Stark States Research Articles

Measurement of the three-dimensional velocity distribution of Stark-decelerated Rydberg atoms

The full three-dimensional velocity distributions of decelerated and accelerated particles in a Stark decelerator for Rydberg atoms and molecules have been measured. In the experiment, argon atoms in a supersonic beam are excited to low-field and high-field seeking Stark states with principal quantum number in the range n=15 to 25 and are decelerated in a 3 mm long decelerator consisting of four electrodes on which time-dependent voltages are applied. The time dependence of the resulting inhomogeneous electric field is chosen such that the decelerating force acting on the high-field seeking states is maximized at each point along the trajectories. The three-dimensional velocity distribution of the atoms before and after the deceleration is determined by measuring times of flight and two-dimensional images of the atomic cloud on the detector. Under optimal deceleration conditions, the decrease in kinetic energy in the longitudinal dimension amounts to 1.0×10-21 J and the increase in mean kinetic energy in the transverse dimensions is only 1.0×10-23 J. The corresponding temperatures of 100 mK and 300 mK in the two transverse dimensions are sufficiently low that trapping can be envisaged. The possibility of focusing a Rydberg atom beam is demonstrated experimentally.

Ionization of Rydberg atoms colliding with a metal surface

We report on a theoretical study of the ionization process of ${\mathrm{Xe}}^{*}$ Rydberg atoms colliding with a metal surface, in the presence of an external electric field. The evolution of the ${\mathrm{Xe}}^{*}$ outer electron is studied by a wave packet propagation approach, allowing to include all dynamical aspects of the collision, in particular nonadiabatic inter-Rydberg transitions. We investigate how the different ${\mathrm{Xe}}^{*}$ Stark states formed in the external field couple together and ionize on the surface and how the different polarizations of the electronic cloud in the ${\mathrm{Xe}}^{*}$ states are reflected in their ionization properties. We show that the presence of the external electric field can significantly perturb the dynamics of the ionization process. Our results account for recent results from Dunning et al. [Nucl. Inst. Meth. B 203, 69 (2003)]. In particular, it is explained how the external electric field present in the experimental procedure of Dunning et al. leads to the apparent absence of a polarization effect in the ionization process.

Local density of states of dc-biased superlattices with time-dependent imperfection

The single-particle Green's function for a dc-biased superlattices with single impurity potential varying harmonically with time has been obtained in the framework of U(t,t') method and Floquet–Green's function. The calculation of the local density of states shows that new states will emerge between the resonant Wannier–Stark states as a result of the intervention of time-dependent impurity potential and the increase in electric field strength of impurity will result in the growing of the number of new states between the gaps of neighbouring Stark ladders.

Homogeneous line width of rare-earth-doped glasses for levels in a Stark level ladder: A new simple rule

Concentrating on higher Stark states in multiplet ladders of lanthanide-doped glasses, we show that at 10 K, the homogeneous widths along Stark ladders vary quasi-linearly with the energy position in the ladder. Assuming that the mechanisms creating the homogeneous widths are first-order processes, either direct processes or two-phonon non-radiative transitions, the observed behavior is simply explained by the fact that energy separations between Stark levels for usual glasses are less, i.e., not far from kT D, the Debye energy. This in turn provides quasi-equal phonon transition probabilities between individual Stark states. A simple theoretical relationship is derived and compared with experiments.

Stark shift of the A2Π1∕2 state in YbF174

We have measured the Stark shift of the A2Π1∕2–X2Σ+ transition in YbF. We use a molecular beam triple resonance method, with two laser transitions acting as pump and probe, assisted by an rf transition that tags a single hyperfine transition of the X state. After subtracting the known ground state Stark shift, we obtain a value of 70.3(1.5)Hz∕(V∕cm)2 for the static electric polarizability of the state A2Π1∕2(J=12,f) by fitting our data to a purely quadratic Stark shift in the interval 0–5kV∕cm. A more exact analysis that does not assume a perfectly quadratic Stark effect yields the value μe=2.48(3)D for the electric dipole moment of the A2Π1∕2(v=0) state.

