## Abstract

Electrons in atoms and molecules are versatile physical systems allowing a vast range of light–matter interactions. Spontaneous emission, which appears in a wide variety of applications, depends crucially on the bound electron energy levels. The discrete nature of these electron energy levels and the ionization threshold constrain the energy scale of all light–matter interactions involving bound electrons. To bypass these constraints, we take ideas from optical and electronic beam shaping and propose creating new electron states as superpositions of extended states above the ionization threshold. We show that such superpositions enable the control of spontaneous emission with tunable spectra in the eV–keV range. We find that the specific shaping lengthens the diffraction and radiative lifetimes of the wavepackets in exchange for increasing their spatial spreads. Our approach could have applications toward developing novel kinds of light emitters at hard-to-access spectral ranges.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The rich physics of bound electrons in atoms and molecules is typically limited by the discrete nature of the energy spectrum and by the ionization energy threshold. This is why many phenomena considered in atomic physics are usually confined to the IR–UV spectral range. The upper limit is set by the typical ionization threshold of the outer valence electrons. Going beyond these energy ranges usually involves free electrons that can be manipulated with a strong electromagnetic field [1,2]. Another promising approach to access light–matter interactions at the UV frequencies and above is by high-harmonic generation, also using strong fields, yet applied on initially bound electrons [3–8]. In either case, the interaction has to involve extended electron states out of equilibrium, manipulated by the strong fields. It seems inevitable then that accessing light–matter interactions at such high frequencies always involves states that are not bound in space and are therefore not considered relevant for atomic physics phenomena.

One could conceive of a completely different way to deal with this problem by engineering the potential to allow for the existence of bound states in the continuum (BICs) [9]. Von Neumann and Wigner were the first to introduce such a construction, defying the conventional wisdom that bound states must be spectrally separated from the extended states [10]. However, the kinds of potentials that support BICs have to be specially designed, while the potentials of atoms and molecules generally cannot be designed [11–14].

We will consider another approach by which electron states in the continuum can behave as bound states: shaping the electron as a superposition of extended states to create a localized wavepacket. For example, in 1979, Berry and Balázs proposed a solution of the free-space Schrödinger equation that appears to be localized in 1D and whose shape remains time invariant as with bound electron states (while it also exhibits self-acceleration) [15]. In recent years, this idea has been extensively explored in the optics community, with paraxial and non-paraxial optical beams [16–19].

More generally, these kinds of optical beams, such as Bessel [20–23] and Airy [16,17] beams, are propagation-invariant wave functions and have the intriguing property of self-healing—restoring their original shape after encountering an obstacle [24]. Such beams can also remain propagation invariant in curved spaces [25,26] and in nonlinear media [27,28]. While the above wavepackets are localized mostly only in 2D, the concept of optical beam shaping can also be applied to create “light-bullets,” which are localized in 3D [29,30]. Importantly, due to the mathematical analogies between optical wavepackets and electron wavepackets, similar concepts are applicable to electrons [31–37].

However, the well-known problem with all of the above time-invariant or propagation-invariant wavepackets is that their probability density is not square integrable (hence, without a physical interpretation as a probability density). To circumvent this, one can truncate the wavepackets at a finite distance. In this case, there is no exact time invariance or propagation invariance, but the non-diffracting properties are still present for a finite time and distance, which can be long enough for the desired interaction in experiments [17,32,38–40]. Another approach that makes the wavepacket square integrable is shaping a superposition of a range of energies/frequencies, which can create a localized wavepacket, in exchange for limiting its range/duration for which it is non-diffracting [30]. Therefore, wavepacket shaping in space and time offers localized and long-lived electron wavepackets in the continuum of energy levels, created from superpositions of extended states.

With the above wavepackets in mind, we now ask: can shaped electron wavepackets mimic optical phenomena of conventional bound electron states in atoms, such as electron transitions by light emission/absorption? Can shaped electrons give us access to light–matter interactions beyond the ionization threshold that limits bound electron systems? For example, can one create engineered spontaneous emission dynamics (engineered rates, engineered optical spectra, etc.) by shaping a superposition of extended states in the energy continuum to have a tailored spatial profile and energy spectrum?

