## Abstract

We demonstrate that chirped light is effective in shrinking or expanding a class of nonspreading electron eigenstate wave packets in Rydberg atoms.

© Optical Society of America

Modification of the evolution of quantum systems and quantum processes is a step toward deterministic command of highly non-classical configurations of fields [1] and matter [2]. This becomes more attractive if it can be done by external (nonquantum mechanical) forces without destroying quantum properties. An elegant avenue toward such command is through the adiabatic principle.

Standard applications involving adiabatically controlled sweeps of field strength (magnetic or electric) in spin physics and resonance physics have a long history [3]. Control of phase provides another method for manipulating physical systems coherently [4]. Recent discussions of control of chemical systems have been numerous [5, 6, 7, 8]. Key steps toward understanding situations in which several external fields are independently manipulated were taken in the last decade [9, 10]. In this connection we can also mention striking developments associated with electromagnetically induced transparency (EIT) recently [11].

Here we report a new result on classical guidance of quantum mechanical electrons and their spatial structures via control of another external parameter, namely field frequency *ω*. We will describe the consequences of adiabatically chirping the frequency of the same external field that is responsible for the ‘capture’ of an atomic electron in a nonspreading quantum mechanical eigen-packet that has been called a Trojan state [12, 13, 14, 15].

First we review briefly the nature of a Trojan state. The Trojan eigenbasis states apply to a hydrogenic electron in a circularly polarized (CP) radiation field of strength *ε*. They are solutions to the rotating frame Schrödinger equation with the hamiltonian

in the near-circular pendular approximation [10]. Trojan states can be decomposed into superpositions of circular states near to the state satisfying the principal ‘classical’ resonance condition:

They move with uniform angular velocity along the corresponding classical circular Newtonian trajectory. A snapshot of the lowest Trojan state’s probability distribution is shown in Fig. 1 for *n*
_{0} = 24, and Fig. 2 shows the energy spectrum [calculated in a truncated basis of aligned (*l* = *m*) states] of the hamiltonian (1) near *n*
_{0} = 20, *l*
_{0} = 19, as a function of the electric field strength *ε*.

The preparation of either angularly or radially localized classical packet states has been demonstrated [16]. Simultaneous angular and radial localization has not yet been accomplished experimentally. Previous numerical simulations [13] indicate that Trojan states can be obtained from a field-free circular state by rapid adiabatic increase of the CP field strength *ε* if the circular state has the correct angular momentum *l*
_{0} (1/*ω*) ≡ ${n}_{0}^{3}$ ≡ (*l*
_{0} + 1)^{3}). Subsequent changes in the strength of the CP field, within a wide range around the optimum field strength, do not affect the packet radius much because in this range of parameter values the classical nonlinear force balance is basically a ‘Coulomb balance’, while the ‘Volkov’ contribution to the balance is much less significant. However, the packet radius is quite sensitive to the field frequency and we report here the first results of simulations in which adiabatic frequency changes are monitored.

We consider a CP light field **E**(*t*) with a linearly chirped frequency:

where 2*μ* is obviously the chirping rate. As long as the process remains adiabatic we do not expect the choice of the chirping function to have a strong influence on the compression. A set of adiabatic conditions for the instantaneous frequency *ω*(*t*) = *ω*
_{0} + 2/*μt* can be derived. The energies of Trojan states can be found from the Schrödinger equation 𝛨*f*_{j}
= *ε*_{j}*f*_{j}
with the effective hamiltonian

where Δ = 1/${n}_{0}^{3}$ - *ω*), *r*
_{0} = ${n}_{0}^{2}$, and *n*
_{0} is choosen such that ∣Δ∣ is the smallest possible detuning for all *n*
_{0}. We note that the hamiltonian (4) is equivalent to the one for a 1D solid with the lattice potential *εr*
_{0} cos *ϕ*. The eigenfunctions *f*_{j}
are the periodic parts of the Bloch functions, with the wave vector *k* = Δ${r}_{0}^{2}$/3. The tight- and weak-binding approximations known from the theory of condensed systems allow us to derive strong-and weak-field dispersion relations for the energies *ε* in the frequency (Δ) domain. These permit us to estimate adiabaticity conditions which will be fully described elsewhere [17].

In the strong field limit *ε*_{sc}
= *εω*
^{-4/3} > 3/8${n}_{0}^{2}$ we obtain

This leads to the following local condition for adiabatic passage near the resonant value of *n*
_{0} in terms of scaled variables *τ* = *tω*/2*π* and *ω̃* = ${\omega n}_{0}^{3}$

In the weak field limit *ε*_{sc}
< 3/8${n}_{0}^{2}$ the corresponding condition is

which in terms of the scaled variables near the resonant *n*
_{0} gives

One should realize that the term “adiabatic” used here means that the frequency sweep is adiabatic-rapid, since the energy of the Trojan state as a function of the frequency has a series of weakly avoided crossings with weakly interacting levels [17]. The transition through such a crossing is described by Zener theory [13] and if the passage is too slow the packet may be guided to a different dressed non-Trojan state which is not localized, and the compression process will be terminated. On the other hand when the frequency sweep is too fast a similar transition may occur in a non-adiabatic way. The failure of compression for a chirp that is too great is a direct counterpart of the Zener transition which occurs in driven condensed-matter systems when interband transitions abruptly start to dominate over intraband transitions, but this will be described elsewhere.

In order to confirm our predictions of the adiabatic compression of the atom, we have solved the time dependent Schrödinger equation with the hamiltonian (1). We have used the so-called split operator method [18]. This allows one to integrate the Schrödinger equation very efficiently on a spatial lattice using a fast Fourier transform. We performed our integration on a square lattice with 512×512 grid points. We imposed non-absorbing boundaries since ionization is negligible during the chirping process for the field strength used in the simulation. The initial state was chosen to be the Gaussian packet Trojan state for *ω*
_{0} = 1/24^{3} and *ε* = 0.016/24^{3}. Then *ω*(*t*) was changed linearly to the final value 1/18^{3} during a time equal to 60 original cycles while the amplitude of the field was kept steady. Figure 3 shows the wave packet at the begining and end of the compression process.

One can predict the gross behaviour during adiabatic chirping easily. The spatial extent of Trojan packets scales during adiabatic chirping like 1/√*ω*. Therefore, with increasing frequency, the region of electron localization becomes smaller and smaller and the probability density becomes compressed. We display the details of the entire process in the movie linked from Fig. 3, whose first and last frames are given in Fig. 3. The center of the packet is seen to move inward on a helical trajectory. The opposite adiabatic process, with increasing frequency, would lead to expansion of the packet.

Note that this process, together with the adiabatic generation of Trojan states from circular states [13], may also be used for coherent population redistribution from one circular state to another or, more generally, between two states of small and fixed deviation from circularity. In order to achieve this one should adiabatically excite a Trojan packet from one circular state and then adiabatically increase or decrease the frequency of the field, and finally decrease adiabatically the strength of the field to zero, holding the frequency constant.

Finally, regarding experimental realization, we can say that our simulations indicate that our proposal is viable for all *n*
_{0} > 10, and almost any pulse longer than a few cycles will avoid ionization damage of the packet. Thus a wide flexibility in experimental set-ups appears available. Taking into account rapid progress in experimental methods of circular state preparation [19], ionization control of Rydberg states [20], as well as the development of ultra-short pulse wave packet spectroscopy [21], we suspect that the generation and detection of Trojan states may be feasible in the reasonably near future.

## Acnowledgements:

The authors are pleased to thank E.A. Shapiro for a number of conversations on topics related to this note. The work was financially supported by NSF Grants PHY94-15583 and PHY95-11582.

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