Electric-field induced helium-helium resonances

TL;DR

This paper explores how external electric fields can induce resonances in helium-helium interactions, providing theoretical predictions that could inform future experimental studies on quantum control of helium systems.

Contribution

It presents the first detailed calculations of electric-field induced resonances in helium-helium systems, including bound states and implications for wave packet dynamics.

Findings

Electric-field induced resonances are predicted for helium isotopic pairs.

Qualitative agreement with previous literature for certain helium systems.

Implications for laser-induced wave packet dynamics are discussed.

Abstract

The tunability of the helium-helium interaction through an external electric field is investigated. For a static external field, electric-field induced resonances and associated electric-field induced bound states are calculated for the $^4$He-$^4$He, $^3$He-$^4$He, and $^3$He-$^3$He systems. Qualitative agreement is found with the literature for the $^3$He-$^4$He and $^3$He-$^3$He systems [E. Nielsen, D. V. Fedorov, and A. S. Jensen, Phys. Rev. Lett. {\bf{82}}, 2844 (1999)]. The implications of the predicted electric-field induced resonances for $^4$He-$^4$He on the wave packet dynamics, initiated by intense laser pulses, are investigated. Our results are expected to guide next generation experiments.

Figures

Equations

$$V_{\text{lm}}(r,\theta ,t)=-\frac{1}{2}|ϵ(t){|}^2\left[{\alpha }_{\parallel }(r){\cos }^2\theta +{\alpha }_{\perp }(r){\sin }^2\theta \right],$$

$${\alpha }_{\perp }(r)=2{\alpha }_0-\frac{2{\alpha }_0^2}{4\pi {\mathcal{E}}_0{r}^3}+\frac{2{\alpha }_0^3}{(4\pi {\mathcal{E}}_0{)}^2{r}^6}$$

$${\alpha }_{\parallel }(r)=2{\alpha }_0+\frac{4{\alpha }_0^2}{4\pi {\mathcal{E}}_0{r}^3}+\frac{8{\alpha }_0^3}{(4\pi {\mathcal{E}}_0{)}^2{r}^6},$$

$$V_{\text{lm}}(r,\theta ,t)$$

$$\displaystyle H=-\frac{\hbar^{2}}{2\mu}\Bigg{[}\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial}{\partial r}\right)+$$

$$\displaystyle\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)\Bigg{]}+V_{\text{2b}}(r)+V_{\text{lm}}(r,\theta,t).$$

$$\displaystyle\epsilon_{\text{SG}}(t)=\left\{\begin{array}[]{lll}\epsilon_{\text{G}}(t)&\mbox{for}&t\leq 0\\ \epsilon_{0,\text{G}}&\mbox{for}&0<t<t_{\text{hold}}\\ \epsilon_{\text{G}}(t-t_{\text{hold}})&\mbox{for}&t_{\text{hold}}\leq t,\end{array}\right.$$

$${ϵ}_{\text{G}}(t)={ϵ}_{0,\text{G}}\exp \left(-2\ln (2)\frac{t^2}{{\tau }^2}\right)$$

$$K_{l,l^{\prime }}(k)=\tan \left({\delta }_{l,l^{\prime }}(k)\right),$$

$$a_{l,l^{\prime }}=\underset{k\to 0}{\lim }\frac{-K_{l,l^{\prime }}(k)}{k}.$$

$$\psi (r,\theta )=\sum _{l^{\prime }}\frac{u_{l^{\prime }}(r)}{r}{Y}_{l^{\prime },0}(\cos \theta ),$$

$$\left(-\frac{{\hslash }^2}{2\mu }\frac{{\partial }^2}{\partial r^2}+V_{\text{2b}}(r)\right)u_l(r)-$$

$$\frac{1}{2}|{ϵ}_{0,\text{S}}{|}^2\sum _{l^{\prime }}{W}_{l,l^{\prime }}(r)u_{l^{\prime }}(r)=E{u}_l(r).$$

$$\Psi (r,\theta ,t)=\sum _{l^{\prime }}\frac{U_{l^{\prime }}(r,t)}{r}{Y}_{l^{\prime },0}(\cos \theta ).$$

$$\left(-\frac{{\hslash }^2}{2\mu }\frac{{\partial }^2}{\partial r^2}+V_{\text{2b}}(r)\right)U_l(r,t)-$$

$$\frac{1}{2}|ϵ(t){|}^2\sum _{l^{\prime }}{W}_{l,l^{\prime }}(r)U_{l^{\prime }}(r,t)=ı\hslash \frac{\partial U_l(r,t)}{\partial t},$$

$$C_2(r,t)=\frac{{\int }_0^{\pi }{\Psi }^{\ast }(r,\theta ,t){\cos }^2\theta \Psi (r,\theta ,t)\sin \theta d\theta }{{\int }_0^{\pi }|\Psi (r,\theta ,t){|}^2\sin \theta d\theta }.$$

$$V_{\text{lm}}(r,\theta ,t)=g(t)\left[{\alpha }_{0,0}(r)Y_{0,0}+{\alpha }_{2,0}(r)Y_{2,0}(\cos \theta )\right],$$

$$g(t)=-\frac{|ϵ(t){|}^2}{2},$$

$${\alpha }_{0,0}(r)=\frac{\sqrt{4\pi }}{3}\left[{\alpha }_{\parallel }(r)+2{\alpha }_{\perp }(r)\right],$$

