Stark state

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

In detection by degenerate superposed states, the detector is a gas of hydrogenic atoms or ions or a layer of donor doped isotropic semiconductor (e.g., GaAs and CdTe). An electrostatic field is applied in the z direction, combining the degenerate hydrogenic states 2y) and 2po) into the Stark states ... 

Fig. 6.2 Stark structure and field ionization properties of the m = 1 states of the H atom. The zero field manifolds are characterized by the principal quantum number n. Quasidiscrete states with lifetime r > 10-6 s (solid line), field broadened states 5 x 10 10 s < x < 5 x 10-6 s (bold line), and field ionized states r < 5 x 10 10 s (broken line). Field broadened Stark states appear approximately only for W > ITC. The saddle point limit Wc = -2 /E is shown by a heavy curve (from ref. 3).

An equivalent form is given by Englefield.11 It is possible to find quite a variety of phases for the transformation coefficients of Eq. (6.18).10-13 The phase depends on the phase conventions established for the spherical and parabolic states. The choice of phase in Eq. (6.18) is for spherical functions with an /, as opposed to (-r)e, dependence at the origin and the spherical harmonic functions of Bethe and Salpeter. A few examples of the spherical harmonics are given in Table 2.2. The parabolic functions are assumed to have an ( n) ml/2 behavior at the origin and an e m angular dependence. This convention means, for example, that for all Stark states with the quantum number m, the transformation coefficient (nni>i2m nmm) is positive. To the extent that the Stark effect is linear, i.e. to the extent that the wavefunctions are the zero field parabolic wavefunctions, the transformation of Eqs. (6.17) and (6.18) allows us to decompose a parabolic Stark state in a field into its zero field components, or vice versa. 

For the extreme red Stark state of high n Eq. (6.38) reduces to Eq. (6.35) since Z2 1. For this Stark state the Stark shift increases the binding energy, and for an m = 0 state the energy is adequately given using the linear Stark effect as... 

Fig. 6.10 Calculated SFI profile for diabatic ionization of the H like m a 3 states. Top, extreme members of the n = 31, m = 3 Stark manifold. The crosses represent the points at which each m a 3 Stark state achieves an ionization rate of 10 s I. Bottom, calculated SFI profile for diabatic ionization of a mixture containing equal numbers of atoms in each m a 3 Stark level for n = 31 at a slew rate of 109 V/cm s. (from ref. 26).

The second form of ionization is similar to autoionization.31 In a nonhydrogenic atom is not a good quantum number and bound states of high nt are coupled to Stark continua of low n j. This form of ionization applies to those states other than the reddest Stark states. The extreme red Stark states have nx = 0 and ionize, as do the red H states, at the classical ionization limit given by Eq. (6.35), modified by Eq. (6.37) for m 0 states. This point has been demonstrated explicitly by Littman et al,32 who measured the time resolved ionization of Na m = 2 states subsequent to pulsed laser excitation. Their measured rates are in good agreement with the rates obtained by extrapolation of the rates of Bailey et al.20 shown in Fig. 6.8. 

Since Vd(r) is only nonzero near r = 0 the matrix element of Eq. (6.51) reflects the amplitude of the wavefunction of the continuum wave at r 0. Specifically, the squared matrix element is proportional to C, the density of states defined earlier and plotted in Fig. 6.18. From the plots of Fig. 6.18 it is apparent that the ionization rate into a continuum substantially above threshold is energy independent. However, as shown in Fig. 6.18, there is often a peak in the density of continuum states just at the threshold for ionization, substantially increasing the ionization rate for a degenerate blue state of larger This phenomenon has been observed experimentally by Littman et al.32 who observed a local increase in the ionization rate of the Na (12,6,3,2) Stark state where it crosses the 14,0,11,2 state, at a field of 15.6 kV/cm, as shown by Fig. 6.19. In this field the energy of the... 

Observations to date indicate that Na m = 0,1, and 2 Stark states of n 20 above the classical ionization limit have widths of 30-90 GHz, 1-3 GHz, and 100 MHz,34 in rough accord with Eq. (6.55). 

Because it can be efficient and selective, field ionization of Rydberg atoms has become a widely used tool.1 Often the field is applied as a pulse, with rise times of nanoseconds to microseconds,2"4 and to realize the potential of field ionization we need to understand what happens to the atoms as the pulsed field rises from zero to the ionizing field. In the previous chapter we discussed the ionization rates of Stark states in static fields. In this chapter we consider how atoms evolve from zero field states to the high field Stark states during the pulse. Since the evolution depends on the risetime of the pulse, it is impossible to describe all possible outcomes. Instead, we describe a few practically important limiting cases. 

In Fig. 7.2 the ionization fields shown correspond to an ionization rate of 106 s 1. The fields for the extreme blue and red states are taken from the calculations of Bailey et al,5 and the fields for intermediate states are simply interpolated. We have also not shown any other n states since the Stark states of different n, which cross each other, have different values of nu and do not interact, as described in Chapter 6. 

What conditions must be fulfilled for the ionization process to occur in the adiabatic fashion we have just described First, the transition from the zero field ntm states to the intermediate field Stark states must be adiabatic. Second, the traversal of the avoided crossings in the strong field regime, E > l/3n5 must be adiabatic as well. Finally, ionization only occurs at E > W2/4 if the ionization rate exceeds the inverse of the time the pulse spends with E > W2/4. 

The passage from the zero field nd state to a single Stark state is evidently adiabatic. A hydrogen nd /n =2 state would pass diabatically from zero field to many nnjn22 Stark states and exhibit multiple ionization fields. 

Exposure of blue Stark states to rapidly rising fields also results in diabatic, or H like, ionization, at fields far above the classical field for ionization. This point has been demonstrated by Neijzen and Donszelmann12 using high lying, n = 66, states of In and by Rolfes et al.13 using Na atoms of n = 34 and m = 2. 

In light alkali atoms, Li and Na, the fine structure splitting of a low state is typically much larger than the radiative decay rate but smaller than the interval between adjacent states. In zero field the eigenstates are the spin orbit coupled tsjnij states in which and s are coupled. However, in very small fields and s are decoupled, and the spin may be ignored. From this point on all our previous analysis of spinless atoms applies. How the passage from the coupled to the uncoupled states occurs depends on how rapidly the field is applied. It is typically a simple variant of the question of how the m states evolve into Stark states. When... 

As an example, we consider first the excitation of the n = 15 Stark states from the ground state in a field too low to cause significant ionization of n = 15 states. From Chapter 6 we know the energies of the Stark states, and we now wish to calculate the relative intensities of the transitions to these levels. One approach is to calculate them in parabolic coordinates. This approach is an efficient way to proceed for the excitation of H however, it is not easily generalized to other atoms. Another, which we adopt here, is to express the n = 15 nn n2m Stark states in terms of their nfm components using Eqs. (6.18) or (6.19) and express the transition dipole moments in terms of the more familiar spherical nim states.1,2... 

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