Journal of Physics B Atomic Molecular and Optical Physics

Wave packet fractional revivals in a one-dimensional Rydberg atom

Rita Veilande, Imants Bersons

Published May 24, 2007• 20 citations

Abstract

We investigate many characteristic features of revival and fractional revival phenomena via derived analytic expressions for an autocorrelation function of a one-dimensional Rydberg atom with weighting probabilities modelled by a Gaussian or a Lorentzian distribution. The fractional revival phenomenon in the ionization probabilities of a one-dimensional Rydberg atom irradiated by two short half-cycle pulses is also studied. When many states are involved in the formation of the wave packet, the revival is lower and broader than the initial wave packet and the fractional revivals overlap and disappear with time.

DOI: 10.1088/0953-4075/40/11/013

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Journal of Physics B Atomic Molecular and Optical Physics

Wave packet fractional revivals in a one-dimensional Rydberg atom

Rita Veilande, Imants Bersons

Published May 24, 2007• 20 citations

Publisher

Abstract

DOI: 10.1088/0953-4075/40/11/013

Related papers

Fractional Revival of Rydberg Wave Packets in Twice-Kicked One-Dimensional Atoms

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We study revival and fractional revival phenomena of wave packets in a one-dimensional Rydberg atom irradiated by two time-delayed half-cycle pulses using an autocorrelation function characterized by electronic transition probabilities as weighting factors rather than modeling them by a Gaussian or Lorentzian distribution.If the momentum (q2) delivered to the atom by the second kick is much smaller than that (q1) imparted by the first one, the times of revival and fractional revival coincide with those of the single kicked atom.For q2 ≥ q 1 4 appearance of revival and fractional revival depends on both the values of q2 and time delay t1 between the pulses but more sensitively on t1.The number of fractional revivals tends to become numerous as the value of t1 increases.We study revival and fractional revival phenomena of wave packets in a one-dimensional Rydberg atom irradiated by two time-delayed half-cycle pulses using an autocorrelation function characterized by electronic transition probabilities as weighting factors rather than modeling them by a Gaussian or Lorentzian distribution.If the momentum (q2) delivered to the atom by the second kick is much smaller than that (q1) imparted by the first one, the times of revival and fractional revival coincide with those of the single kicked atom.For q2 ≥ q 1 4 appearance of revival and fractional revival depends on both the values of q2 and time delay t1 between the pulses but more sensitively on t1.The number of fractional revivals tends to become numerous as the value of t1 increases.

Acta Physica Polonica ASep 1, 20112 citations

A time–frequency analysis of wave packet fractional revivals

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We show that the time frequency analysis of the autocorrelation function is, in many ways, a more appropriate tool to resolve fractional revivals of a wave packet than the usual time domain analysis. This advantage is crucial in reconstructing the initial state of the wave packet when its coherent structure is short-lived and decays before it is fully revived. Our calculations are based on the model example of fractional revivals in a Rydberg wave packet of circular states. We end by providing an analytical investigation which fully agrees with our numerical observations on the utility of time-frequency analysis in the study of wave packet fractional revivals.We show that the time frequency analysis of the autocorrelation function is, in many ways, a more appropriate tool to resolve fractional revivals of a wave packet than the usual time domain analysis. This advantage is crucial in reconstructing the initial state of the wave packet when its coherent structure is short-lived and decays before it is fully revived. Our calculations are based on the model example of fractional revivals in a Rydberg wave packet of circular states. We end by providing an analytical investigation which fully agrees with our numerical observations on the utility of time-frequency analysis in the study of wave packet fractional revivals.