Here, we propose shape-invariant wavepackets in the presence of a general potential (e.g., the Coulomb potential) that simultaneously suppress their own diffraction, while enabling access to a customizable spectrum of transitions, ranging from the visible to the hard x-ray, via radiative decay to bound states. We develop the analytic tools to maintain the shape invariance of wavepackets and the analytic tools to calculate the radiative transitions of such states into bound states of the potential. Our methods can be extended to a variety of potentials, including the transitions of free electrons illuminated by general time-dependent fields. Specifically for the Coulomb potential, we derive the shape-invariant electron wavepackets and study their dynamics. For example, we find that a shape-invariant wavepacket also affects the behavior of the electron in the Coulomb potential. We show that the presence of a Coulomb potential changes the physics of the system drastically relative to free electrons by allowing shaped wavepackets to decay to bound states through radiative capture. We monitor the “competition” between spontaneous emission and diffraction by developing a quantum electrodynamics (QED) formalism that quantifies the rate of decay by the excited wavepacket, studying specifically how the wavepacket shape alters the emitted radiation into the far field. We find that in all these cases, the wavepacket’s lifetime is limited by the diffraction dynamics of its wavepacket. Even though, in general, the electron wavepackets we consider are diffracting and subject to spontaneous emission, we find parameters that can suppress both effects significantly. Hence, we concisely refer to the shaped wavepackets, which are almost propagation invariant and time invariant, as being *quasi-shape invariant*.

## 2. RESULTS

#### A. Constructing Quasi-Shape-Invariant Wavepackets

To illustrate the concept of quasi-shape-invariant quantum wavepackets above the ionization threshold, we begin from the textbook example of the hydrogen atom, consisting of the Schrödinger equation with the Coulomb potential $V(r)=-{e}^{2}/4\pi {\u03f5}_{0}r$. The hydrogen atom is one of the most famous problems in quantum mechanics; its electron bound states are well studied, and analytic expressions for the extended states exist in the literature [41]. We introduce a dimensionless parameter $x$, defined by $x=(2/{a}_{0})r$ with spherical radius $r$ and the Bohr radius ${a}_{0}$. Likewise, let $\kappa =k{a}_{0}$ be the dimensionless parameter from momentum $k$.

Figure 1 presents the shaping of the electron wavepacket that is created from superpositions of extended eigenstates, which are called the Whittaker functions ${w}_{\kappa}(x,t)$ and can be found in [42]. These eigenstates are of the form

#### B. Spatial and Temporal Dynamics

Having shaped the wavepackets, we now characterize their spatial and temporal dynamics. We do this by defining a spatial spread $\mathrm{\Delta}r$, given by a standard deviation,

#### C. Profile of the Quasi-Shape-Invariant Wavepackets

We now study the profile of the Whittaker wavepacket and how it evolves in time in a quasi-shape-invariant manner. Just as wavepacket shaping is known to extend the lifetime of Dirac fermions [44], wavepacket shaping extends both the diffraction and radiative lifetimes of a Schrödinger electron wavepacket, as the Whittaker wavepacket considered here. For example, Fig. 3(a) presents the Whittaker wavepacket’s radial wavefunction at time zero, marking its nodal structure that highlights the shape-invariant properties, in a fashion similar to the shaped packets in related works [15–18,20,21,24,44].

In particular, as time evolves, the nodes vanish sequentially, which happens via “lifting” of the wavepacket profile due to continuity. The amplitude of the spatial oscillations decreases in time until the nodes vanish and the wavepacket starts to spread out in space. Originally, the Whittaker modes ${w}_{\kappa}$ are in phase. Their closely spaced zeros define the nodal structure of the Whittaker wavepacket (see Supplement 1). As we vary time $t$, the factors $\mathrm{exp}(-i\omega t{\kappa}^{2})$ put the modes ${w}_{\kappa}$ out of phase; thus, the zeros of the Whittaker wavepacket continuously disappear. In this way, the diffraction eventually destroys the original nodal structure, and having no nodes means that the wavepacket is free to spread in space like a free wavepacket, as shown in Fig. 3(b). In Fig. 3(b), the upper and lower envelopes converge, which destroys the nodal structure. This process of “node lifting” bottlenecks the diffraction of the wavepacket and enables the quasi-shape-invariant property. Notice the similarity between the dynamics of the envelopes in (a) and (b). Finally, in Fig. 3(c), we show that free-particle wavepackets also have a similar nodal structure at time zero, which yields approximately the same overlap function as that of Whittaker wavepackets. A way to see why this holds true is to look at the limiting behavior of the extended mode (1) for large $r$ ($x\gg 1$) [45]: ${w}_{\kappa}(x,0)\sim \mathrm{exp}(i\kappa x)\mathrm{exp}(i(1/2\kappa )\text{ln}(x))/x$. The logarithmic dependence on $x$ is due to the Coulomb potential. Thus, the electron wavepacket does not approximate a free-particle wave exactly; however, at large values of $x$, the oscillations from $\mathrm{exp}(-i\kappa x)$ are dominant and form the free-particle modes. This means that the Whittaker wavepacket at large radii is similar to that of a free particle; hence their time evolutions should be similar. This similarity shows that our methods and conclusions also apply for free electron wavepackets (possibly with a time-dependent perturbation, such as pulsed laser excitation).

