## Abstract

We investigate the possibility that Stark wave packets can be used as a source of shaped terahertz radiation. Calculations for the sodium atom reveal that the frequency and intensity of the THz emission can be controlled over a broad range by varying the parameters of the excitation pulse.

© Optical Society of America

## 1. Introduction

When a static DC field is applied to an atom, the orbital angular momentum degeneracy is lifted, and the energy levels are split by the Stark effect. For small DC fields, the Stark shift is linear, and the energy levels are given by ${E}_{\mathit{nk}}=-\frac{1}{2}{n}^{2}+\frac{3}{2}{E}_{0}k,$, where *n* is the principle quantum number, *E*
_{0} is the magnitude of the DC field, and *k* = *n*
_{1} - *n*
_{2}, where *n*
_{1} and *n*
_{2} are parabolic quantum numbers (non-negative integers). The parabolic quantum numbers are related to the principle quantum number by *n* = *n*
_{1} + *n*
_{2} + ∣*m*∣ + 1. We use atomic units (ħ = *e* = *m* = 1) throughout. This simple linear dependence of the energy versus static field breaks down for field strengths larger than *E*
_{0} ≈ 1/3*n*
^{5}, when adjacent Rydberg manifolds exhibit real or avoided crossings.^{1} Since ℓ is not a good quantum number, the conventional dipole selection rules no longer apply, and a laser pulse with sufficient frequency bandwidth excites a coherent superposition of ∣*k*⟩ states, which creates a time-dependent wave packet.^{2,3} The time evolution of this wave packet is quite complex when viewed in laboratory coordinates. Fig. 1 shows a snapshot of the probability distribution for a Stark wave packet in Na, in which the excitation is from the 3*p* state to the vicinity of *n* = 15, with a static DC field of 400 V/cm.

The complex spatial form of the Stark wave packets, along with the extreme mixing of the oscillator strength, results in a complicated time-dependent dipole moment, which causes a burst of coherent photon emission. For typical experimental conditions, the emission is in the infrared (THz) regime. In this paper we investigate the possibility of using Stark wave packets as a tunable source of THz radiation. Though our calculations are at a preliminary stage, we demonstrate that, at least in principle, features of the emitted spectrum can be varied over a broad range by altering the amplitude and phase of the excitation pulse. We also discuss the possibility that the THz spectrum can be shaped, using various high-order optimization procedures. Finally, we speculate on the possibility that shaped Stark wave packets can be used to control electronic motion in atoms.

## 2. Method and results

In this letter we employ a simple model for calculating the THz radiation from a Stark wave packet (SWP). For concreteness all of the calculations presented here are performed for sodium. The initial state is assumed to be a low-lying *p* state which can be resonantly excited from the ground (3*s*) state. We use an ℓ-dependent pseudopotential,^{4} and a nonlinear finite difference grid for diagonalizing the Stark Hamiltonian. The excitation pulse is assumed to be linearly polarized along the *ẑ* direction. We calculate the complex, time-dependent amplitudes of the Stark states, ∣*k*⟩, using the rotating wave approximation in the weak response (perturbative) limit. Under these approximations the coherence of the laser field is directly transfered to the electronic wave function. The time-dependent polarization (dipole moment) induced by the excitation pulse has two contributions. The first includes terms that originate from the coherence between the ground and excited states and leads to radiation at or near the optical excitation frequency. The second contribution, which gives rise to radiation at THz frequencies, is due to the coherence between excited states and is given by

where Ω
_{ki}
is the transition moment between the initial state and the *k*th Stark state, *ω*^{k}
is the detuning of the Stark state from the central frequency of the excitation laser, *ϵ̃*(*ω*_{k}
) is the Fourier component of the laser envelope at frequency *ω*_{k}
, *Z*_{k,k′}
is the dipole matrix element between two Stark states, and *ω*_{k,k′}
= *ω*_{k}
- *ω*_{k′}
. The radiated electric field *ε*(*t*) is proportional to the acceleration *a*(*t*) = *∂*
^{2}⟨*z*⟩/*∂*
*t*
^{2}. Note that this expression depends on a product of two electric fields, and thus THz emission is a second-order (*χ*
^{2}), nonlinear process, involving difference frequency mixing. A proper treatment of the emitted THz signal should take into account not only the polarization induced by the excitation laser but also the propagation of the THz field in the medium (*i.e.*, phase matching). We shall not pursue this point further in this preliminary study except to note that the induced dipoles are all created in response to the same excitation pulse which leads, in the limit of low density, to proper phase relationships among the radiating dipoles.

As an example, consider the electric field produced by the wave packet in Fig. 1. Excitation is via a 2 ps, transform-limited pulse centered at 485 cm^{-1} below the zero field ionization threshold, which is near the center of the *n* = 15 manifold.