Ionization thresholds and positions of avoided crossing of potassium Stark Rydberg states: a Stark-adapted quantum defect orbital treatment

We have calculated the positions of the avoided level crossings between (n+2)s, np states and nd, k Stark states in the Rydberg Stark states of the potassium atom with principal quantum number n comprised between 12 and 17. We have also studied the adiabatic electric field ionization thresholds for the above Rydberg states. Both the ionization thresholds and the positions of avoided crossings have been calculated using the recently developed Stark-adapted quantum defect orbital (SQDO) formalism. The presently reported values appear to be in very good agreement with the available theoretical and experimental data.

On the electrostatic deceleration of argon atoms in high Rydberg states by time-dependent inhomogeneous electric fields

Argon atoms in a pulsed supersonic expansion are prepared in selected Stark components of Rydberg states with effective principal quantum number in the range n* = 15–25. When traversing regions of inhomogeneous electric fields, these atoms get accelerated or decelerated depending on whether the Stark states are low- or high-field seeking states. Using a compact electrode design, which enables the application of highly inhomogeneous and time-dependent electric fields, the Rydberg atoms experience kinetic energy changes of up to 1.2 × 10−21 J (i.e. 60 cm−1 in spectroscopic units) in a single acceleration/deceleration stage of 3 mm length. The resulting differences in the velocities of the low- and high-field seeking states are large enough that the corresponding distributions of times of flight to the Rydberg particle detector are fully separated. As a result, efficient spectral searches of the Rydberg states best suited for acceleration/deceleration experiments are possible. Numerical simulations of the particle trajectories are used to analyse the time-of-flight distributions and to optimize the time dependence of the inhomogeneous electric fields. The decay of the Rydberg states by fluorescence, collisions and transitions induced by black-body radiation takes place on a timescale long enough not to interfere significantly with the deceleration during the first ∼5 µs.

Heavy Rydberg states

This paper discusses the nature of heavy Rydberg states, i.e. quantum states in molecular systems that are bound by the almost pure Coulomb potential between pairs of ions. A theoretical framework is developed in terms of mass–scaling laws for heavy Rydberg systems of certain reduced mass, so that the physics of electronic Rydberg states can be straightforwardly applied to heavier ion–pair systems, or any other system bound by a 1/r potential. The general description of such quantum systems is supported by an experimental investigation of the energy region near the H+H− ion–pair dissociation limit, using a 1 XUV + 1 UV laser excitation scheme. Such a scheme allows for preparation of a single intermediate rovibrational quantum state, from which the ion–pair threshold region can be explored with a narrowband Fourier–transform limited laser. Field–induced lowering of the H+H− dissociation limit was observed in the presence of an electric field. Using a combination of DC and pulsed electric fields two–photon induced threshold ion–pair production spectroscopy (TIPPS) was performed for a variety of field strengths. Below the field–dissociation limit coherent wave packets of bound heavy Rydberg states are excited in an electric field. The coherent evolution of the Stark states, giving rise to oscillations in angular momentum space, can be quantitatively understood in terms of the linear Stark effect in the hydrogen atom, where the light electron is replaced by the heavier H− particle. The dynamics of wave packets of heavy Rydberg states is slower than in ordinary electronic Rydberg states, according to their larger mass. For a wide variety of binding energy and external electric field, the observed oscillation frequencies of the Stark wave packets match this model. An important parameter describing the specific properties of the H+H− Rydberg states is the scaled lifetime, which may be considered as a material constant, and it is determined atτ= (5.8±2.0)×10−21 s n 4.

Conversion factors and widths of avoided crossings in Rydberg Stark states of K

In the experiment concerning the level anticrossing spectroscopy, the experimental widths are measured in the unit of V/cm, but the theoretical ones are presented in the unit of 1/cm. To facilitate the comparison of the experimental data with theoretical values, it is necessary to know the conversion factors that give the relation between the units of V/cm and 1/cm. In this paper, the results are presented for the conversion factors of Rydberg potassium based on the theory of the anticrossing signals and a kind of potential model, which make it possible to directly compare the theoretical widths with experimental ones. The method described here for obtaining the conversion factors of Rydberg potassium is suitable for any alkaline metal atom.