$${\alpha }_{2,0}(r)=\frac{\sqrt{16\pi }}{3\sqrt{5}}\left[{\alpha }_{\parallel }(r)-{\alpha }_{\perp }(r)\right].$$

$$W_{l,l^{\prime }}(r)={\alpha }_{0,0}(r)⟨Y_{l,0}|Y_{0,0}|Y_{l^{\prime },0}⟩+$$

$${\alpha }_{2,0}(r)⟨Y_{l,0}|Y_{2,0}|Y_{l^{\prime },0}⟩,$$

Full text

I Introduction

The helium atom is a chemically inert rare gas atom. Helium has two naturally occuring isotopes: 4He, a composite boson, and 3He, a composite fermion. Whether or not these isotopes form diatomic molecules was debated for a long time in the literature. It is now agreed upon that the 4He-4He system supports a single rotationless bound state with an extremely small binding energy of about $1.3$ mK schoellkopf1994 ; tang1995 ; janzen1995 ; luo1996 ; schoellkopf1996 ; grisenti2000 ; zeller2016 . Neither the 3He-4He nor the 3He-3He system support, in the absence of external fields, molecular bound states.

The extremely small binding energy of the 4He-4He dimer is associated with a large positive $s$-wave scattering length. The 3He-4He system, in contrast, is characterized by a negative and large, in magnitude, $s$-wave scattering length. Motivated by the tunability of many of the alkali dimers through the application of an external magnetic field in the vicinity of a Fano-Feshbach resonance chin2010 , one may ask if the helium-helium interaction can be tuned as well, with the external magnetic field replaced by an external electric field. If such a tunability existed, this would open up many new research directions related to the study of extremely weakly-bound molecular states for a system that is amenable to ab initio calculations. Indeed, Ref. nielsen1999 pointed out the tunability of the 3He-4He and 3He-3He systems by a static external electric field. Moreover, Ref. nielsen1999 explored the consequences of this tunability for the three-body sector in the context of Efimov physics efimov70 ; braaten2006 ; naidon2016 . The tunability of the 4He-4He system by a static electric field and by laser pulses strong enough to involve electronically excited potential curves was very recently pointed out in Ref. QiWei .

Working in the opposite regime of short laser pulses, a recent molecular beam experiment kunitski2018 demonstrated that a short 310fs laser pulse with an intensity of a few times $10^{14}$W/cm2 can induce dissociative wave packet dynamics of the 4He-4He dimer, including interferences between the $l=0$ and $l=2$ partial wave channels. Here, $l$ denotes the orbital angular momentum quantum number. While Ref. kunitski2018 provided no evidence for the existence of electric field-induced bound states or hybridized states such as those predicted in Ref. friedrich1998 , the experimental results clearly show that the laser-molecule coupling is strong enough to trigger measurable changes such as a clean alignment signal. Moreover, the excellent agreement between the experimental and theoretical results in Ref. kunitski2018 suggests that the laser-molecule interaction, which included the lowest Born-Oppenheimer potential curve and assumed inertness of the electronic degrees of freedom, provides a reliable description, at least in the short-pulse regime for the intensities considered.

The present theoretical work considers laser pulses that are longer than those utilized in Ref. kunitski2018 . As a first exploration, our theoretical framework neglects, as in Ref. kunitski2018 , the electronic degrees of freedom. It is expected that corrections due to the electronic motion (see, e.g., Ref. becker ) need to be accounted for in follow-up work. One of the goals is to explore under which conditions the electric-field induced resonances of the helium-helium systems, first investigated in Ref. nielsen1999 for the 3He-4He and 3He-3He systems and for the 4He-4He system in Ref. QiWei , can be observed experimentally in time-dependent set-ups. To interpret the dynamic wave packet studies, the static field case is revisited and some quantitative discrepancies with the literature nielsen1999 , which we have no explanation for, are pointed out. To observe the electric-field induced resonances, the associated bound states have to be populated with sufficiently high probability and some signature that this has been achieved needs to be recorded. Our time-dependent calculations show, owing to the extremely floppy and highly quantum mechanical nature of the helium dimers, that revival dynamics competes with dissociative dynamics. Probing this intricate dynamics experimentally is expected to be possible but quite challenging due to the need of realizing long, intense laser pulses.

The remainder of this article is organized as follows. Section II introduces the system Hamiltonian and relevant theoretical background. Sections III and IV present our results for a static external field and a time-dependent external field, respectively. Last, Sec. V concludes.

II System Hamiltonian and Theoretical Background

This section describes the theoretical framework employed to investigate the tunability of the effective helium-helium interaction strength. Section II.1 introduces the system Hamiltonian. The determination of the scattering and bound states of the static Hamiltonian are discussed in Secs. II.2 and II.3. Last, Sec. II.4 summarizes how the wave packet propagation is done when the Hamiltonian is time-dependent.

II.1 System Hamiltonian

We consider two helium atoms, either two 4He atoms, a 3He-4He pair, or two 3He atoms, with reduced mass $\mu$ interacting through the spherically-symmetric state-of-the-art “electronic ground state” potential $V_{\text{2b}}(r)$ from Ref. cencek2012 , where $\vec{r}$ denotes the internuclear distance vector and $r$ is equal to $|\vec{r}|$. Due to the adiabatic beyond Born-Oppenheimer correction term cencek2012 , the interaction potentials for 4He-4He, 3He-4He, and 3He-3He are slightly different.

