Stereodynamical control of a quantum scattering resonance in cold molecular collisions

TL;DR

This paper demonstrates that the outcome of cold molecular collisions can be fully controlled by aligning the rotational angular momentum of molecules, effectively switching a key quantum resonance on or off.

Contribution

It introduces a method to control quantum scattering resonances in cold molecular collisions through molecular alignment, a novel approach in collision dynamics.

Findings

Resonance can be switched on or off by molecular alignment.

Complete control over collision outcomes is achievable.

Alignment affects the dominance of specific partial wave resonances.

Abstract

Cold collisions of light molecules are often dominated by a single partial wave resonance. For the rotational quenching of HD(v=1,j=2) by collisions with ground state para-H2, the process is dominated by a single L=2 partial wave resonance centered around 0.1 K. Here, we show that this resonance can be switched on or off simply by appropriate alignment of the HD rotational angular momentum relative to the initial velocity vector, thereby enabling complete control of the collision outcome.

Equations

$${\sigma }_{\alpha \to {\alpha }^{\prime }}$$

$$S_q^{(k)}(\theta )$$

$$\frac{d{\sigma }_{\alpha }^{\beta }}{d\omega }=\sum _{k=0}^{2j_{{}_{\mathrm{HD}}}}\sum _{q=-k}^k(2k+1){\left[S_q^{(k)}(\theta )\right]}^{\ast }{A}_0^{(k)}{C}_{kq}(\beta ,\alpha )$$

Full text

At cold ($<$ 1 K) and ultracold ($<$ 1 $\mu$K) temperatures molecules can be prepared in precisely defined quantum states and interrogated with unprecedented precision. Recent developments in molecule cooling and trapping technologies Wynar et al. (2000); Regal et al. (2003); Sawyer et al. (2007); Shuman et al. (2010); Hummon et al. (2013); Akerman et al. (2017); Anderegg et al. (2017); Truppe et al. (2017) as well as merged or co-expanding beam techniques Henson et al. (2012); Jankunas et al. (2014); Klein et al. (2017); Perreault et al. (2018a); Amarasinghe and Suits (2017); Naulin and Costes (2014) have made it increasingly possible to study molecular systems at these low temperatures. Such systems have even been used in the frontiers of particle physics, DeMille et al. (2017) for example in the search for the electric dipole moment of the electron. Hudson et al. (2011); Baron et al. (2014); Cairncross et al. (2017) Cold and ultracold molecules therefore offer an ideal platform on which to precisely study fundamental aspects of molecular dynamics Bell and Softley (2009); Carr et al. (2009); Balakrishnan (2016); Bohn et al. (2017) such as the role of quantum statistics, Ospelkaus et al. (2010) threshold law,s Balakrishnan and Dalgarno (2001) and geometric-phase effects. Kendrick et al. (2015)

One of the most fundamental questions in molecular dynamics is the dependence of a collision outcome on the relative orientation and/or alignment of the reactants – the stereodynamics of a collision process. Bernstein et al. (1987); Levine and Bernstein (1987); Orr-Ewing and Zare (1994); Orr-Ewing (1996); Aoiz et al. (2015); Sharples et al. (2018); Heid et al. ; Wang et al. (2012, 2014) At cold and ultracold temperatures, where collisions proceed through just one or a few partial waves, their stereodynamics can be studied at their most fundamental level – the single quantum state level. In a recent series of papers Perreault et al. have examined the role that the initial alignment of HD plays in cold collisions with H2 and D2 Perreault et al. (2017, 2018b). Control over rotational quenching rates was demonstrated, and subsequent theoretical studies revealed that for certain states the scattering dynamics of cold HD+n-H2 collisions is determined by a single ($L=2$) partial-wave shape resonance at around 1 K Croft et al. (2018); Croft and Balakrishnan (2019).

While the stereodynamics of atom-diatom collisions has been explored in previous theoretical studies de Miranda and Clary (1997); Kandel et al. (2000); Jambrina et al. (2016); Aldegunde et al. (2008); Jambrina et al. (2019); Brouard et al. (2014), collisions between oriented and/or aligned molecules in cold conditions remain largely unexplored. In this work we apply the theoretical methods required to describe the stereodynamics of inelastic molecule-molecule collisions, specifically, to one of the prototypical systems recently studied by Perreault et al. – rotational quenching of HD in cold collisions with H2. In particular, we will demonstrate how the stereodynamics of cold molecule-molecule collisions can be determined by a single partial wave shape resonance and how it can be used to obtain exquisite control of the collision outcome.