An accurate quantum expression for radiative transitions between the Stark levels of nearby Rydberg states

The paper presents a new alternative exact quantum expression of the x-component of the dipole matrix element between the Stark states of a hydrogen atom in terms of the Jacobi polynomials by transforming the hypergeometric functions appearing in the standard quantum formula. The new quantum formula readily leads to analytic study and numerical computation for such large values of the parabolic quantum numbers for which difficulties had earlier been encountered. The paper goes on to derive an approximate but simple quantum formula of the dipole matrix element in terms of the ordinary Bessel functions and demonstrates its remarkable accuracy for transitions ranging from that between the Stark levels of the lowest lying states to that between the Stark levels of nearby Rydberg states. The formula enables accurate numerical computation to be performed over an extended range of large parabolic quantum numbers that had earlier defied evaluation. The expressions given in this paper in essence solve the problem of determination of analytic behaviour and numerical computation of the dipole matrix element for transitions between the Stark levels of nearby Rydberg states. The paper also presents, for the first time, a derivation of the formula of the correspondence principle method from the quantum expression without appealing to any classical or semiclassical argument, and clarifies the conditions of its applicability.

Controlling the motion of hydrogen molecules: design of a two-dipole Rydberg decelerator

Hydrogen molecules are excited by 1 + 1′ (VUV + UV) laser excitation to Rydberg states ( n = 15–20) in the inhomogeneous field generated by an electrostatic dipole. High-field or low-field seeking Stark states are selected spectroscopically and experience a force either towards or away from the dipole, respectively. Deflection of the molecular beam is observed if the dipole is oriented parallel to the beam direction, whereas acceleration or deceleration is observed for a perpendicular orientation. The measured deflections and acceleration/deceleration are consistent with trajectory simulations based on the known field and Stark map. A two-dipole device is proposed that creates a travelling potential minimum and would enable deceleration of an H 2 beam seeded in argon to a mean velocity of zero. Simulations are presented of its characteristics.

Absence of Wannier–Stark ladders in finite superlattices

Precise theoretical transmission coefficients calculations provide an extremely good account of individual Wannier–Stark states splitting and localization effects, resolved recently in a highly accurate experiment for four and five period finite superlattices. At the same time, we show that the so-called Wannier–Stark ladders, predicted for nfinite systems are absent for finite superlattices. The observed energy levels are nothing else than the intra miniband energy levels.

Long-range interaction between polar Rydberg atoms

The most attractive and the most repulsive potential-energy curves for interaction between two Rydberg atoms in a broad superposition of internal angular momentum states are studied. The extreme Stark states have the largest dipole moments and provide the dominant contribution to the interaction which is then expressed as a long-range expansion involving the permanent multipole moments Qj of each polar atom. Analytical expressions are obtained for all Qj associated with principal quantum number n of H(n) and permit the long-range expansion for the H(n)–H(n′) first-order interaction to be explicitly expressed analytically in terms of n, n′ and internuclear distance R. Possible quasi-molecular formation is investigated. Direct calculations show that the H(n = 2)–H(n′ = 2) interaction is capable of supporting 47 bound vibrational levels. As n increases, the long-range interaction becomes increasingly attractive so that molecular formation at large internuclear distances is expected to be scarcely possible for these extreme Stark levels.

Probabilities of Radiative Transitions between Stark States in Orthohelium

The wavefunctions, matrix elements, and probabilities of radiative transitions between Stark sub-levels of atomic multiplets are calculated as a function of the strength of a dc electric field. The general expressions for the wavefunction of a multiplet state in a field obtained by perturbation theory for close-lying levels with the use of completely reduced Green’s function allow one to determine the field dependence of both the dipole-allowed and dipole-forbidden radiation amplitudes. A decomposition of the second-order amplitude for the transition between fine-structure sublevels of two levels of equal parity into irreducible components is obtained. Numerical calculations of the probabilities of radiative transitions between triplet states of helium show the possibility of experimental observation of the emergence and vanishing of Stark lines of radiative transitions in the vicinity of anticrossing fields.

The experimental study of Ba 6p jnk | M| = 0, 1 autoionizing series

The spectra of the Ba 6p nk autoionizing Stark states with | M| = 0, 1, converging to the 6p 1/2 + and 6p 3/2 + ionization thresholds, are measured as a function of the electric field strength F. Several 6p j nk Stark manifolds with n = 13–15 have been systematically studied in order to explore their characteristics of configuration interaction. Experimental results are analyzed by fitting them to the Lorentzian profile, from which the positions and widths are determined. Different spectroscopic properties between the Ba 6p 1/2 nk and 6p 3/2 nk autoionizing Stark states are investigated. Comparison between the Ba 6p j nk autoionizing Stark states with | M| = 0 and those with | M| = 1 are made.