Throughout we assume that the electric field of the laser is oriented along the $z$-axis. Moreover, we assume that the oscillations of the electric field are so fast that they can be integrated over. With these assumptions, the time-dependent laser-molecule interaction $V_{\text{lm}}(r,\theta,t)$ reads friedrich1995

[TABLE]

where $\theta$ denotes the angle between the $z$-axis and the internuclear distance vector $\vec{r}$ (in spherical coordinates, this is the azimuthal angle), $\epsilon(t)$ characterizes the shape of the laser pulse, and $\alpha_{\perp}(r)$ and $\alpha_{\parallel}(r)$ denote the polarizabilities perpendicular and parallel to the molecular axis. The difference between these polarizabilities is responsible for the intriguing dynamics discussed in Sec. IV.

Following the pioneering work of Buckingham and Watts buckingham1973 , analytic expressions for $\alpha_{\perp}(r)$ and $\alpha_{\parallel}(r)$ read

[TABLE]

and

[TABLE]

where $\alpha_{0}$ denotes the atomic polarizability, $\alpha_{0}=1.383a.u.$ (“$a.u.$” stands for “atomic units”), and ${\cal{E}}_{0}$ the permittivity ($4\pi{\cal{E}}_{0}=1a.u.$; note that the symbols $\epsilon$ and ${\cal{E}}_{0}$ refer to different physical quantities). To discuss the physics, we rewrite $V_{\text{lm}}$ (in doing so, we drop the $r$-independent terms, which only contribute an energy shift and/or an overall phase),

[TABLE]

The first term in square brackets shows that the laser-molecule interaction increases the $C_{6}$ van der Waals coefficient of the helium-helium potential. The second term in square brackets shows that the laser-molecule interaction introduces an angle-dependence into the $C_{6}$ coefficient. Finally, the third term in square brackets corresponds to the interaction between two point dipoles, yielding a repulsive interaction energy for a side-by-side configuration and an attractive interaction energy for a head-to-tail configuration. These analytic expressions agree well with the state-of-the-art ab initio results from Ref. cencek2011 in the large $r$ region but not in the small $r$ region (see Fig. 1).

We find that the analytic expressions and the ab initio parametrization yield predictions that differ quantitatively but not qualitatively. The analytic polarizability model, for example, supports field-induced bound states for somewhat smaller field strengths than the ab initio parametrization. Similarly, the dynamical results presented in Sec. IV are dominated by the polarizabilities around $4a.u.$ to $10a.u.$ for which the two sets of polarizabilities agree quite well. Since the results for the two models agree qualitatively, the majority of the results presented in this work employs the polarization model from Ref. cencek2011 .

Since the Hamiltonian $H$ is in our set-up independent of the polar angle $\phi$, the projection quantum number $m_{l}$, which is associated with the $z$-component of the orbital angular momentum operator $\vec{l}$, is a good quantum number. We restrict ourselves to the $m_{l}=0$ channel in this work. Combining the interaction terms, the time-dependent Hamiltonian $H$, written in spherical coordinates, reads

[TABLE]

We consider two parametrizations of $\epsilon(t)$:

Static field with $\epsilon(t)=\epsilon_{0,\text{S}}$, where $\epsilon_{0,\text{S}}$ is a constant. Even though some of the field strengths $\epsilon_{0,\text{S}}$ considered in this work can only be realized for a relatively short time with present day technology, the results for the static field provide a useful framework for understanding the results for time-dependent pulses. 2. 2.

A “stretched” Gaussian pulse with $\epsilon(t)=\epsilon_{\text{SG}}(t)$,

[TABLE]

where

[TABLE]

with $\ln(2)=0.6931\cdots$. At $t=0$, the pulse is maximal for the first time and $\tau$ is the FWHM, which determines the rise and fall-off of the Gaussian pulse. While the stretched Gaussian pulse shape may not be realizable experimentally, the ensuing dynamics is comparatively straightforward to interpret and thus serves as a guide to what might be expected for stretched pulses with somewhat different profiles.

II.2 Scattering States

In the absence of the external field ($V_{\text{lm}}=0$), the $s$-wave scattering length $a_{s}$ of the 4He-4He system is positive and large ($a_{s}=170.9a.u.$), signaling the existence of a weakly-bound molecular $s$-wave state. In fact, this is the only bound state supported in the field-free case; no rotationally or vibrationally excited states exist. The $s$-wave scattering length of the 3He-4He system, in contrast, is negative and large in magnitude ($a_{s}=-34.2a.u.$), signaling that the system is just short of supporting a weakly-bound $s$-wave bound state. No deep-lying bound states are supported. The $s$-wave scattering length of the 3He-3He system is equal to $-13.73a.u.$ in the absence of an external electric field; in this case, the nuclear spins form a singlet, thereby enforcing the anti-symmetry of the full wave function under the exchange of two identical 3He atoms. The magnitude of the generalized higher partial wave scattering lengths such as the $p$-wave scattering volume for the 3He-4He and 3He-3He systems and the $d$-wave scattering hypervolume for the 4He-4He system are small.

