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Annual Meeting Optical Society of America

Generation of macroscopic superpositions in a two-photon micromaser

Pierre Meystre, Martin Wilkens

Published Jan 1, 1990• 0 citations
Publisher

Abstract

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.

DOI
10.1364/oam.1990.fgg8

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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.
We study the preparation of coherent quantum states in a two-photon micromaser for applications in quantum metrology. While this setting can be in principle realized in a host of physical systems, we consider atoms interacting with the field of a cavity. We focus on the case of the interaction described by the Jaynes-Cummings Hamiltonian, which cannot be achieved by the conventional approach with three-level atoms coupled to the cavity field at two-photon resonance. We find that additional levels are required in order to cancel Stark shifts emerging in the leading order. Once this is accomplished, the dynamics of the cavity features a degenerate stationary state manifold of pure states. We derive the analytic form of these states and show that they include Schr\"odinger cat states with a tunable mean photon number. We also confirm these states can be useful in phase estimation protocols with their quantum Fisher information exceeding the standard limit. To account for realistic imperfections, we consider single-photon losses from the cavity, finite lifetime of atom levels, and higher order corrections in the far-detuned limit, which result in metastability of formerly stationary cavity states and long-time dynamics with a unique mixed stationary state. Despite being mixed, this stationary state can still feature quantum Fisher information above the standard limit. Our work delivers a comprehensive overview of the two-photon micromaser dynamics with particular focus on application in phase estimation and, while we consider the setup with atoms coupled to a cavity, the results can be directly translated to optomechanical systems.
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.
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 report a numerical simulation of the action of the one 85 Rb atom micromaser which shows that the micromaser field evolves towards a steady state equilibrium when the condition that Tp t int satisfies. In this steady state regime the cavity field reaches a steady state after a sufficient number of atoms have passed through the cavity. The cavity temperature T and the cavity quality factor Q can produce a quantitive change when one of them or the two are changed but they are not responsible for the qualitative changes in the micromaser field. Evidently the only responsible parameter for these qualitative changes is the repetition time T p namely if T p t int then the micromaser field evolution is towards a steady state equilibrium. In contrast when T p reduced to the order of t int then the micromaser field evolution is controlled by the trapping state dynamics – no steady state is apparent in this case. The first going-up trapping state at n=3 plays a significant role in the early dynamics of the micromaser field. Moreover the state of the field ultimately evolves towards a mixed state rather than a pure state.
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.