#### D. Radiative Decay

Importantly, however, the physics of free particles is different from that of electrons in a potential. The potential may cause scattering processes between the proton and electron, which in turn could lower the energy of the electron through radiative decay, and thus reduce its stability. We now quantitatively evaluate the competition between diffraction and radiative decay by quantifying the radiative decay. We further show how the shape (specifically spatial extent) of the wavepacket changes the rate of photon emission. Moreover, the spectrum of decay by the electronic wavepacket can be controlled by controlling the mean energy of the wavepacket, and thus the spectral peaks can be continuously tuned if the excitation energy of the electron wavepacket is so tuned.

To show this, we calculate the probability of decay ${P}_{n}(t)$to a bound state $|n\u27e9$ at a given time $t$; we develop a formalism (described in Section 4 Methods) through the $S$-matrix approach [41]. The total decay probability is simply the sum $P(t)=\sum _{n=1}^{\infty}{P}_{n}(t)$. Having the total probability, we define the *average rate* of decay over the time of two diffraction lifetimes $\mathrm{\Delta}t$ via the formula

Since the radiative lifetime is much longer than the diffraction lifetime, we conclude that formulas (6) and (7) give a good parameterization of the stability and the large $\mathrm{\Delta}t$ versus large $\mathrm{\Delta}r$ trade-off of the Whittaker wavepackets, and show how those properties can be customized.

#### E. Quantitative Examples

We now present quantitative examples of parameters achievable with Whittaker wavepackets, summarized in Table 1. For a quasi-shape-invariant wavepacket to be considered stable enough for optical transitions, its diffraction lifetime $\mathrm{\Delta}t$ should be longer than the period of the optical cycle of light emitted in the transition. For wavepackets designed to have transitions in the x-ray frequencies, the shortest lifetime we consider in Table 1 is 53 as (comparable to the shortest x-ray pulse duration measured [46] with high-harmonic generation [47–49]). For the largest $\mathrm{\Delta}r$, considered in Table 1, we take 143 nm (comparable to the recently observed [50] Rydberg state at the $n=52$ state). We note that by shaping electrons in transmission electron microscopes, coherent wavepackets over spatial extents of tens of micrometers have been observed [51]; hence, one can consider much wider electron wavepackets as well. To give specific examples from Table 1, if we fix $\mathrm{\Delta}t=53\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{as}$, then according to formulas (6) and (7) a soft x-ray Whittaker wavepacket ($E=200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$) has an energy spread $\mathrm{\Delta}E=0.033\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{eV}$ and a spatial spread $\mathrm{\Delta}r=0.72\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, which all fit within the limits established here. If we consider a hard x-ray Whittaker wavepacket ($E=10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{keV}$), then the energy spread has to shrink to $\mathrm{\Delta}E=660\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu eV}$; for the same lifetime, however, $\mathrm{\Delta}r$ gets to 5.1 nm, much smaller than 143 nm. We also studied Whittaker wavepackets with lifetimes on the order of nanoseconds, which require spread $\mathrm{\Delta}r$ on the order of micrometers.

## 3. DISCUSSION

We have proposed shaping quasi-shape-invariant wavepackets in the continuum of the hydrogen atom. By studying the dynamics of the wavepackets, we utilize the large lifetime versus large spatial spread trade-off of the wavepackets to extend the transition lifetime (both radiative and diffraction lifetimes). These wavepackets exhibit unique phenomena in their decay dynamics, such as a stark change from radiative decay similar to bound states at short times, to saturation at long times (once the electron’s probability density spreads away).

Our approach involves the reduction of the Scrödinger equation to a confluent hypergeometric equation above the ionization threshold, which is known in the literature to lead to the non-singular Whittaker solutions [52,53]. To the best of our knowledge, the quasi-shape-invariant applications of Whittaker wavepackets have been previously unexplored. More importantly, the methods utilized here can be extended directly to other potentials to design additional long-lived wavepackets.

As a proof of concept, we demonstrate our method for the one-electron case. In principle, we could extend our work to many-electron wavepackets: consider the joint wavefunction of a multiple-electron system. With some mean-field techniques, such a possibility has been considered before [54], and it could be an interesting idea for future research to see whether it can alter the light–matter interactions of the many-electron system as in our current paper. Many of the concepts from our paper will still apply in the many-electron case. Altogether, it is clear that the hydrogen atom does not capture all of the physics in any possible system, as, for example, there are no additional electrons. Nevertheless, some competing decay mechanisms could be captured phenomenologically by a master equation approach, to take into account decoherence induced by electron–electron interaction.

It would also be interesting to consider whether or not the energy bandwidth of the shaped electron would be converted into the spectral bandwidth of the emitted radiation. Similarly, the coherence of the radiation could be related to the coherence function of the electron. For example, a partially mixed electron state would probably result in a partially mixed radiation state.