Quantum Mechanical (QM) inelastic scattering calculations were carried out using the time-independent coupled-channel formalism within the total angular momentum (TAM) representation of Arthurs and Dalgarno Arthurs and Dalgarno (1960), which has previously been successfully applied to collisions of H2 with H2 Schaefer and Meyer (1979); Pogrebnya and Clary (2002); Quéméner et al. (2008) and HD  Schaefer (1990); Flower (1999); Balakrishnan et al. (2018). The scattering calculations were performed using a modified version of the TwoBC code Krems ; Quéméner and Balakrishnan (2009) on the full-dimensional potential surface of Hinde Hinde (2008). In the TAM representation the rotational angular momenta of the dimers, $\bm{j}_{{}_{\rm H_{2}}}$ and $\bm{j}_{{}_{\rm HD}}$, are coupled to form $\bm{j}_{12}=\bm{j}_{{}_{\rm H_{2}}}+\bm{j}_{{}_{\rm HD}}$ which is in turn coupled with the orbital angular momentum $\bm{L}$ to form the total angular momentum $\bm{J}=\bm{L}+\bm{j}_{12}$. Scattering calculations are performed separately for each value of the total angular momentum $J$ and parity $I=(-1)^{j_{{}_{\rm H_{2}}}+j_{{}_{\rm HD}}+L}$ that reflects the inversion symmetry of the wavefunction Alexander and DePristo (1977), yielding the Scattering matrix, $S^{J}_{\gamma,\gamma^{\prime}}$, labeled by the asymptotic entrance and exit channels $\gamma$ and $\gamma^{\prime}$ respectively. The composite index $\gamma=v_{{}_{\rm HD}}\,j_{{}_{\rm HD}}\,v_{{}_{\rm H_{2}}}\,j_{{}_{\rm H_{2}}}\,L\,j_{12}$ denotes the vibrational and rotational quantum numbers of each dimer ($v$ and $j$ respectively), the orbital angular momentum ($L$), and the sum of the angular momenta of the dimers ($j_{12}$). The integral cross section for state-to-state rovibrationally inelastic collisions is then given in terms of the S-matrix by

[TABLE]

where $\alpha$ is the combined molecular state, $\alpha=v_{{}_{\rm H_{2}}}\,j_{{}_{\rm H_{2}}}\,v_{{}_{\rm HD}}\,j_{{}_{\rm HD}}$, $k^{2}=2\mu E_{\rm col}/\hbar^{2}$ is the square of the wave vector, $E_{\rm col}$ the collision energy, and $\mu$ the reduced mass. From the elements of the S-matrix, the scattering amplitudes, $f_{\alpha^{\prime}\Omega^{\prime},\alpha\Omega}$ where $\Omega$ ($\Omega^{\prime}$) are the helicities, the projection of $j$ ($j^{\prime}$) into the approach (recoil) direction, were determined using the procedure described in Ref 37.

Inelastic collisions of HD($v=1,j=2$) with H2($v=0,j=0$) at low collision energies are dominated by $\Delta j=-1$ and $-2$ transitions in HD leading to HD($v^{\prime}=1,j^{\prime}=1$)+H2 and HD($v^{\prime}=1,j^{\prime}=0$)+H2, respectively. Vibrational de-excitation of HD is energetically allowed, but the cross section for vibrational relaxation is around 5-6 orders of magnitude smaller at these collision energies. Energetically, two-quanta rotational excitation of H2 is not allowed.

The energy dependence of the rotational quenching cross sections is shown in the top panel of Fig. 1. It is seen that at the lowest energies considered, the cross section for $\Delta j=-1$ is about a factor of seven larger than for $\Delta j=-2$. Both show the onset of the Wigner threshold regime below about 0.01 K ($\propto E_{\rm coll}^{-1/2}$ for pure $s$–wave collisions). The most salient feature for $\Delta j=-1$ is the presence of a sharp resonance at 0.1 K, where the cross section increases by almost a factor of four. This is an $L=2$ shape resonance (Fig. 1 shows the contribution from $L=2$ separately), that is caused by a single S-matrix element corresponding to $L=2$ and $J=3$ in the TAM representation. As a consequence of this, the resonance has a defined parity, in this case the block that does not include $\Omega$=0, which as we will show later has important consequences for the collision mechanism. This particular resonance is not observed for $\Delta j=-2$, even though most of the scattering also comes from $L=2$. Such resonances are ubiquitous features of inelastic and reactive collisions, especially in the cold regime. Here we show how they can be used to reveal the collision mechanism and, perhaps more importantly, control the collision outcome.

