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

Annual Meeting Optical Society of America

Cotangent states of the electromagnetic field

John J. Slosser, P. Meystre

Published Jan 1, 1989• 0 citations
Publisher

Abstract

We consider a simple quantized harmonic oscillator driven by a current of spin 1/2 particles resonant with the oscillator frequency. A possible realization of this model might be achieved in a micromaser. Our analysis shows that the system may evolve to pure states j of the field even for mixed state initial conditions. Specifically, writing the field state as |f⟩ = Σ n |n⟩ we find the two possible solutions: For this reason we call these states cotangent and tangent states of the harmonic oscillator, respectively.

DOI
10.1364/oam.1989.wdd5

Related papers

Tangent and cotangent states of the electromagnetic field

John J. Slosser, Pierre Meystre

We consider a lossless micromaser in which a monoenergetic, low-density beam of two-level atoms in a coherent superposition of their upper and lower states is injected inside a single-mode high-Q cavity. This is a realization of a harmonic oscillator driven by a quantum current. We find that under appropriate conditions the field may evolve to pure states, which we call tangent and cotangent states, even for mixed-state initial conditions. In various limits, they exhibit nonclassical properties such as sub-Poissonian photon statistics, or even more interestingly acquire the characteristics of ``macroscopic'' quantum superpositions. The conditions under which these states are reached, as well as the dynamical approach of the system to steady state, are discussed in detail.

Physical Review A•Apr 1, 1990•112 citations

Harmonic oscillator driven by a quantum current

John J. Slosser, Pierre Meystre, Samuel L. Braunstein

We show that under certain circumstances a simple quantum harmonic oscillator driven by a quantum current evolves to unique pure states even if started as a mixed state. In various limits, these states exhibit nonclassical properties such as sub-Poissonian statistics, or more interestingly resemble macroscopic superpositions.

Physical Review Letters•Aug 28, 1989•153 citations

Dynamics of a coherently driven micromaser.

John J. Slosser

This dissertation considers both a lossless and a dissipative micromaser in which a monoenergetic beam of two-level atoms in a coherent superposition of their upper and lower states is injected inside a single mode, high-Q cavity. In the lossless case, we find that under appropriate conditions a field initially in a mixed state will evolve to previously unknown pure states, which we call the tangent and cotangent states. In various limits, these states exhibit interesting properties such as sub-Poissonian photon statistics and squeezing, and most importantly they acquire the characteristics of "macroscopic" quantum superpositions. When dissipation of the cavity mode is incorporated into the model, we find that although the field no longer evolves to a pure state, the mixed steady-state field may still retain the properties of a macroscopic superposition under experimentally realizable damping rates. We then evaluate the experimental conditions necessary for the preparation and detection of such macroscopic superpositions.

•Jan 1, 1990•0 citations

Phase operators of the harmonic oscillator and suppression of a number state

C V Sukumar

The phase properties of the quantized electromagnetic field have been studied using operators ${P}^{\ifmmode\pm\else\textpm\fi{}}$ of the harmonic oscillator whose classical analogs are exp(\ensuremath{\mp}i\ensuremath{\theta}), \ensuremath{\theta} being the classical phase. ${P}^{+}$ and ${P}^{\mathrm{\ensuremath{-}}}$ are used to construct a unitary operator ${U}_{p}$(\ensuremath{\alpha})=exp(\ensuremath{\alpha}${P}^{+}$-${\ensuremath{\alpha}}^{\mathrm{*}}$${P}^{\mathrm{\ensuremath{-}}}$) and the properties of the state generated by ${U}_{p}$ from a number state are studied. It is shown that by tuning \ensuremath{\alpha} it is possible to produce a pure state ${\ensuremath{\psi}}_{0}$=${U}_{p}$(\ensuremath{\alpha})\ensuremath{\Vert}0〉 in which a specific number-state component is suppressed.