Deflection and deceleration of hydrogen Rydberg molecules in inhomogeneous electric fields.

Hydrogen molecules are excited in a molecular beam to Rydberg states around n=17-18 and are exposed to the inhomogeneous electric field of an electric dipole. The large dipole moment produced in the selected Stark eigenstates leads to strong forces on the H2 molecules in the inhomogeneous electric field. The trajectories of the molecules are monitored using ion-imaging and time of flight measurements. With the dipole rods mounted parallel to the beam direction, the high-field-seeking and low-field-seeking Stark states are deflected towards and away from the dipole, respectively. The magnitude of the deflection is measured as a function of the parabolic quantum number k and of the duration of the applied field. It is also shown that a large deflection is observed when populating the (17d2)1 state at zero field and switching the dipole field on after a delay. With the dipole mounted perpendicular to the beam direction, the molecules are either accelerated or decelerated as they move towards the dipole. The Rydberg states are found to survive for over 100 micros after the dipole field is switched off before being ionized at the detector and the time of flight is measured. A greater percentage change in kinetic energy is achieved by initial seeding of the beam in helium or neon followed by inhomogeneous field deceleration/acceleration. Molecular dynamics trajectory simulations are presented highlighting the extent to which the trajectories can be predicted based on the known Stark map. The spectroscopy of the populated states is discussed in detail and it is established that the N+=2, J=1, MJ=0 states populated here have a special stability with respect to decay by predissociation.

Nonhydrogenic effects in the deceleration of Rydberg atoms in inhomogeneous electric fields.

Argon atoms in Stark states at n approximately 25 have been decelerated and accelerated in inhomogeneous electric fields. The acceleration and deceleration behavior can be understood only by considering the adiabatic Landau-Zener dynamics that take place at the avoided crossings between the Stark states and the limited fluorescence lifetimes of approximately 10 micros.

Doppler-detuned CW excitation of then = 4 Stark manifold of atomic hydrogen

The excitation of metastable hydrogen towards then = 4 Stark manifold is investigated both theoretically and experimentally. The Stark Hamiltonian isdiagonalized to obtain the energy levels, oscillator strengths and relaxation rates of then = 2 and n = 4 Stark states. In a fast beam experiment, the transitions are excited by the blue line of aCW argon ion laser with appropriate Doppler detuning, and the measured transitionfrequencies agree well with the calculated values. The population of the 4s state, asmeasured by its fluorescence, is maximum for an electric field strength between 50 and100 V cm−1, and a laser polarization perpendicular to the static field.

Ionization of Stark states with half-cycle pulses: Interference effects in the continuum

We study the ionization of extreme Stark states in sodium by THz electromagnetic half-cycle pulses. The results of our full-quantum calculations reveal the presence of an oscillatory pattern in the ionization spectra of extreme red (downhill) states that have been kicked away from the ion core (downhill with respect to the potential imposed by the Stark field). We find no oscillations in the spectra of extreme blue (uphill) states that have been kicked towards the ion core (also downhill with respect to the Stark potential). The oscillatory pattern in the red state ionization spectra is explained with a one-dimensional semiclassical model in terms of interferences between two classical paths. This model also predicts that the blue state ionization spectra should also show oscillatory behavior. The absence of the oscillations in the full calculations can be regarded as a breakdown of the one-dimensional model in representing the ionization dynamics of these states. We find that the one-dimensional model fails when the duration of the THz pulse is comparable to the classical Kepler orbit time.

Study of Ba 6 snk Rydberg states in electric field

The autoionization detection is employed to measure the Stark spectra of Ba 6 snk Stark states with n from 13 to 15. The Stark maps with M=0 and | M|=1 can be selectively excited by two lasers with different combinations of polarization under different strengths of electric fields. General features of Ba Stark states observed in the experiment are discussed in detail, and the comparisons between M=0 and | M|=1 cases are made. Theoretical calculations based on the perturbation theory are performed to simulate the corresponding experiments, results of which are satisfactory.