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The Revival Structure of Rydberg Wave Packets Beyond the Revival Time

Robert Bluhm, V. Alan Kostelecký

After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time $t_{\rm rev}$ , at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than $t_{\rm rev}$ . We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period $t_{\rm sr} \gg t_{\rm rev}$ with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time $t_{\rm rev}$ . Furthermore, at times that are rational fractions of $t_{\rm sr}$ , we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time $t_{\rm rev}$ . These periodicities indicate a new type of fractional revival, occurring for times much greater than $t_{\rm rev}$ . We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time $t_{\rm rev}$, at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than $t_{\rm rev}$. We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period $t_{\rm sr} \gg t_{\rm rev}$ with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time $t_{\rm rev}$. Furthermore, at times that are rational fractions of $t_{\rm sr}$, we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time $t_{\rm rev}$. These periodicities indicate a new type of fractional revival, occurring for times much greater than $t_{\rm rev}$. We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.

arXiv:quant-ph/9508025Aug 29, 19950 citations

Long-term evolution and revival structure of Rydberg wave packets for hydrogen and alkali-metal atoms

Robert Bluhm, V. Alan Kostelecký

This paper begins with an examination of the revival structure and long-term evolution of Rydberg wave packets for hydrogen. We show that after the initial cycle of collapse and fractional or full revival, which occurs on the time scale ${\mathit{t}}_{\mathrm{rev}}$ , a new sequence of revivals begins. We find that the structure of the new revivals is different from that of the fractional revivals. The new revivals are characterized by periodicities in the motion of the wave packet with periods that are fractions of the revival time scale ${\mathit{t}}_{\mathrm{rev}}$ . These long-term periodicities result in the autocorrelation function at times greater than ${\mathit{t}}_{\mathrm{rev}}$ having a self-similar resemblance to its structure for times less than ${\mathit{t}}_{\mathrm{rev}}$ . The new sequence of revivals culminates with the formation of a single wave packet that more closely resembles the initial wave packet than does the full revival at time ${\mathit{t}}_{\mathrm{rev}}$ , i.e., a superrevival forms. Explicit examples of the superrevival structure for both circular and radial wave packets are given. We then study wave packets in alkali-metal atoms, which are typically used in experiments. The behavior of these packets is affected by the presence of quantum defects that modify the hydrogenic revival time scales and periodicities. Their behavior can be treated analytically using supersymmetry-based quantum-defect theory. We illustrate our results for alkali-metal atoms with explicit examples of the revival structure for radial wave packets in rubidium.This paper begins with an examination of the revival structure and long-term evolution of Rydberg wave packets for hydrogen. We show that after the initial cycle of collapse and fractional or full revival, which occurs on the time scale ${\mathit{t}}_{\mathrm{rev}}$, a new sequence of revivals begins. We find that the structure of the new revivals is different from that of the fractional revivals. The new revivals are characterized by periodicities in the motion of the wave packet with periods that are fractions of the revival time scale ${\mathit{t}}_{\mathrm{rev}}$. These long-term periodicities result in the autocorrelation function at times greater than ${\mathit{t}}_{\mathrm{rev}}$ having a self-similar resemblance to its structure for times less than ${\mathit{t}}_{\mathrm{rev}}$. The new sequence of revivals culminates with the formation of a single wave packet that more closely resembles the initial wave packet than does the full revival at time ${\mathit{t}}_{\mathrm{rev}}$, i.e., a superrevival forms. Explicit examples of the superrevival structure for both circular and radial wave packets are given. We then study wave packets in alkali-metal atoms, which are typically used in experiments. The behavior of these packets is affected by the presence of quantum defects that modify the hydrogenic revival time scales and periodicities. Their behavior can be treated analytically using supersymmetry-based quantum-defect theory. We illustrate our results for alkali-metal atoms with explicit examples of the revival structure for radial wave packets in rubidium.

arXiv:quant-ph/9506009Physical Review AJun 1, 199570 citations

Manifestations of wave packet fractional revivals in a Morse-like anharmonic system