We now include a time-independent laser-molecule Hamiltonian (parametrization 1. in Sec. II.1), which couples different orbital angular momentum channels. For the bosonic 4He-4He system, only even-$l$ channels contribute because the spatial wave function has to be symmetric under the exchange of the two 4He atoms. For the 3He-4He system, in contrast, no symmetry constraints exist, implying that even- and odd-$l$ channels contribute (due to the nature of the laser-molecule interaction, the even- and odd-$l$ channels are decoupled). Last, for the fermionic 3He-3He system, even-$l$ channels contribute when the nuclear spins form a singlet and odd-$l$ channels when the nuclear spins form a triplet. The long-range nature of the laser-molecule interaction modifies the threshold law in the non-zero partial wave channels weiner1999 ; marinescu1998 ; yi2001 ; deb2001 . In particular, since the K-matrix elements $K_{l,l^{\prime}}(k)$,

[TABLE]

are proportional to the wave vector $k$ as $k$ goes to zero ($k$ is defined in terms of the scattering energy $E$ through $\sqrt{2\mu E}/\hbar$), the zero-energy scattering length matrix elements $a_{l,l^{\prime}}$ are defined through

[TABLE]

For short-range interactions (interactions that fall off faster than $1/r^{3}$ at large internuclear distances), the denominator in Eq. (12) reads $k^{l+l^{\prime}+1}$ instead of $k$ newton_book ; the modification of the power of $k$ reflects the modified threshold behavior. The threshold laws, Eqs. (11) and (12), depend crucially on the angle dependence of the $-r^{-3}$ potential. If the angle dependence was absent, one would not be able to define an $s$-wave scattering length.

The phase shifts $\delta_{l,l^{\prime}}(k)$ [see Eq. (11)] are obtained by matching the inside solution to the large-$r$, free-particle solution, with the relative importance of the regular solutions [the spherical Bessel functions $j_{l}(kr)$] and the irregular solutions [the Neumann functions $n_{l}(kr)$] given by the tangent of the phase shifts $\delta_{l,l^{\prime}}(k)$. The scattering solutions in the presence of a static external field are thus characterized by an, in general, non-diagonal scattering length matrix. Even though the determination of the scattering solutions requires the entire scattering length matrix, the emergence of a new zero-energy bound state that is even (odd) in the relative coordinate $z$ is accompanied by the $a_{0,0}$ ($a_{1,1}$) matrix element going to infinity ticknor2005 ; kanjilal2008 .

We determine the K-matrix by decomposing the full wave function $\psi(r,\theta)$ into partial waves,

[TABLE]

where the sum over $l^{\prime}$ includes all angular momentum values allowed by symmetry and where the spherical harmonics $Y_{l^{\prime},m_{l}^{\prime}}$ are independent of $\phi$ since $m_{l}^{\prime}$ is assumed to be zero throughout. Inserting Eq. (13) into the Schrödinger equation $H\psi=E\psi$ and projecting onto the $Y_{l,0}^{*}$ states, we obtain a set of coupled differential equations for the radial components $u_{l}(r)$,

[TABLE]

An explicit expression for the coupling matrix elements $W_{l,l^{\prime}}(r)$, which arise from integrating over the angular degrees of freedom, is given in Appendix A. Equation (A) shows that the laser-molecule interaction $V_{\text{lm}}$ couples only channels with the same $l$ or channels whose indices differ by two. Imposing that the $u_{l}(r)$ vanish at small $r$, the logarithmic derivative matrix is propagated using the Johnson algorithm johnson with adjustable step size. Matching the large-$r$ solution to the asymptotic free-particle solution, the K-matrix is extracted. We find that a scattering energy of $10^{-12}a.u.$ approximates the zero-energy limit accurately; for this energy, we choose the large-$r$ matching point to be $10^{6}a.u.$. We find that the inclusion of about $8$ even and/or 8 odd partial wave channels yields converged results for the field strengths considered in this work.

II.3 Bound States

In addition to the scattering states, we calculate the bound states of the helium-helium systems in a static external field. Since a time-dependent external field can, at each time, be thought of as being static, the solutions for the static Hamiltonian provide guidance for interpreting our time-dependent results. In the extreme case of an adiabatically changing external field, the full dynamics can be readily extracted from the static results by, e.g., performing a Landau-Zener analysis.

To determine the bound state spectrum, we express the eigen states $\psi(r,\theta)$ in terms of a B-spline basis using non-linear grids in $r$ and $\theta$. The largest $r$ is adjusted so that the most weakly-bound state is fully covered by the numerical grid. For 4He-4He, we calculate eigen states that are even in the relative coordinate $z$. For the 3He-3He and 3He-4He systems, both even and odd states in $z$ are considered. In the case of 3He-3He, the even and odd partial waves must be combined with singlet and triplet nuclear spin states, respectively. Even though the bound states cannot be labeled by a single $l$ quantum number due to the $\theta$-dependence of $V_{\text{lm}}$, the weakly-bound states are typically dominated by a single partial wave. The dominant character can be obtained by projecting the eigen states onto different $l$ channels.

II.4 Dynamics

If the laser-molecule interaction is time dependent, we have to solve the time-dependent Schrödinger equation for a given initial state $\Psi(r,\theta,t=-\infty)$. In practice, the initial state is prepared at a time where the laser-molecule interaction can be neglected, i.e., at a time much smaller than [math].