The concept of a collision mechanism can be at times somewhat vague, relying on qualitative rather than on quantitative results, which can lead to misinterpretations. To avoid any ambiguities we use the three-vector correlation $\bm{k}-\bm{j}_{{}_{\rm HD}}-\bm{k}^{\prime}$ (where $\bm{k}$ and $\bm{k}^{\prime}$ are the initial and final relative wave vectors) which is especially well-suited to characterizing collision mechanisms within a purely quantum-mechanical framework. More explicitly we use the set of reactant polarization parameters, $s^{\{k\}}_{q}$, of rank $k$ and component $q=-k\ldots k$, which define the vector correlation Aldegunde et al. (2005a). For the present purposes, the most relevant of these parameters is $s^{\{2\}}_{0}$, the first alignment moment of $\bm{j}$ about the incoming relative velocity. Its value specifies the direction of the rotational angular momentum relative to the initial approach. Negative values of $s^{\{2\}}_{0}$ indicate a preference for head-on collisions (rotational angular momentum $\bm{j}_{{}_{\rm HD}}$ perpendicular to $\bm{k}$), whereas positive values indicate a preference for side-on collisions ($\bm{j}_{{}_{\rm HD}}$ mostly parallel to $\bm{k}$). The polarization parameters are calculated from the integration of the polarization-dependent differential cross sections, $S^{\{k\}}_{q}(\theta)$, over the scattering angle, $\theta$. For the $\bm{k}-\bm{j}_{{}_{\rm HD}}-\bm{k}^{\prime}$ correlation the $S^{\{k\}}_{q}(\theta)$ can be determined from $f_{\alpha^{\prime}\Omega^{\prime},\alpha\Omega}$:Aldegunde et al. (2005b)

[TABLE]

where $\langle..,..|..\rangle$ denotes the Clebsch-Gordan coefficient. In the particular case of $q$=0, the $s^{\{k\}}_{q}$ can also be obtained directly from the S-matrix elements Aldegunde et al. (2012).

The middle panel of Fig. 1 shows the polarization moment $s^{\{2\}}_{0}$ as a function of the collision energy for both the $\Delta j=-1$ and $-2$ transitions. At the lowest energies, the moment goes to zero, as required for ultracold collisions Aldegunde et al. (2006). With increasing collision energy, s${}^{\{2\}}_{0}$ takes negative values for both transitions, showing a preference for head-on encounters. However, at the proximity of the resonance, $s^{\{2\}}_{0}$ exhibits markedly different behavior for the two transitions. It turns positive for $\Delta j=-1$ peaking at the energy of the resonance (denoted with a vertical dashed line), while for $\Delta j=-2$ it remains negative. This shows that the resonance for $\Delta j=-1$ is associated with a specific mechanism that is not shared by the $\Delta j=-2$ transition. At energies above the resonance, s${}^{\{2\}}_{0}$ again shows the same trend for both transitions, with a small change around 4.75 K caused by a second resonance (present in both $\Delta j$ transitions) that does not change the mechanism significantly.

To unambiguously analyze the effect of the resonance, the $L=2$ contribution to $s^{\{2\}}_{0}$ is shown in the bottom panel of Fig. 1. It is calculated by including only the $L=2$ elements of the scattering matrix (without considering their coherence with other $L$ values). Regardless of $\Delta j$, the $L=2$ contribution to $s^{\{2\}}_{0}$ goes to zero at ultracold energies, as does $\sigma_{L=2}$ (see top panel). Moreover, up to 0.5 K, including the resonance, the sign of the $L=2$ contribution to $s^{\{2\}}_{0}$ is positive (favoring side-on collisions) while it is negative for higher collision energies. Although at the resonance the sign of the $L$=2 contribution to $s^{\{2\}}_{0}$ is positive for both $\Delta j$=-1 and -2, its magnitude is much larger for the former. Since, $L=2$ collisions dominate around 0.1 K for both $\Delta j$ (see top panel of Fig. 1), these results indicate that the overall change of $s^{\{2\}}_{0}$, and hence of the collision mechanism, is caused by the resonance and not due to a larger contribution of $L=2$.

