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

Manipulation of semiclassical photon states

Michael VanValkenburgh, Abstract Gabriel F. Calvo

Published Jan 1, 2009• 2 citations
OpenAlex

Abstract

Abstract. Gabriel F. Calvo and Antonio Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite-Gaussian modes HT = {|0, 0〉, |1,0〉, |0, 1〉}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all selfadjoint extensions. And secondly we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier Integral Operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite-Gaussian modes, since near the origin the Hamilton trajectories are periodic, tracing out the bounded components of elliptic curves. 1.

Related papers

Manipulation of semiclassical photon states

Michael VanValkenburgh

Calvo and Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite–Gaussian modes HT={|0,0⟩,|1,0⟩,|0,1⟩}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And second, we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite–Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.

arXiv:0810.0786•Journal of Mathematical Physics•Feb 1, 2009•1 citation

Manipulation of Semiclassical Laguerre-Gaussian Modes: a Model Case

Michael VanValkenburgh

We continue the study, from a semiclassical viewpoint, of Calvo and Picon's operators, as manipulating photon states in quantum communication. In a previous paper, we defined a one-dimensional model operator and studied it analytically before moving on to Calvo and Picon's full two-dimensional operators. In the present paper, we show how the one-dimensional operator may also be useful as an experimental model, since it allows manipulations of two-dimensional Laguerre-Gaussian modes; the intensity distributions (in physical space) of the Laguerre-Gaussian modes then approximately flow along the elliptic curves studied earlier. Since the Wigner transform is fundamental in the study of Laguerre-Gaussian modes, we give a slightly expanded and improved treatment of the semiclassical Wigner transform, which was only briefly mentioned in the previous paper.

arXiv:0902.1976•Feb 11, 2009•0 citations

On the derivation of the semiclassical approximation to the quantum propagator

S. Fischer, Andreas Buchleitner

In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these difference equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.

Journal of Mathematical Physics•Jul 1, 2015•2 citations

Semiclassical approximations in wave mechanics

Michael Berry, Keith Mount

We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.

Reports on Progress in Physics•Jan 1, 1972•1,424 citations

Moyal quantum mechanics: the semiclassical Heisenberg dynamics

T. A. Osborn, Molzahn, F H

The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.

•Jan 1, 1994•46 citations

Semiclassical structure of Hamiltonians

Mario Feingold, David M. Leitner, Oreste Piro

The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A${^}_{1}$(q^,p^),A${^}_{2}$ (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A${^}_{2}$ takes the form of a banded matrix in the ordered eigenrepresentation of A${^}_{1}$. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=${H}_{0}$+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.

Physical Review A•Jun 1, 1989•52 citations
Abstract. Gabriel F. Calvo and Antonio Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite-Gaussian modes HT = {|0, 0〉, |1,0〉, |0, 1〉}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all selfadjoint extensions. And secondly we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier Integral Operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite-Gaussian modes, since near the origin the Hamilton trajectories are periodic, tracing out the bounded components of elliptic curves. 1.
Calvo and Picón defined a class of operators, for use in quantum communication, that allows arbitrary manipulations of the three lowest two-dimensional Hermite–Gaussian modes HT={|0,0⟩,|1,0⟩,|0,1⟩}. Our paper continues the study of those operators, and our results fall into two categories. For one, we show that the generators of the operators have infinite deficiency indices, and we explicitly describe all self-adjoint realizations. And second, we investigate semiclassical approximations of the propagators. The basic method is to start from a semiclassical Fourier integral operator ansatz and then construct approximate solutions of the corresponding evolution equations. In doing so, we give a complete description of the Hamilton flow, which in most cases is given by elliptic functions. We find that the semiclassical approximation behaves well when acting on sufficiently localized initial conditions, for example, finite sums of semiclassical Hermite–Gaussian modes, since near the origin the Hamilton trajectories trace out the bounded components of elliptic curves.
We continue the study, from a semiclassical viewpoint, of Calvo and Picon's operators, as manipulating photon states in quantum communication. In a previous paper, we defined a one-dimensional model operator and studied it analytically before moving on to Calvo and Picon's full two-dimensional operators. In the present paper, we show how the one-dimensional operator may also be useful as an experimental model, since it allows manipulations of two-dimensional Laguerre-Gaussian modes; the intensity distributions (in physical space) of the Laguerre-Gaussian modes then approximately flow along the elliptic curves studied earlier. Since the Wigner transform is fundamental in the study of Laguerre-Gaussian modes, we give a slightly expanded and improved treatment of the semiclassical Wigner transform, which was only briefly mentioned in the previous paper.
In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these difference equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
The structure of few-body Hamiltonian matrices is studied in the semiclassical regime. Given (A${^}_{1}$(q^,p^),A${^}_{2}$ (q^,p^)), a pair of operators, it can be shown that, under quite general conditions, A${^}_{2}$ takes the form of a banded matrix in the ordered eigenrepresentation of A${^}_{1}$. Moreover, the bandwidth depends only on \ensuremath{\Elzxh} and certain generalized microcanonical averages. In particular, if H=H0{H}_{0}H0​+\ensuremath{\epsilon}V, this implies that the perturbed Hamiltonian is banded in the appropriately ordered unperturbed basis.