To solve the time-dependent Schrödinger equation, we decompose the wave packet $\Psi(r,\theta,t)$, similar to what we did in Sec. II.2 to obtain the time-independent scattering states, into partial wave components,

[TABLE]

Inserting Eq. (15) into the time-dependent Schrödinger equation $\imath\hbar\partial\Psi/\partial t=H\Psi$, we obtain a set of coupled time-dependent equations for the radial components $U_{l}(r,t)$,

[TABLE]

where the coupling elements $W_{l,l^{\prime}}(r)$ are given in Eq. (A).

To solve the coupled set of time-dependent radial equations, we discretize the $r$ coordinate (we typically use about 32,000 points) and propagate the $U_{l}(r,t)$ by expanding the radial propagator in terms of Chebychev polynomials kosloff . The time step $\Delta t$ is chosen such that $|\epsilon(t)|^{2}$ can be considered, to a very good approximation, as time independent during each time step. We use $\Delta t\approx 0.24~{}\text{fs}$ to $0.60~{}\text{fs}$ and about $30$ terms in the expansion into Chebychev polynomials. For the pulses considered, accounting for about eight partial wave channels yields converged results.

III Time-Independent Field Strength

This section discusses the characteristics of the helium-helium systems in the presence of a static external field (parametrization 1. in Sec. II.1). Figure 2 shows the dimer binding energy $E_{\rm{bind}}$ for (a) 4He-4He, (b) 3He-4He, and (c) 3He-3He as a function of the field strength $\epsilon_{0,\text{S}}$. The binding energy associated with states that are even in $z$ is shown by solid lines and that associated with states that are odd in $z$ is shown by dashed lines. The 4He-4He system supports new $s$-wave ($l=0$) dominated bound states for field strengths larger than about $\epsilon_{0,\text{S}}=0.0715a.u.$ and larger than about $\epsilon_{0,\text{S}}=0.0976a.u.$. No evidence for the existence of these states is reported in Ref. QiWei . For field strengths larger than about $\epsilon_{0,\text{S}}=0.10962a.u.$, a new bound state with appreciable $d$-wave admixture is being supported [also notice the related avoided crossing between the $s$-wave dominated and $d$-wave dominated states at $(\epsilon_{0,\text{S}},E_{\rm{bind}})\approx(0.11a.u.,5\times 10^{-7}a.u.)$]. Owing to the orbital angular momentum barrier, this bound state acquires an appreciable binding energy over a fairly small variation of the field strength $\epsilon_{0,\text{S}}$. Reference QiWei refers to this $d$-wave dominated state as a “pendular state”.

The 3He-4He system [see Fig. 2(b)] supports its first $s$-wave dominated bound state for field strengths larger than about $\epsilon_{0,\text{S}}=0.0311a.u.$, a second $s$-wave dominated bound state for field strengths larger than about $\epsilon_{0,\text{S}}=0.0776a.u.$, and a third $s$-wave dominated bound state for field strengths larger than about $\epsilon_{0,\text{S}}=0.1054a.u.$. Owing to the smaller reduced mass, the latter two field strenths are a bit larger than the critical field strengths for the 4He-4He system. The first field-induced bound state, which first appears at $\epsilon_{0,\text{S}}=0.0311a.u.$, has no analog in the 4He-4He system since this system already supports a weakly bound state in the absence of an external electric field. The 3He-4He system additionally supports bound states that are odd in the relative coordinate $z$ [see the dashed lines in Fig. 2(b)]. Interestingly, these odd-$z$ bound states first appear at field strengths that are just a bit larger than the field strengths at which the even-$z$ bound states first appear. As the binding energy increases, the energy difference, normalized by the binding energy itself, between the pairs of even-$z$ and odd-$z$ states decreases. This is not unlike the tunneling splitting in a double-well potential, where the tunneling is much smaller for deep-lying states than for states that lie near or above the barrier.

Last, the 3He-3He system first supports even-$z$ bound states at field strengths larger than about $\epsilon_{0,\text{S}}=0.0388a.u.$, larger than about $0.0832a.u.$, and larger than about $0.1126a.u.$, respectively. Odd-$z$ bound states are first supported at field strengths larger than about $\epsilon_{0,\text{S}}=0.0492294a.u.$, larger than about $\epsilon_{0,\text{S}}=0.0875679a.u.$, and larger than about $\epsilon_{0,\text{S}}=0.116342a.u.$. Owing to the smaller reduced mass, these field strenths are a bit larger than the corresponding critical field strengths for the 3He-4He system.