Physical Review A•Nov 1, 1989•1 citation

Field oscillations in a micromaser with injected atomic coherence

Mark S. Hillery, Jozef Škvarček

The electric field in a lossless, regularly-pumped micromaser with injected atomic coherence can undergo a period-two oscillation in the steady state. The field changes its value after a single atom passes through the micromaser cavity, but returns to its original value after a second atom travels through. We give a simple explanation for this phenomenon in terms of tangent and cotangent states. We also examine the effect of cavity damping on this steady state.

arXiv:quant-ph/9804017•Journal of Modern Optics•Aug 1, 1998•4 citations

Quantum entanglement of a harmonic oscillator with an electromagnetic field

Dmitry N. Makarov

At present, there are many methods for obtaining quantum entanglement of particles with an electromagnetic field. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic field) interacting with a quantized electromagnetic field is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic field. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy.

arXiv:1709.04716•Scientific Reports•May 22, 2018•30 citations
We consider a simple quantized harmonic oscillator driven by a current of spin 1/2 particles resonant with the oscillator frequency. A possible realization of this model might be achieved in a micromaser. Our analysis shows that the system may evolve to pure states j of the field even for mixed state initial conditions. Specifically, writing the field state as |f⟩ = Σ n |n⟩ we find the two possible solutions: For this reason we call these states cotangent and tangent states of the harmonic oscillator, respectively.
We consider a lossless micromaser in which a monoenergetic, low-density beam of two-level atoms in a coherent superposition of their upper and lower states is injected inside a single-mode high-Q cavity. This is a realization of a harmonic oscillator driven by a quantum current. We find that under appropriate conditions the field may evolve to pure states, which we call tangent and cotangent states, even for mixed-state initial conditions. In various limits, they exhibit nonclassical properties such as sub-Poissonian photon statistics, or even more interestingly acquire the characteristics of ``macroscopic'' quantum superpositions. The conditions under which these states are reached, as well as the dynamical approach of the system to steady state, are discussed in detail.
This dissertation considers both a lossless and a dissipative micromaser in which a monoenergetic beam of two-level atoms in a coherent superposition of their upper and lower states is injected inside a single mode, high-Q cavity. In the lossless case, we find that under appropriate conditions a field initially in a mixed state will evolve to previously unknown pure states, which we call the tangent and cotangent states. In various limits, these states exhibit interesting properties such as sub-Poissonian photon statistics and squeezing, and most importantly they acquire the characteristics of "macroscopic" quantum superpositions. When dissipation of the cavity mode is incorporated into the model, we find that although the field no longer evolves to a pure state, the mixed steady-state field may still retain the properties of a macroscopic superposition under experimentally realizable damping rates. We then evaluate the experimental conditions necessary for the preparation and detection of such macroscopic superpositions.
The phase properties of the quantized electromagnetic field have been studied using operators ${P}^{\ifmmode\pm\else\textpm\fi{}}$ of the harmonic oscillator whose classical analogs are exp(\ensuremath{\mp}i\ensuremath{\theta}), \ensuremath{\theta} being the classical phase. P+{P}^{+}P+ and ${P}^{\mathrm{\ensuremath{-}}}$ are used to construct a unitary operator Up{U}_{p}Up​(\ensuremath{\alpha})=exp(\ensuremath{\alpha}P+{P}^{+}P+-${\ensuremath{\alpha}}^{\mathrm{*}}$${P}^{\mathrm{\ensuremath{-}}}$) and the properties of the state generated by Up{U}_{p}Up​ from a number state are studied. It is shown that by tuning \ensuremath{\alpha} it is possible to produce a pure state ${\ensuremath{\psi}}_{0}$=Up{U}_{p}Up​(\ensuremath{\alpha})\ensuremath{\Vert}0〉 in which a specific number-state component is suppressed.
At present, there are many methods for obtaining quantum entanglement of particles with an electromagnetic field. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic field) interacting with a quantized electromagnetic field is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic field. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy.