suggest an energy dependent value ${b=b}_{E}$ that results in considerable improvement over other choices. This value for the width will be generally different from the widths ${\ensuremath{\sigma}}^{\ensuremath{'}}$ or ${\ensuremath{\sigma}}^{\ensuremath{''}}$ of the initial and final states $|{z}^{\ensuremath{'}}〉$ and $|{z}^{\ensuremath{''}}〉.$

Physical Review A•Dec 30, 2003•10 citations

Ordered quantization and the Ehrenfest time scale

R. M. Angelo, L. Sanz, K. Furuya

We propose a prescription to quantize classical monomials in terms of symmetric and ordered expansions of noncommuting operators of a bosonic theory. As a direct application of such quantization rules, we quantize a classically time evolved function O(q,p,t), and calculate its expectation value in coherent states. The result can be expressed in terms of the application of a classical operator that performs a Gaussian smoothing of the original function O evaluated at the center of the coherent state. This scheme produces a natural semiclassical expansion for the quantum expectation values at a short time scale. Moreover, since the classical Liouville evolution of a Gaussian probability density gives the same form for the classical statistical mean value, we can calculate the first-order correction in h entirely from the associated classical time evolved function. This allows us to write a general expression for the Ehrenfest time in terms of the departure of the centroid of the quantum distribution from the classical trajectory, provided we start with an initially coherent state for each subsystem. In order to illustrate this approach, we have calculated analytically the Ehrenfest time of a model with N-coupled nonlinear oscillators with nonlinearity of even order.

arXiv:quant-ph/0302020•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 11, 2003•12 citations
For time-independent Hamiltonians, the dynamics of quantum expectation values of observables in coherent states |AˉT\bar A_TAˉT​| can be easily represented as an integral formula involving forward and backward propagators |K±K_\pmK±​|⁠. In the semiclassical regime, an approximate formula |AˉTsc\bar{A}_T^{\rm sc}AˉTsc​| can be constructed via the replacement of |K±K_\pmK±​| by their semiclassical versions, followed by a consistent integration procedure. Alternatively, one can keep the original propagators and rewrite the integral formula for |AˉT\bar A_TAˉT​| as a truncated series expansion, thus introducing a new approximate formula |AˉTse\bar A_T^{\rm se}AˉTse​|⁠. Yet a third approximation |AˉTcl\bar A_T^{\rm cl}AˉTcl​| can be derived by use of a classical statistical approach based on the Liouville equation and Gaussian probability distributions. In the present paper, we develop these three approximate formulas for expectation values, apply them to simple systems, and evaluate their accuracy.
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.
Semiclassical mechanics allows for a description of quantum systems which preserves their phase information, and thus interference effects, while using only the system's classical dynamics as an input. In particular one of the strengths of a semiclassical description is to present a coherent picture which (to negligible higher-order ℏ corrections) is independent of the particular canonical coordinates used. However, this coherence relies heavily on the use of the stationary phase approximation. It turns out, however, that in some important cases, a brutal application of stationary phase approximation washes out all interference, and thus quantum, effects. In this paper, we address this issue in detail in one of its simplest instantiations, namely the evaluation of the time evolution of the expectation value of an operator. We explain why it is necessary to include contributions which are not in the neighborhood of stationary points and provide new semiclassical expressions for the evolution of the expectation values. The efficiency of our approach is based on the fact that we treat analytically all the integrals that can be performed within the stationary phase approximation, implying that the remaining integrals are simple integrals, in the sense that the integrand has no significant variations on the quantum scale (and thus they are very easy to perform numerically). This to be contrasted with other approaches such as the ones based on initial value representation, popular in chemical and molecular physics, which avoid a root search for the classical dynamics, but at the cost of performing numerically integrals whose evaluation requires a sampling on the quantum scale, and which are therefore not well designed to address the deep semiclassical regime. Along the way, we get a deeper understanding of the origin of these interference effects and an intuitive geometric picture associated with them.
We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.
In this work, we derived a semiclassical approximation for the matrix elements of a quantum propagator in coherent states (CS) basis that avoids complex trajectories; it only involves real ones. For that purpose, we used the symplectically invariant semiclassical Weyl propagator obtained by performing a stationary phase approximation (SPA) for the path integral in the Weyl representation. After that, for the transformation to CS representation SPA is avoided; instead a quadratic expansion of the complex exponent is used. This procedure also allows us to express the semiclassical CS propagator uniquely in terms of the classical evolution of the initial point without the need of any root search typical of van Vleck--Gutzwiller-based propagators. For the case of chaotic Hamiltonian systems, the explicit time dependence of the CS propagator has been obtained. The comparison with a realistic chaotic system that derives from a quadratic Hamiltonian, the cat map, reveals that the expression here derived is exact up to quadratic Hamiltonian systems.
We derive a semiclassical approximation for the coherent state propagator $〈{z}^{\ensuremath{''}}|{e}^{\ensuremath{-}iHt/\ensuremath{\Elzxh}}|{z}^{\ensuremath{'}}〉$ using a path integral formulation in which the intermediate coherent states can have arbitrary widths. Our semiclassical formula involves complex trajectories of the smoothed Hamiltonian $\mathcal{H}(q,p,b)=〈z|\ifmmode \hat{H}\else \^{H}\fi{}|z〉$ where b, the width of the coherent state ∣z〉,|z〉,∣z〉, is a free function that can be chosen conveniently. The generality of this formalism enables us to derive a semiclassical approximation which contains, as particular cases, other similar approximations known in the literature, providing a natural link between them. We present numerical results showing that the semiclassical propagation can be very sensitive to the choice of b and we suggest an energy dependent value b=bE{b=b}_{E}b=bE​ that results in considerable improvement over other choices. This value for the width will be generally different from the widths ${\ensuremath{\sigma}}^{\ensuremath{'}}$ or ${\ensuremath{\sigma}}^{\ensuremath{''}}$ of the initial and final states $|{z}^{\ensuremath{'}}〉$ and $|{z}^{\ensuremath{''}}〉.$
We propose a prescription to quantize classical monomials in terms of symmetric and ordered expansions of noncommuting operators of a bosonic theory. As a direct application of such quantization rules, we quantize a classically time evolved function O(q,p,t), and calculate its expectation value in coherent states. The result can be expressed in terms of the application of a classical operator that performs a Gaussian smoothing of the original function O evaluated at the center of the coherent state. This scheme produces a natural semiclassical expansion for the quantum expectation values at a short time scale. Moreover, since the classical Liouville evolution of a Gaussian probability density gives the same form for the classical statistical mean value, we can calculate the first-order correction in h entirely from the associated classical time evolved function. This allows us to write a general expression for the Ehrenfest time in terms of the departure of the centroid of the quantum distribution from the classical trajectory, provided we start with an initially coherent state for each subsystem. In order to illustrate this approach, we have calculated analytically the Ehrenfest time of a model with N-coupled nonlinear oscillators with nonlinearity of even order.