As mentioned in Sec. II.2, the emergence of a new even-$z$ bound state is accompanied by a diverging $a_{0,0}$ and the emergence of a new odd-$z$ bound state by a diverging $a_{1,1}$. Solid lines in Fig. 3 show the scattering length matrix element $a_{0,0}$ as a function of the field strength $\epsilon_{0,\text{S}}$ for (a) 4He-4He, (b) 3He-4He, and (d) 3He-3He while dashed lines show the scattering length matrix element $a_{1,1}$ for (c) 3He-4He and (e) 3He-3He. Comparison with Fig. 2 shows that $a_{0,0}$ and $a_{1,1}$ go through infinity at the field strengths at which new, respectively, even-$z$ and odd-$z$ bound states are first being supported. We checked that the generalized scattering lengths $a_{l,l^{\prime}}$, except for $a_{0,0}$, are well described—as they should be for potentials that are purely dipolar at large internuclear distances yi2001 —by the Born approximation for field strengths where resonances are absent. This is illustrated in Figs. 3(c) and 3(e), where the Born approximation results (solid circles) reproduce the full coupled-channel calculations (dashed lines) reliably. In the Born approximation, $a_{1,1}$ is given by $-2\mu|\epsilon_{0,\text{S}}|^{2}|\alpha_{0}|^{2}/[5(4\pi{\cal{E}}_{0})^{2}\hbar^{2}]$ kanjilal2007 . Figure 4 shows an enlargement of the scattering length matrix element $a_{1,1}$ for 3He-4He near the third resonance shown in Fig. 3(c), i.e., near $\epsilon_{0,\text{S}}\approx 0.10885a.u.$. The Born approximation values do not capture the resonance; instead, they continue to change quadratically with $\epsilon_{0,\text{S}}$ across the resonance.

The calculations presented thus far employ the ab initio polarization model from Ref. cencek2011 . If we use the simpler analytical polarization model [Eqs. (2) and (3)], the first electric-field induced resonance for 4He-4He occurs at $\epsilon_{0,\text{S}}=0.0699a.u.$ instead of at $\epsilon_{0,\text{S}}=0.0715a.u.$ and the first electric-field induced resonance for 3He-4He occurs at $\epsilon_{0,\text{S}}=0.0304a.u.$ instead of at $\epsilon_{0,\text{S}}=0.0311a.u.$. The deviation between the results for the two different polarization models increases with increasing field strength.

Our results for the scattering properties of the 3He-4He and 3He-3He systems in the presence of a static field disagree quantitatively with those presented in Ref. nielsen1999 ; footnote . Repeating the calculations for the interactions employed in Ref. nielsen1999 (i.e., using the LM2M2 potential aziz1991 and the analytic polarizability model), we find that the first bound state for 3He-4He is supported for $\epsilon_{0,\text{S}}=0.0305a.u.$ as opposed to $0.053a.u.$ as reported in Ref. nielsen1999 and the first $s$-wave dominated bound state for 3He-3He first appears at $\epsilon_{0,\text{S}}=0.0382a.u.$ as opposed to $\epsilon_{0,\text{S}}=0.067a.u.$ as reported in Ref. nielsen1999 . We have no insight into what might be the reason for the discrepancies.

The results presented so far indicate that the helium-helium interaction strength can be varied through the application of a static external field. While the field strengths required are attainable with present-day technology, they can only be realized for a relatively short time. For the field to be considered truly static, the pulse duration has to be longer than the internal or characteristic time scale of the helium-helium system. If we convert the 4He-4He binding energy in the absence of an external field, we find a time scale of about $29.71$ ns. Clearly, the realization of such temporally extended, high-intensity laser pulses is presently out of reach. Alternatively, the minimal energy of the He-He interaction potential corresponds to a time scale of about $4.364$ ps. This time scale estimate looks much more promising from an experimental point of view. Alternatively, we can estimate the time scale associated with the field-induced bound states. Energies of $10^{-10}a.u.$, $10^{-8}a.u.$, and $10^{-6}a.u.$ correspond to time scales of about $1,520$ ns, $15.20$ ns, and $0.1520$ ns. If one were to populate the new bound state and if the system could be held at a particular field strength longer than the time given above, one should be able to see revival signatures corresponding to the above time scale in the dynamical evolution of an appropriately chosen observable. While challenging, realizing such a scenario experimentally does not seem entirely out of reach. Ultimately, one has to analyze the full dynamics to see which pulse shapes and lengths yield observable signatures of the electric-field induced tunability of the helium-helium interaction strength. Exploratory calculations along these lines are presented in the next section. While earlier work bruch2000 employed a perturbative framework to address this question, we employ a full coupled-channel treatment. Our calculations employ peak electric field strengths of $\epsilon_{0,\text{G}}=0.0843949a.u.$ and $0.11a.u.$ (corresponding to $2.5\times 10^{14}$W/cm2 and $4.247\times 10^{14}$W/cm2, respectively). These field strengths are significiantly lower than those employed in “realm II” of Ref. QiWei .

IV Time-Dependent Field Strength

This section summarizes our results for the stretched Gaussian pulse (parameterization 2. in Sec. II.1). Our studies are motivated by two questions: What, if any, are the signatures of the field-induced resonances discussed in Sec. III that could be measured experimentally in pump-probe experiments? Do the field-induced resonances lead to revival dynamics, somewhat reminiscent of what has been observed in pump-probe experiments for stiff, rigid rotor-like diatomic molecules RMP_alignment ; review_lemeshko ? To address these questions, we focus on the 4He-4He system. We assume that the dimer is prepared in the absence of an external field in its $l=0$ ground state, as is being done in molecular beam experiments schoellkopf1994 ; schoellkopf1996 ; grisenti2000 ; zeller2016 ; kunitski2018 . The laser pulse is then turned on and the system is assumed to be imaged via COLTRIMS after a delay time kunitski2018 ; coltrims . In this technique, an extremely short and intense probe pulse, which “rips off” one electron of each of the helium atoms, is applied and the ions are imaged. Since the probe pulse, to a very good approximation, instantaneously projects the helium atoms to one particular configuration, we do not simulate the imaging part of the experiment. Repeated experimental measurements for the same time delay provide access to the quantum mechanical density distribution of the wave packet. In what follows, the delay time is defined such that it is zero when the stretched Gaussian pulse first reaches its maximum. Our calculations scan the delay time from zero to many times $t_{\text{hold}}$.

