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Physical Review A

Coherent states of the driven Rydberg atom: Quantum-classical correspondence of periodically driven systems

Luz V. Vela-Arevalo, Ronald F. Fox

Published Jun 9, 2005• 20 citations
Publisher

Abstract

A methodology to calculate generalized coherent states for a periodically driven system is presented. We study wave packets constructed as a linear combination of suitable Floquet states of the three-dimensional Rydberg atom in a microwave field. The driven coherent states show classical space localization, spreading, and revivals and remain localized along the classical trajectory. The microwave strength and frequency have a great effect in the localization of Floquet states, since quasienergy avoided crossings produce delocalization of the Floquet states, showing that tuning of the parameters is very important. Using wavelet-based time-frequency analysis, the classical phase-space structure is determined, which allows us to show that the driven coherent state is located in a large regular region in which the $z$ coordinate is in resonance with the external field. The expectation values of the wave packet show that the driven coherent state evolves along the classical trajectory.

DOI
10.1103/physreva.71.063403

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Physical Review A•Dec 11, 2003•21 citations
A methodology to calculate generalized coherent states for a periodically driven system is presented. We study wave packets constructed as a linear combination of suitable Floquet states of the three-dimensional Rydberg atom in a microwave field. The driven coherent states show classical space localization, spreading, and revivals and remain localized along the classical trajectory. The microwave strength and frequency have a great effect in the localization of Floquet states, since quasienergy avoided crossings produce delocalization of the Floquet states, showing that tuning of the parameters is very important. Using wavelet-based time-frequency analysis, the classical phase-space structure is determined, which allows us to show that the driven coherent state is located in a large regular region in which the zzz coordinate is in resonance with the external field. The expectation values of the wave packet show that the driven coherent state evolves along the classical trajectory.
Both quantum and classical analyses are performed to study the barrier crossing dynamics in a driven quartic oscillator. The regions of phase space with regular classical motion are found to be smaller than the size of a quantum state. However, coherent tunnelling is still possible due to the existence of Floquet states localized on two small regular islands. The quantum evolution of localized wavepackets and the classical evolution of the corresponding distributions show similar coherence properties although the degree of coherence is quantally enhanced.
We investigate the Floquet spectrum and excitation properties of a two-ultracold-atom system with periodically driven interaction in a three-dimensional harmonic trap. The interaction between the atoms is changed by varying the s-wave scattering length in two ways, the cosine and the square-wave modulations. It is found that as the driving frequency increases, the Floquet spectrum exhibits two main features for both modulations, the accumulating and the spreading of the quasienergy levels, which further lead to different dynamical behaviors. The accumulation is associated with collective excitations and the persistent growth of the energy, while the spread indicates that the energy is bounded at all times. The initial scattering length, the driving frequency and amplitude can all significantly change the Floquet spectrum as well as the dynamics. However, the corresponding relation between them is valid universally. Finally, we propose a mechanism for selectively exciting the system to one specific state by using the avoided crossing of two quasienergy levels, which could guide preparation of a desired state in experiments.
Summary form only given. The quantum driven pendulum can be implemented using laser cooled rubidium atoms which are exposed to a far detuned amplitude modulated standing wave. This system allows controlled variation of the effective Planck's constant and the amount of coupling to the environment (decoherence). Therefore it is possible to undertake experiments in both classical and quantum regimes of the dynamics of these atoms. The potential can be extremely well approximated as one-dimensional, making the system ideal for simulation using a master equation or quantum trajectories. The dynamics of the system feature distinct regions of regular motion (second order resonances) bounded by a sea of chaos. The resonances are observed as side peaks in the atomic momentum distribution, which each contain up to 30% of the atoms initially in the trap. We report on the results of our experiments to achieve quantum phase space tunneling. After adiabatic elimination of the excited state, the Hamiltonian describing the system is given. The evolution of the quantum driven pendulum can be described with the Floquet operator. Eigenstates of the Floquet operator may be identified with fixed points in phase space. Starting the evolution in a superposition of two Floquet states which is localised in one resonance, time evolution leads to localisation in the other fixed point. This is a coherent tunneling process forbidden by classical dynamics. To observe this tunneling process it is important to localise the superposition of Floquet states in one fixed point. We discuss the effectiveness of alternative methods of loading most of the atoms into a single resonance in order to observe quantum phase space tunneling.
We simulate periodically driven Rydberg arrays with purely imaginary-phase driving, showing that it reshapes both internal dynamics and the structure of coupled vacuum fluctuations. In a driven Rydberg array, this interplay produces\emph{hybrid vacuum--photonic quasienergy channels}---coherent,directional modes arising from Floquet-induced modulation of theelectromagnetic local density of states (LDOS). These channels enable a coherent mapping of correlation structure far beyond the nominal dipolar range andexhibit stability characteristic of attractor-like Floquet fixed points. Using a microscopic Hamiltonian with spatiotemporally structureddipole--vacuum coupling, we derive a Floquet--Magnus expression for theeffective LDOS:ρeff(ω)=∑mJm2(A) ρ0(ω+mΩ),\rho_{\rm eff}(\omega)=\sum_m J_m^2(A)\,\rho_0(\omega+m\Omega),ρeff​(ω)=∑m​Jm2​(A)ρ0​(ω+mΩ),revealing how the drive imprints anisotropic vacuum structure.Large-scale open-system simulations confirm that this LDOS reshapingproduces long-lived, quasienergy-resolved photonic corridors, alongwhich correlation fronts propagate with direction-dependent velocities. We further show that the system's long-time behavior is well describedby an \emph{effective geometric picture}: anisotropic correlationfronts define a synthetic light cone, LDOS-induced anisotropies definean emergent spatial metric, and wavepacket bundle separation obeys ageodesic-deviation-like equation whose curvature is determined directlyfrom simulated trajectories over thousands of drive periods. We conclude with a speculative perspective: if driven matter can imprintstable geometric structure onto the vacuum LDOS, then controlled analogspacetime engineering---without material waveguides or externalphotonic scaffolds---may be achievable.
We study the three-dimensional (3D) unidirectionally kicked Rydberg atom. For parabolic initial states elongated in the direction of the kicks we show that the ionization of the quantum system is suppressed as compared to the classical counterpart and that the quantum wave function is localized along all degrees of freedom, whereas the classical system is globally diffusive. We discuss the connection to the previously studied one-dimensional (1D) model of the kicked Rydberg atom and verify that the 1D model is a good approximation to the 3D quantum case in the limiting case of the most elongated initial states. We further study the quantum phase-space distribution (Husimi distribution) of the eigenstates of the period-one time-evolution (Floquet) operator and show that the eigenstates are localized in phase space. For the most elongated parabolic initial state, we are able to identify the unstable periodic orbits around which Floquet states localize. We discuss the possibility of observing quantum localization in high Rydberg states in n>100.n>100.n>100.