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

arXiv:2605.15420

Coherent States of Non-Null Torus Knots

Gabriel Canadas da Silva, Ion Vasile Vancea

quant-phhep-thmath-phSubmitted May 14, 2026• Announced May 18, 2026
arXivHTMLPDF

Abstract

We construct coherent states for the quantized electromagnetic field that correspond to the classical non-null torus knot solutions of Maxwell's equations in vacuum. We derive the displacement operators from the general relation between classical fields and coherent state amplitudes and verify the defining properties of coherent states through direct computation. We determine the observables of the model: field expectation values, energy density, Poynting vector, helicity, photon number, quadrature uncertainties, and correlation functions, and calculate their expectation values in the knotted coherent states in terms of the integer parameters $(n,m,l,s)$ of the classical solutions. As an example, we particularize the construction in the case of the Hopfion coherent state. These results establish the quantum-classical correspondence for this type of vacuum topological electromagnetic systems.

Related papers

Coherent and squeezed states of the radiation field

Marlan O. Scully, Muhammad Suhail Zubairy

Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.

Cambridge University Press eBooks•Sep 4, 1997•3 citations

Topological quantum hodographs

Nikolay Rosanov, Sergey Fedorov, Mikhail V. Arkhipov

In quantum mechanics, the wave function encodes all information about a particle through its probability density and phase. While stationary states are characterized by conserved quantum numbers such as energy and, in central potentials, angular momentum, non-stationary superpositions - particularly in anisotropic or time-dependent fields - generally lack equally universal descriptors of their spatiotemporal dynamics. Here we introduce quantum hodographs: the trajectories traced in time by the expectation value of observable vector quantities. For a free electron in a superposition of three plane waves, all hodographs of the probability current lie on a universal cubic surface with conical singularities. Rational frequency (energy) difference ratios produce non-contractible loops with well-defined winding numbers. In anisotropic harmonic oscillators the Ehrenfest trajectories - hodographs of the expectation value of particle radius-vector - form three-dimensional Lissajous knots, echoing the classical Thomson vortex-atom model. Externally driven quantum systems allow controllable initiation of knotted hodographs. We propose an optical modulation spectroscopy scheme for reconstructing these topological features in trapped ions and single-electron systems. The topological indices of the loops and knots are robust to parameter variations, offering a new tool for characterizing complex quantum dynamics.

arXiv:2606.08528•Jun 7, 2026•0 citations

Interpreting quantum states of electromagnetic field in time-dependent linear media

Jeong Ryeol Choi

Recently, Pedrosa and Rosas [Phys. Rev. Lett. 103, 010402 (2009)] investigated the quantum states of an electromagnetic field in time-dependent linear media using a Hermitian linear invariant. The wave function obtained by them is represented in terms of an arbitrary weight function $g({\ensuremath{\lambda}}_{l})$. Since the type of wave function varies depending on the choice of $g({\ensuremath{\lambda}}_{l})$ in their problem, it may be a difficult task to construct a coherent state that resembles the classical state from their theory. We suggest, on the basis of a non-Hermitian linear invariant, another quantum state that is a kind of coherent state. The expectation value of canonical variables in this alternate state follows an exact classical trajectory. For a simple case in which the time dependence of the parameters $\ensuremath{\epsilon}(t)$, $\ensuremath{\mu}(t)$, and $\ensuremath{\sigma}(t)$ disappears, we showed that the quantum energy expectation value in the alternate quantum state recovers exactly to the classical energy in the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$. This alternate state leads to the correspondence between the quantum and the classical behaviors of physical observables in a high-energy limit.

Physical Review A•Nov 24, 2010•26 citations

The quantum electromagnetic field in the absence of sources

G. Compagno, Roberto Passante, Franco Persico

Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.

Cambridge University Press eBooks•Aug 3, 1995•0 citations

Quantum Theory of the Electromagnetic Field

Peter W. Milonni

Abstract The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.

•Jan 31, 2019•1 citation

Topological aspects of quantum entanglement

Louis Hirsch Kauffman, Eshan Mehrotra

Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the $$ER = EPR$$ E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.