V.V. Eryomin, S. I. Vetchinkin, И. М. Уманский

The general properties of the time-dependent characteristics of one-dimensional vibrational wave packets in an anharmonic system with the quadratic spectrum are discussed. The local behavior of the autocorrelation function and its square modulus (the survival function) is studied near the moments of fractional revival t=(p/q)Trev, where Trev is the period of the complete revival, p/q is an irreducible fraction, and q is the order of the revival. It is shown that all the wave packets have the same values of the survival function at the moments of fractional revival provided that the number of states in the packet exceeds the order of the revival q. It is also found that for the given packet, all the fractional revivals are identical to each other in the sense that their lifetimes do not depend on the order of the revival. The lifetimes of all fractional revivals are expressed in terms of the parameters of the initial packet (the effective number of states and the period of classical motion of the packet center) and the anharmonicity of the molecular system.The general properties of the time-dependent characteristics of one-dimensional vibrational wave packets in an anharmonic system with the quadratic spectrum are discussed. The local behavior of the autocorrelation function and its square modulus (the survival function) is studied near the moments of fractional revival t=(p/q)Trev, where Trev is the period of the complete revival, p/q is an irreducible fraction, and q is the order of the revival. It is shown that all the wave packets have the same values of the survival function at the moments of fractional revival provided that the number of states in the packet exceeds the order of the revival q. It is also found that for the given packet, all the fractional revivals are identical to each other in the sense that their lifetimes do not depend on the order of the revival. The lifetimes of all fractional revivals are expressed in terms of the parameters of the initial packet (the effective number of states and the period of classical motion of the packet center) and the anharmonicity of the molecular system.

The Journal of Chemical PhysicsDec 15, 199419 citations

Superrevivals of Rydberg Wave Packets

Robert Bluhm, V. Alan Kostelecký

The revival structure and evolution of Rydberg wave packets are studied on a time scale much greater than the revival time $t_{\rm rev}$ . We find a new level of revival structure and periodic motion different from that of the known fractional revivals. The new sequence of revivals culminates with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time $t_{\rm rev}$ . We refer to such a revival as a superrevival. We also show that an initial radial wave packet may be described as a type of squeezed state known as a radial squeezed state. Our results apply not only for hydrogenic wave packets, but for wave packets in alkali-metal atoms as well in the context of quantum defect theory.The revival structure and evolution of Rydberg wave packets are studied on a time scale much greater than the revival time $t_{\rm rev}$. We find a new level of revival structure and periodic motion different from that of the known fractional revivals. The new sequence of revivals culminates with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time $t_{\rm rev}$. We refer to such a revival as a superrevival. We also show that an initial radial wave packet may be described as a type of squeezed state known as a radial squeezed state. Our results apply not only for hydrogenic wave packets, but for wave packets in alkali-metal atoms as well in the context of quantum defect theory.

arXiv:quant-ph/9508026Aug 29, 19950 citations

After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time

t_{\rm rev}

, at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than

t_{\rm rev}

. We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period

t_{\rm sr} \gg t_{\rm rev}

with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time

t_{\rm rev}

. Furthermore, at times that are rational fractions of

t_{\rm sr}

, we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time

t_{\rm rev}

. These periodicities indicate a new type of fractional revival, occurring for times much greater than

t_{\rm rev}

. We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.

{\mathit{t}}_{\mathrm{rev}}

, a new sequence of revivals begins. We find that the structure of the new revivals is different from that of the fractional revivals. The new revivals are characterized by periodicities in the motion of the wave packet with periods that are fractions of the revival time scale

{\mathit{t}}_{\mathrm{rev}}

. These long-term periodicities result in the autocorrelation function at times greater than

{\mathit{t}}_{\mathrm{rev}}

having a self-similar resemblance to its structure for times less than

{\mathit{t}}_{\mathrm{rev}}

. The new sequence of revivals culminates with the formation of a single wave packet that more closely resembles the initial wave packet than does the full revival at time

{\mathit{t}}_{\mathrm{rev}}

, i.e., a superrevival forms. Explicit examples of the superrevival structure for both circular and radial wave packets are given. We then study wave packets in alkali-metal atoms, which are typically used in experiments. The behavior of these packets is affected by the presence of quantum defects that modify the hydrogenic revival time scales and periodicities. Their behavior can be treated analytically using supersymmetry-based quantum-defect theory. We illustrate our results for alkali-metal atoms with explicit examples of the revival structure for radial wave packets in rubidium.

The revival structure and evolution of Rydberg wave packets are studied on a time scale much greater than the revival time

t_{\rm rev}

. We find a new level of revival structure and periodic motion different from that of the known fractional revivals. The new sequence of revivals culminates with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time

t_{\rm rev}

. We refer to such a revival as a superrevival. We also show that an initial radial wave packet may be described as a type of squeezed state known as a radial squeezed state. Our results apply not only for hydrogenic wave packets, but for wave packets in alkali-metal atoms as well in the context of quantum defect theory.