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:quant-ph/9809049 · Physical Review A

High-order nonlinearities in the motion of a trapped atom

Sascha Wallentowitz, W. Vogel, P. L. Knight

Published Jan 1, 1999• 46 citations
arXivHTMLPDF

Abstract

We study the counterpart to the multiphoton down-conversion in the quantized motion of a trapped atom. The Lamb-Dicke approximation leads to a divergence of the mean motional excitation in a finite interaction time for k-quantum down-conversions with $k>~3,$ analogous to the situation in the parametric approximation of nonlinear optics. We show that, in contrast to the Lamb-Dicke approximation, the correct treatment of the overlap of the atomic center-of-mass wave function and the driving laser waves leads to a proper dynamics without any divergence problem. That is, the wavy nature of both matter and light is an important physical property that cannot be neglected for describing the motional dynamics of a trapped atom, even for small Lamb-Dicke parameters.

DOI
10.1103/physreva.59.531

Related papers

Nonlinear squeezing in the motion of a trapped atom

Sascha Wallentowitz, W. Vogel

We study a Raman-driven two-quantum interaction in the motion of a trapped atom, which in the Lamb-Dicke limit corresponds to a classically driven parametric interaction known from nonlinear optics. When the Lamb-Dicke approximation fails, interference effects between the driving laser waves and the spatially extended motional wave function of the atom lead to nonlinear modifications of the motional dynamics. Destructive interferences occur for particular values of the motional amplitude. They lead to a partitioning of the motional phase space accompanied by interesting dynamical effects. Starting the dynamics in the motional ground state of the atom, a quadrature-squeezed state emerges for short interaction times, in agreement with recent experiments [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)]. For longer interaction times the phase-space partitioning gives rise to a reduction of the variance of the motional excitation number and to phase-locking effects leading to a partial revival of the state towards the origin of the phase space.

Physical Review A•Jul 1, 1998•44 citations

Dynamics of a trapped two-level and three-level atom interacting with classical electromagnetic field

Aditi Ray

The dynamics of a two-level atom driven by a single laser beam and three-level atom (Lambda configuration) irradiated by two laser beams are studied taking into account of the quantized center-of-mass motion of the atom. It is shown that the trapped atom system under appropriate resonance condition exhibits the large time-scale revivals when the index of the vibrational sideband responsible for the atomic electronic transition is greater than unity. The revival times are shown to be dependent on the initial number of vibrational excitations and the magnitude of the Lamb-Dicke parameter. The sub-Poissonian statistics in vibrational quantum number is observed at certain time intervals. The minimum time of interaction for which the squeezed states of motional quadrature are generated is found to be decreasing with the increase in the Lamb-Dicke parameter.

Physical Review A•Mar 10, 2004•2 citations

Dynamical properties of a trapped two-level ion interacting with a single-mode quantized field in the nonlinear regime

Navid Yazdanpanah, Mohammad Kazem Tavassoly

Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.

Journal of Modern Optics•Dec 3, 2014•9 citations

Cooling atomic motion with quantum interference

Giovanna Morigi

We theoretically investigate the quantum dynamics of the center of mass of trapped atoms, whose internal degrees of freedom are driven in a $\ensuremath{\Lambda}$-shaped configuration with the lasers tuned at two-photon resonance. In the Lamb-Dicke regime, when the motional wave packet is well localized over the laser wavelength, transient coherent population trapping occurs, canceling transitions at the laser frequency. In this limit the motion can be efficiently cooled to the ground state of the trapping potential. We derive an equation for the center-of-mass motion by adiabatically eliminating the internal degrees of freedom. This treatment provides the theoretical background of the scheme presented in [G. Morigi et al., Phys. Rev. Lett. 85, 4458 (2000)] and implemented in [C.F. Roos et al., Phys. Rev. Lett. 85, 5547 (2000)]. We discuss the physical mechanisms determining the dynamics and identify parameters regimes, where cooling is efficient. We discuss implementations of the scheme to cases where the trapping potential is not harmonic.

arXiv:quant-ph/0211043•Physical Review A•Mar 13, 2003•79 citations

K-quantum downconversion in the motion of a trapped atom

Sascha Wallentowitz, W. Vogel, P. L. Knight

Summary form only given.The phenomenon of multiquantum downconversion is of great interest in quantum optics. In the limit of a classical driving field it can be described by an interaction Hamiltonian (in the interaction picture). For k=2 an interaction of this type is of interest in the context of squeezing. For multiquantum processes of higher order (k>2), however, the Hamiltonian is known to lead to a divergent dynamics. This problem is resolved by taking into account the quantization of the pump mode. More recently, the realization of similar interactions also became possible in the quantized motion of a trapped atom, where the operator it represents the annihilation of a motional quantum of the center-of-mass motion. For k=2 such a dynamic has been used to generate motional squeezing, and other interactions of that type have been considered.

Conference on Lasers and Electro-Optics•Nov 27, 2002•0 citations

Nonlinear Jaynes-Cummings dynamics of a trapped ion

W. Vogel, R. L. de Matos Filho

We show that the vibronic motion of a trapped ion far from the Lamb-Dicke regime can be described, under appropriate resonance conditions, by a strongly nonlinear, multiquantum Jaynes-Cummings model. In contrast to previous nonlinear or multiphoton generalizations of the Jaynes-Cummings model, it could presently be realized in experiments.