arXiv:1611.08047•Quantum Information Processing•Feb 2, 2019•20 citations
We construct coherent states for the quantized electromagnetic field that correspond to the classical non-null torus knot solutions of Maxwell's equations in vacuum. We derive the displacement operators from the general relation between classical fields and coherent state amplitudes and verify the defining properties of coherent states through direct computation. We determine the observables of the model: field expectation values, energy density, Poynting vector, helicity, photon number, quadrature uncertainties, and correlation functions, and calculate their expectation values in the knotted coherent states in terms of the integer parameters (n,m,l,s)(n,m,l,s)(n,m,l,s) of the classical solutions. As an example, we particularize the construction in the case of the Hopfion coherent state. These results establish the quantum-classical correspondence for this type of vacuum topological electromagnetic systems.
Following the development of the quantum theory of radiation and with the advent of the laser, the states of the field that most nearly describe a classical electromagnetic field were widely studied. In order to realize such ‘classical’ states, we will consider the field generated by a classical monochromatic current, and find that the quantum state thus generated has many interesting properties and deserves to be called a coherent state . An important consequence of the quantization of the radiation field is the associated uncertainty relation for the conjugate field variables. It therefore appears reasonable to propose that the wave function which corresponds most closely to the classical field must have minimum uncertainty for all times subject to the appropriate simple harmonic potential. In this chapter we show that a displaced simple harmonic oscillator ground state wave function satisfies this property and the wave packet oscillates sinusoidally in the oscillator potential without changing shape as shown in Fig. 2.1. This coherent wave packet always has minimum uncertainty, and resembles the classical field as nearly as quantum mechanics permits. The corresponding state vector is the coherent state |α〉, which is the eigenstate of the positive frequency part of the electric field operator, or, equivalently, the eigenstate of the destruction operator of the field. Classically an electromagnetic field consists of waves with welldefined amplitude and phase. Such is not the case when we treat the field quantum mechanically.
In quantum mechanics, the wave function encodes all information about a particle through its probability density and phase. While stationary states are characterized by conserved quantum numbers such as energy and, in central potentials, angular momentum, non-stationary superpositions - particularly in anisotropic or time-dependent fields - generally lack equally universal descriptors of their spatiotemporal dynamics. Here we introduce quantum hodographs: the trajectories traced in time by the expectation value of observable vector quantities. For a free electron in a superposition of three plane waves, all hodographs of the probability current lie on a universal cubic surface with conical singularities. Rational frequency (energy) difference ratios produce non-contractible loops with well-defined winding numbers. In anisotropic harmonic oscillators the Ehrenfest trajectories - hodographs of the expectation value of particle radius-vector - form three-dimensional Lissajous knots, echoing the classical Thomson vortex-atom model. Externally driven quantum systems allow controllable initiation of knotted hodographs. We propose an optical modulation spectroscopy scheme for reconstructing these topological features in trapped ions and single-electron systems. The topological indices of the loops and knots are robust to parameter variations, offering a new tool for characterizing complex quantum dynamics.
Recently, Pedrosa and Rosas [Phys. Rev. Lett. 103, 010402 (2009)] investigated the quantum states of an electromagnetic field in time-dependent linear media using a Hermitian linear invariant. The wave function obtained by them is represented in terms of an arbitrary weight function $g({\ensuremath{\lambda}}_{l})$. Since the type of wave function varies depending on the choice of $g({\ensuremath{\lambda}}_{l})$ in their problem, it may be a difficult task to construct a coherent state that resembles the classical state from their theory. We suggest, on the basis of a non-Hermitian linear invariant, another quantum state that is a kind of coherent state. The expectation value of canonical variables in this alternate state follows an exact classical trajectory. For a simple case in which the time dependence of the parameters $\ensuremath{\epsilon}(t)$, $\ensuremath{\mu}(t)$, and $\ensuremath{\sigma}(t)$ disappears, we showed that the quantum energy expectation value in the alternate quantum state recovers exactly to the classical energy in the limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$. This alternate state leads to the correspondence between the quantum and the classical behaviors of physical observables in a high-energy limit.
Introduction . The purpose of this chapter is to present the quantum theory of the electromagnetic field in the absence of charges and currents. Thus the classical field discussed in the previous chapter is subjected to canonical quantization in Section 2.1, where creation and annihilation operators for plane-wave and spherical wave modes, as well as their commutation relations, are derived along with various field-field commutators related to field propagators. In the next section we introduce the concept of the photon as an elementary excitation of the electromagnetic field. The attention is focused on the ground state of the quantized electromagnetic field in the absence of sources, which is the photon vacuum. The amplitude fluctuations, or zero-point fluctuations of this vacuum, are evaluated. Excited states of the field are examined in the next sections. In particular, Section 2.3 is concerned with number states and coherent states, the latter being obtained by a Glauber transformation of the vacuum, and with their statistical properties. Squeezed states of the field are introduced in Section 2.4 by a unitary transformation leading from the normal to the squeezed vacuum, whose statistical properties are compared with those of a coherent state. Section 2.5 is devoted to a brief discussion of thermal states. The final section of this chapter is dedicated to a discussion of the nonlocalizability of the photon.
Abstract The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.