Fig. 2 shows the time-dependent dipole induced by the pulse, and the resulting THz emission. The time between the bursts of radiation is set by the mean frequency difference in the manifold, which is controlled, in turn, by the static electric field. Evidently the resulting THz spectrum depends sensitively on the static field strength and the initial state, as well as the Fourier components of the laser field at the excitation energies. Notice also that the magnitude of the dipole moment is enormous.

For a laser pulse with a bandwidth that overlaps *N* states there are on the order of *N*
^{2} frequency differences, but only of order *N* adjustable parameters. Optimizing the THz spectrum to meet a desired control objective is clearly an over-specified problem. Future work will incorporate advanced optimization routines to analyze which aspects of the system are most sensitive, and the robustness of various possible control schemes. For the purposes of this work, we illustrate the range of variation that can be achieved in the THz spectrum by varying the spectral intensity, ∣*a*(*ω*)∣^{2}. Under the same assumptions as discussed for Eq. 1,

where *P*_{k}
is the population in the *k*th state. The populations, and hence the frequency components of the emitted radiation, are not sensitive to the coherence of the excitation field. Note that in the weak response limit, ∣*a*(*ω*)∣^{2} scales as the square of the excitation laser intensity.

In Fig. 3 we compare the spectra for two wave packets composed of ∣*k*⟩ states near the *n* = 15 manifold in sodium. The first spectrum is for the case illustrated in Fig. 2 and exhibits a band of frequencies between 10–15 cm^{-1}. The second spectrum is calculated using a slightly different central frequency and a somewhat broader bandwidth. It is dominated by a single line at about 30 cm^{-1}, or 1 THz. The motion of this wave packet in *z* is almost purely sinusoidal. Notice also that this peak is approximately 200 times as intense as the peaks in the first spectrum. An analysis of the wave packets
that create these spectra reveals that the first wave packet has several states populated, and thus produces several THz peaks. In the second wave packet, the central frequency, -461 cm^{-1} lies between the 16*p* and 17*s* states which have a large dipole coupling (*Z*_{k,k′}
≈ 150 au), and are separated by 30 cm^{-1}. Although the excitation is from a low-lying *p* state, coupling still exists to the (nominally) 16 *p* state, due to the mixing of the ℓ = 2 oscillator strength induced by the static field (in fact, Ω_{16p,3p}* ≈ Ω _{17s,3p}).*

*Finally, we give a hint of the dependence of the THz spectra on the static field strength. Fig. 4 compares the emitted intensity in a 2 cm ^{-1} band of frequencies centered at 20 cm-1 as a function of the static field strength and the central excitation frequency. Once again, small variations in the parameters of the excitation pulse are sufficient to cause large variations in the spectra.*

*3. Discussion and conclusions*

*As a source of tunable THz radiation, SWPs have several desirable characteristics that are not easily duplicated in other sources. For example, one of the factors influencing the coherence time of the radiation is the time scale for dephasing collisions. This can be modified experimentally by choosing both the atomic density and the excitation energy appropriately. Excitation to a lower Stark manifold, for example, populates smaller states which would be less susceptible to dephasing. In addition, the bandwidth of an ultrafast laser pulse encompasses numerous Stark states, many with large state-to-state dipole couplings. This provides a large number of frequency components for use in synthesizing a desired spectrum, and enhances the flexibility in shaping the output. If desired, the static field and laser parameters can be tuned to populate a small number of states, and enhance a single difference frequency. The intrinsic strength of the emission is also quite large, due to the spatial extent of the Rydberg states. For instance,
the intensity at frequency ω_{k,k′}
scales as I_{k,k′}
∝ P_{k}P_{k′}
${Z}_{k\mathit{,}k\mathit{\prime}}^{2}$
. If the population of the individual Rydberg states is of order 1% and Z_{k,k′}
≈ 100 or more, then I_{k,k′}
is of order 1 in atomic units. The emission is limited by the relatively small density of atoms that can be used experimentally, again due to the large size of the Rydberg states.*

*We note that THz emission has been observed from quantum wells, at cryogenic temperatures and low carrier densities. ^{5} The ability to generate and detect THz radiation in semiconductor heterostructures has advanced rapidly over the past few years, and applications are now starting to appear.^{6} There have also been proposals and experiments to use optical pulse shaping to control the charge oscillations in quantum wells, and the resulting THz emission.^{7,8} This encourages us that THz radiation from SWPs will also be observable experimentally.*

*As a final point, we note that just as the THz radiation from a quantum well can be studied to learn about the dynamics of charge carriers in semiconductor heterostructures, ^{9} measuring the amplitude and phase of the SWP emission at many frequencies will allow us to infer details about the motion of the electron. By specifying the amplitude and phase of the various frequency components, we are in effect specifying a trajectory for the electron. Such techniques might be used to control the SWP along a desired trajectory such as a simple periodic orbit, to alter the reactivity of an atom in a scattering experiment, or to control chemical bonding.*

*References*

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