We monitor the correlator or alignment $C_{2}(r,t)$,

[TABLE]

If $\Psi$ was independent of $\theta$ (as it is in the absence of the laser pulse), $C_{2}(r,t)$ would be equal to $1/3$. Deviations from $1/3$ provide a measure of the angle dependence that is introduced to the wave packet by the laser pulse. Importantly, after the laser is “off”, i.e., after its intensity has decayed to a sufficiently small value, the coupling between different $l$ channels vanishes and the populations of the different $l$ channels are independent of time. The wave packet itself, however, continues to change with time since the spatially-dependent phases of the different partial wave components continue to evolve. These phase factors imprint an $r$-dependent interference pattern, which varies with time (see also Ref. kunitski2018 ), onto the correlator $C_{2}(r,t)$.

The upper row of Fig. 5 shows contour plots of $C_{2}(r,t)$ for fixed $\tau$ and $\epsilon_{0,\text{G}}$, $\tau=311$fs and $\epsilon_{0,\text{G}}\approx 0.0844a.u.$ (intensity of $2.5\times 10^{14}$W/cm2), and four different hold times, i.e., for $t_{\text{hold}}=0.5$ps, $2$ps, $4$ps, and $8$ps. The lines in the lower row show cuts, from bottom to top, for $r=3\AA=5.669a.u.$, $r=5\AA=9.449a.u.$, $r=10\AA=18.90a.u.$, and $r=20\AA=37.79a.u.$. For the peak field strength used, the static system supports two bound states that are dominated by the $s$-wave channel (see Figs. 2 and 3).

For the shortest $t_{\text{hold}}$ considered, $C_{2}(r,t)$ is characterized by a fairly regular interference pattern, whose maxima and minima move out with increasing time. Even though $t_{\text{hold}}$ is finite, the interference pattern is quite similar to that observed and interpreted in a very recent joint experiment-theory collaboration, which employed an unstretched Gaussian pulse with $t_{\text{hold}}=0$, the same $\tau$, and comparable field strength kunitski2018 . The pattern of the alignment signal can be traced back to the interference between the dissociating $l=2$ wave packet portion, which gets populated as a consequence of the laser-molecule interaction, and the broad spherically-symmetric background portion (recall, the initial state is a pure $s$-wave state). Close inspection of $C_{2}(r,t)$ in the $t=0.5$ ps to $1$ ps window, however, reveals that the interference pattern is due to two dissociating wave packet portions, one that is emitted starting at $t=0$ and another that is emitted for $t\gtrsim t_{\text{hold}}$. This behavior becomes more prominent for larger $t_{\text{hold}}$ (see below).

The small $r$ behavior changes distinctly when $t_{\text{hold}}$ increases. Figures 5(b)-5(d) display oscillations of $C_{2}(r,t)$ at small $r$ [see also the dashed lines in Figs. 5(f)-5(h)]. These oscillations, which are most prominent for the largest hold time considered [Fig. 5(h)], are roughly governed by the binding energy of the deepest-lying, $s$-wave dominated transient state that is supported by the static Hamiltonian with field strength $\epsilon_{0,\text{G}}$. Its binding energy translates to about $3.800$ ps. The time scale associated with the energy difference between the two $s$-wave dominated transient bound states is equal to about $3.891$ ps, which is very close to the time scale set by the binding energy of the deep-lying transient state. Indeed, we attribute the small-$r$ oscillations of $C_{2}(r,t)$ to two processes, namely the interference between the wave packet portions corresponding to the two transient bound states and the interference between the wave packet portions corresponding to the deep-lying transient bound state and unbound scattering states. These interference processes both contribute to the population transfer between the $l=0$ and $l=2$ channels and thus lead to oscillations in the alignment $C_{2}(r,t)$.

The oscillations of $C_{2}(r,t)$ are reminiscent of revival dynamics in rigid-rotor like molecules due to population transfer between different rotational states. There are, however, important differences. First, unlike for rigid-rotor molecules where multiple eigen energies with spacings set by the rotational constant $B$ exist in the absence of the field, the deep-lying state that sets the time scale in the helium dimer system is transient. Second, the $r$-dependence of the alignment $C_{2}(r,t)$, as highlighted by the “outgoing finger structure” in Fig. 5, is unique to the non-rigid helium dimer. For rigid-rotor molecules, this structure is absent. Third, the broadness of the initial wave packet combined with the fact that the laser-molecule interaction is dominant at small $r$ implies that only a small fraction of the wave packet gets “promoted” to finite $l$ states.

