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Physical Review A

Tangent and cotangent states of the electromagnetic field

John J. Slosser, Pierre Meystre

Published Apr 1, 1990• 112 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.41.3867

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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

Cotangent states of the electromagnetic field

John J. Slosser, P. Meystre

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.

Annual Meeting Optical Society of America•Jan 1, 1989•0 citations

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Harmonic oscillator driven by a quantum current

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Generation of macroscopic superpositions in a two-photon micromaser

Pierre Meystre, Martin Wilkens

We study the dynamics of a two-photon micromaser driven by a sequence of polarized Rydberg atoms. If cavity losses are neglected, the density matrix of the cavity mode consists of four dynamically decoupled components. Under appropriate trapping conditions, similar to those for the one-photon micromaser, it evolves to a stationary state. When the bare two-photon frequency cancels the ac Stark shift we cannot satisfy the trapping conditions for odd and even photon number states simultaneously. For specific initial conditions, where either only odd or only even number states are initially populated, we may find, however, stationary states that are pure and that can lead to macroscopic superpositions. These pure states are analogous to the cotangent and tangent states of the one-photon micromaser. When the two atomic levels are in resonance with the bare two-photon transition, trapping conditions for odd and even photon number states can be fulfilled simultaneously. Although the ac Stark shift does not allow for pure stationary states, the stationary state resembles an incoherent superposition of only two pure states, to within a relative deviation of 10-3 off purity.

Annual Meeting Optical Society of America•Jan 1, 1990•0 citations

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
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.
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.
The dynamics of an exact two-photon Hamiltonian is used to study the time evolution of an initially disentangled pure state of the atom-field system as it goes through cycles of entanglement separated by instances of disentanglement. For specific initial states of the electromagnetic field, the output state is a pure quantum superposition of a squeezed vacuum state and an orthogonal, odd-photon-number state. The odd-photon-number state, which is not a squeezed state, exhibits both nonclassical sub-Poissonian and classical super-Poissonian photon statistics. In the latter case the quantum superposition resembles a macroscopic superposition state. Conditions are obtained on the atom-cavity interaction time for such states to represent the steady states in the injection in a high-QQQ cavity of a monoenergetic, low-density beam of three-level atoms in a coherent state.
We study the dynamics of a two-photon micromaser driven by a sequence of polarized Rydberg atoms. If cavity losses are neglected, the density matrix of the cavity mode consists of four dynamically decoupled components. Under appropriate trapping conditions, similar to those for the one-photon micromaser, it evolves to a stationary state. When the bare two-photon frequency cancels the ac Stark shift we cannot satisfy the trapping conditions for odd and even photon number states simultaneously. For specific initial conditions, where either only odd or only even number states are initially populated, we may find, however, stationary states that are pure and that can lead to macroscopic superpositions. These pure states are analogous to the cotangent and tangent states of the one-photon micromaser. When the two atomic levels are in resonance with the bare two-photon transition, trapping conditions for odd and even photon number states can be fulfilled simultaneously. Although the ac Stark shift does not allow for pure stationary states, the stationary state resembles an incoherent superposition of only two pure states, to within a relative deviation of 10-3 off purity.