Physical Review A•Nov 1, 1995•378 citations
We study the counterpart to the multiphoton down-conversion in the quantized motion of a trapped atom. The Lamb-Dicke approximation leads to a divergence of the mean motional excitation in a finite interaction time for k-quantum down-conversions with k> 3,k>~3,k> 3, analogous to the situation in the parametric approximation of nonlinear optics. We show that, in contrast to the Lamb-Dicke approximation, the correct treatment of the overlap of the atomic center-of-mass wave function and the driving laser waves leads to a proper dynamics without any divergence problem. That is, the wavy nature of both matter and light is an important physical property that cannot be neglected for describing the motional dynamics of a trapped atom, even for small Lamb-Dicke parameters.
We study a Raman-driven two-quantum interaction in the motion of a trapped atom, which in the Lamb-Dicke limit corresponds to a classically driven parametric interaction known from nonlinear optics. When the Lamb-Dicke approximation fails, interference effects between the driving laser waves and the spatially extended motional wave function of the atom lead to nonlinear modifications of the motional dynamics. Destructive interferences occur for particular values of the motional amplitude. They lead to a partitioning of the motional phase space accompanied by interesting dynamical effects. Starting the dynamics in the motional ground state of the atom, a quadrature-squeezed state emerges for short interaction times, in agreement with recent experiments [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)]. For longer interaction times the phase-space partitioning gives rise to a reduction of the variance of the motional excitation number and to phase-locking effects leading to a partial revival of the state towards the origin of the phase space.
The dynamics of a two-level atom driven by a single laser beam and three-level atom (Lambda configuration) irradiated by two laser beams are studied taking into account of the quantized center-of-mass motion of the atom. It is shown that the trapped atom system under appropriate resonance condition exhibits the large time-scale revivals when the index of the vibrational sideband responsible for the atomic electronic transition is greater than unity. The revival times are shown to be dependent on the initial number of vibrational excitations and the magnitude of the Lamb-Dicke parameter. The sub-Poissonian statistics in vibrational quantum number is observed at certain time intervals. The minimum time of interaction for which the squeezed states of motional quadrature are generated is found to be decreasing with the increase in the Lamb-Dicke parameter.
Following the path of Bužek et al. in [Phys. Rev. A 56 2352 1997], we investigate the full intensity-dependent interaction between a motional two-level ion which is trapped in a f-deformed oscillator (so possessing vibrational modes too) with a single-mode nonlinear quantized field in the Lamb-Dicke limit. After finding the explicit form of the entire ion-field state vector in a general framework, we particularly restrict ourselves to an appropriate nonlinearity function (describing intensity-dependent interaction) corresponding to the center-of-mass motion of a trapped ion which explicitly depends on the Lamb-Dicke parameter as well as another well-known nonlinearity function in the form n. Then, we evaluate the effects of two different initial quantized field states as well as the considered nonlinearity function (via varying the Lamb-Dicke parameter) on a few physical properties of the system. For this purpose, we pay attention to the amount of entanglement, mean number of vibrational quanta (phonons), population inversion of the two-level ion and Q-distribution function. Finally, by comparing our results with the previously considered “linear” one which is pointed in the above Ref., we establish that, by tuning the Lamb-Dicke parameter, one can adjust the mentioned nonclassical properties of the system appropriately.
We theoretically investigate the quantum dynamics of the center of mass of trapped atoms, whose internal degrees of freedom are driven in a $\ensuremath{\Lambda}$-shaped configuration with the lasers tuned at two-photon resonance. In the Lamb-Dicke regime, when the motional wave packet is well localized over the laser wavelength, transient coherent population trapping occurs, canceling transitions at the laser frequency. In this limit the motion can be efficiently cooled to the ground state of the trapping potential. We derive an equation for the center-of-mass motion by adiabatically eliminating the internal degrees of freedom. This treatment provides the theoretical background of the scheme presented in [G. Morigi et al., Phys. Rev. Lett. 85, 4458 (2000)] and implemented in [C.F. Roos et al., Phys. Rev. Lett. 85, 5547 (2000)]. We discuss the physical mechanisms determining the dynamics and identify parameters regimes, where cooling is efficient. We discuss implementations of the scheme to cases where the trapping potential is not harmonic.
Summary form only given.The phenomenon of multiquantum downconversion is of great interest in quantum optics. In the limit of a classical driving field it can be described by an interaction Hamiltonian (in the interaction picture). For k=2 an interaction of this type is of interest in the context of squeezing. For multiquantum processes of higher order (k>2), however, the Hamiltonian is known to lead to a divergent dynamics. This problem is resolved by taking into account the quantization of the pump mode. More recently, the realization of similar interactions also became possible in the quantized motion of a trapped atom, where the operator it represents the annihilation of a motional quantum of the center-of-mass motion. For k=2 such a dynamic has been used to generate motional squeezing, and other interactions of that type have been considered.