As already alluded to above, Figs. 5(b)-5(d) show that the decay of the pump pulse from strength $\epsilon_{0,\text{G}}$ to zero (this occurs for times just a bit larger than $t_{\text{hold}}$) triggers the “emission” of a second dissociating wave packet portion, which can be attributed to the fact that the population of the deep-lying transient bound state is no longer bound when the laser intensity is negligible. The second dissociating wave packet produces a new set of outgoing fingers that are delayed by $t_{\text{hold}}$ compared to the first set of fingers and that “collide” with the first set of fingers. The interference of the delayed outgoing wave packet portion with the first dissociating wave packet portion leads, as can be seen nicely in the $r=20\AA$ cuts [solid lines in Figs. 5(f)-5(h)], to “distorsions” of the interference pattern. In particular, it can be seen that $C_{2}(r,t)$ displays a regularly changing wave pattern for $t\leq t_{\text{hold}}$ that changes notably for $t$ just a bit larger than $t_{\text{hold}}$. For $t$ quite a bit larger than $t_{\text{hold}}$, $C_{2}(r,t)$ again displays a regularly changing wave pattern.

Figure 6 shows the same quantities as Fig. 5 but for a larger peak field strength, namely for $\epsilon_{0,\text{G}}=0.11a.u.$. For this field strength, the static 4He-4He system supports three $s$-wave dominated bound states and one $d$-wave dominated bound state. The binding energy of the most strongly-bound transient state translates to a time scale of $1.522$ ps. Indeed, the small-$r$ region of the alignment signal displays close to regular oscillations at roughly this time scale. We do not expect perfect “single-frequency” oscillations since several transient eigen frequencies are expected to contribute to the observed oscillatory pattern. As in the weaker field strength case, the emission of a second dissociating wave packet portion at times just a bit larger than $t_{\text{hold}}$ is clearly visible in the alignment signal. Comparison of Figs. 5 and 6 shows that the larger field strength has two primary effects. First, it leads to a shortening of the oscillation period of the small-$r$ portion of $C_{2}(r,t)$. Second, it enhances the contrast of $C_{2}(r,t)$. Besides these two effects, the overall behavior of $C_{2}(r,t)$ is quite similar.

Figures 5 and 6 demonstrate that pump-probe experiments on the 4He-4He system should provide evidence for the tunability of the bound state spectrum by an external electric field. However, the alignment signal does unfortunately not—or if so rather indirectly—provide access to the number of field-induced bound states since the energy level spacing of the field-induced bound states is highly non-linear, leading to vastly different time scales governing the interference between the more weakly bound states. Moreover, the highly non-linear spacing also makes it difficult to distinguish between oscillations in the alignment $C_{2}(r,t)$ due to the interference of wave packet portions corresponding to the deepest-lying transient state and the most weakly-bound transient state and oscillations in the alignment $C_{2}(r,t)$ due to the interference of wave packet portions corresponding to the deepest-lying transient state and the transient scattering continuum. The latter process contributes also for peak field strengths $\epsilon_{0,\text{G}}$ that are smaller than $0.0715a.u.$, i.e., for peak strengths for which the static field Hamiltonian supports only one bound state. However, in this field strength regime, the large time scale associated with the small binding energy makes the unambiguous experimental observation that the energy of the transient bound state has been tuned essentially impossible.

V Conclusion

This work investigated static and dynamic properties of helium-helium systems in the presence of an external electric field. All three possible combinations of the two isotopes 3He and 4He were investigated, namely the 4He-4He, 3He-4He, and 3He-3He systems. In the absence of an external field, only the 4He-4He system supports a weakly-bound state (and only one). When a static external electric field is applied, all three helium-helium systems display field-induced scattering resonances, which are accompanied by the pulling-in of new two-body bound states. The resonances and their characteristics were analyzed carefully.

Applying a stretched Gaussian laser pulse, the work investigated the signatures imprinted on the dynamics by the field-induced resonances. For this analysis, we focused on the 4He-4He system. Assuming that the system is prepared in its only bound state in the absence of an external field, the time evolution during and after the stretched Gaussian pump pulse was investigated. It was found that the time-evolving wave packet carries fingerprints of the field-induced bound states, in addition to displaying dissociative dynamics that is associated with the fact that the pump laser leads to the population of scattering states with zero and finite angular momenta. It was commented that the experimental realization of the simulated scenarios is technically demanding but not impossible.

The response of diatomic rigid rotor-like molecules to intense laser pulses has been studied extensively in the literature, both experimentally and theoretically. The present dynamical study differs from these earlier works in that the 4He-4He system supports only a single extremely weakly-bound state in the absence of an external field. Thus, the notion of a rotor-like spectrum does not apply. As a consequence, the external field leads to a strong coupling of the vibrational and rotational degrees of freedom, with the populations of finite $l$ states dissociating.

VI Acknowledgement

We are very grateful to R. Dörner and M. Kunitski for extensive discussions, which motivated and inspired this work. We are also very grateful to D. Fedorov for communication related to Ref. nielsen1999 . Support by the National Science Foundation through grant number PHY-1806259 is gratefully acknowledged. This work used the OU Supercomputing Center for Education and Research (OSCER) at the University of Oklahoma (OU).

Appendix A Coupling Matrix Elements $W_{l,l^{\prime}}$

To determine explicit expressions for the coupling matrix elements $W_{l,l^{\prime}}(r)$, we rewrite the laser-molecule interaction $V_{\text{lm}}(r,\theta,t)$ as

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

Using this notation, $W_{l,l^{\prime}}(r)$ becomes

[TABLE]

where the notation $\langle\cdot\rangle$ indicates an integration over the angular degrees of freedom.