We believe that a promising approach for generating quasi-shape-invariant wavepackets is to use quantum optimal control to generate our wavepackets [55]. For example, [56] demonstrates directed electron XUV emission for hydrogen and argon through an optimization process that targets particular photoelectron spectra and angular distributions of the emission in photoionization. Moreover, [57] proposes a gradient-free quantum control method to derive pulses that create tailored superpositions of hole states for argon with predefined properties. In a similar fashion, we could design the optical pulses by solving an optimization problem that aims to match the overlap function and radiative decay of quasi-shape-invariant wavepackets ( [57] even proposes what the target overlap function and radiative decay should be). The vector potential that describes the excitation pulse can be added as a classical potential to the Schrödinger equation, which we can solve by following the same approach and quantified with the same formulas (probably requiring some numerical work).

Other directions toward the design of the quasi-shape-invariant electron wavepacket include: shaping the light excitation with special nanostructures to enable control of the interference between free-electron superposition states [58], control of bound electrons [59,60], and programmable Rydberg wave packets [61]. Our theoretical work in combination with advances in coherent control could navigate research on generation of target electron wavepackets in the future. It may also be of interest to consider how laser-based shaping of an excited electron in the ionization continuum impacts the radiation emission in high-harmonic generation, where propagation of and radiation by an electron in the ionization continuum is a key step in the process.

Our methods, and specifically the time-dependent QED formalism, can be applied to a variety of other systems. For instance, a good candidate is the system of shaped free electrons that interact with time-dependent potentials [58,62–67]. Thus, it will also be interesting to study how more complicated wavepacket shapes (e.g., Airy) alter the spontaneous radiative transition rates.

In most of the above methods, the shaping of the electron is done by fs laser pulses in the visible or infrared spectrum, carrying μJ-mJ pulse energy. This way, we can bring an electron to high energies (EUV, or even soft-x-ray scales) without any need for an x-ray source. For example, the same 1 mJ scale laser pulses that are used in high-harmonic generation [68] and in above-threshold ionization [69], as well as in coherent control in femtochemistry [70], can be used for our purposes. A particularly exciting opportunity for shaping quasi-shape-invariant electron wavepackets could make use of the quantum electron–photon interaction in laser-driven transmission electron microscopes [63–67,71]. For example, recent work has shown the possibility to shape electron pulses into attosecond bunches [66,71], which is possible even on the level of the single electron wavepacket [72]. Very recent work also demonstrated the possibility to shape the orbital angular momentum of the electron through laser interaction [73–75], as well as using multiple laser harmonics for the complete shaping of the electron in time [76].

Quasi-shape-invariant wavepackets can bring atomic physics phenomena to new energy ranges such as soft and hard x rays. Utilizing these phenomena might introduce new quantum light sources and other applications to a diversity of physical systems, including various Dirac particles and free electrons under strong fields, as well as other wave systems that often describe analogous physics, such as water waves, acoustic waves on membranes, and electromagnetic waves.

## 4. METHODS

#### A. Shaping of Quasi-Shape-Invariant Wavepackets

The nature of our analysis (both analytic and numerical) is universal and can be generalized to calculate the transitions of wavepackets under any potential, including time-dependent potentials. Specifically, the derivation of the Whittaker modes, the shaping of the wavepackets, the analytic monitoring of the evolution, and the shaping that enables the quasi-shape-invariant properties of the superpositions (see further details in Supplement 1, Sections I and II) can also be reproduced for electrons in the vicinity of other potentials. However, most cases would require finding the extended states numerically or working with arbitrary wavepackets, without finding the eigenstates at all. Furthermore, to obtain an analytic expression for the probability of transition decay from wavepackets in the continuum to bound states, we use time-dependent spontaneous emission calculations that, in theory, can be applied for any interaction potential.

#### B. QED Formalism for Radiative Decay

Our formalism is based on QED and the $S$-matrix approach [41], for which the probability of transition from the initial state $|i\u27e9$ to the final state $|f\u27e9$, through the emission of a photon with momentum $\mathbf{k}$ and polarization $\lambda $, is given by

## Funding

Marie Curie (328853-MC-BSiCS); Department of Energy Fellowship (DE-FG02-97ER25308); U.S. Department of Energy (DE-SC0001299); Army Research Office (ARO) through the Institute for Soldier Nanotechnologies (W911NF-18-2-0048).

## Acknowledgment

We would like to thank Pamela Siska, Thomas Christensen, Josué López, Peter Lu, and Jamison Sloan for useful comments on the paper, as well as Thomas Beck and Maxim Metlitski for useful discussions. I. K. was supported by the Marie Curie Grant and by the Azrieli Faculty Fellowship. N. R. was supported by the Department of Energy Fellowship DE-FG02-97ER25308.

See Supplement 1 for supporting content.

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