The distinct mechanism for the resonance suggests that it might be possible to suppress its effect by appropriate state-preparation of the HD rotational angular momentum Perreault et al. (2017, 2018b). The cross sections for different extrinsic preparations can be computed following the procedure described in Ref. 57. If HD is prepared in a directed state, $m$=0, where $m$ is the magnetic quantum number, it leads to the alignment of the internuclear axis along the quantization axis (in the case of refs. 35; 36 the polarization vector of the pump and Stokes lasers). By varying the direction of the laboratory-fixed axis with regard to the scattering frame it is possible to change the external preparations generating different relative geometries of the reactants prior the collision. We will label the different extrinsic preparations using $\beta$ and $\alpha$, where $\beta$ is the polar angle between the polarization vector and the initial relative velocity, and $\alpha$ is the azimuthal angle that defines the direction of the polarization vector with respect to the $\bm{k-k^{\prime}}$ frame. Accordingly, $\beta=0^{\circ}$ implies head-on collisions while $\beta=90^{\circ}$ involves side-on collisions. The equations that relate the observed differential cross section (DCS) for a given preparation ($d\sigma_{\alpha}^{\beta}/d\omega$) is given by:Aldegunde et al. (2005b)

[TABLE]

where $C_{kq}(\beta,\alpha)$ are the modified spherical harmonics and extrinsic moments $A^{(k)}_{q}$ that defined the preparation in the laboratory frame are derived in Ref 57. The integral cross section can be obtained by integrating $d\sigma_{\alpha}^{\beta}/d\omega$ over the scattering and the azimuthal angles, hence depending only on $\beta$.

Fig. 2 shows the cross sections for different experimentally achievable extrinsic preparations of the HD rotational angular momentum for an unpolarized H2 molecule. The results for $\Delta j=-2$ are relatively featureless, and are identical to those shown in Ref. 38. In the Wigner threshold regime, no control can be attained for the integral cross section Aldegunde et al. (2006). With increasing collision energy, however, $\beta=0^{\circ}$ always leads to larger cross sections (by up to a factor of 2). The effect of $\beta=90^{\circ}$, and $\beta=$ mag (magic angle) preparations is milder, leading to only small changes in the cross sections with respect to the unpolarized case.

For $\Delta j=-1$ the situation is similar for energies below the resonance. However, at the resonance the collision mechanism changes rather abruptly, and the $\beta=0^{\circ}$ preparation, which implies head-on collision, leads to a sudden decrease of the cross section, by close to a factor of 5, the most extreme effect that could be observed for any preparation of a sharp $j=2$ state. Since the $\beta=0^{\circ}$ preparation is the same as collisions with $\Omega=0$ exclusively, the fact that the S-matrix element that causes the resonance does not contain $\Omega=0$ leads to the disappearance of the resonance.

Well above the resonance, at $E_{\rm coll}\geq$0.6 K, the effect somewhat reverts back to the behavior observed below the resonance, with $\beta=0^{\circ}$ again leading to a slight increase in the cross section. To sum up, the alignment of $\bm{j}_{{}_{\rm HD}}$ perpendicular to $\bm{k}$ slightly enhance the cross section except at the resonance, where it brings about the suppression of the resonance as if it were switched off. The effect of other preparations $\beta=90^{\circ}$ and $\beta=$ mag is relatively minor and, apparently, does not affect the resonance significantly, as far as the integral cross section is concerned.

Up to this point, we have shown that at the resonance there is a change in the collision mechanism, which can be used to control the cross section by changing the preparation of the HD rotational angular momentum. It has been demonstrated recently by Perreault et al. that it is possible to determine the DCS for different reagent preparations Perreault et al. (2017, 2018b), so we now shift our attention to investigating how the DCS is affected by state-preparation of the HD molecule. Fig. 3 shows the DCS as a function of the scattering angle and collision energy for $\Delta j=-1$. The isotropic DCS (with unpolarized collision partners) is shown in panel (a), which features a slight preference for forward scattering. In particular, the resonance appears as a sharp “ridge” with a clear preference for forward scattering. For $\beta=$0*∘*, panel (b), the situation is completely different. First, the resonance completely vanishes, and at 0.1 K there are no marked changes or discontinuities in the energy dependence of the DCS. In addition, the shape of the DCS displays prominent forward and backward peaks irrespective of the collision energy. At low collision energies there is a third peak in the DCS that only survives for energies below 0.03 K. There is also a resonance at higher collision energies, around $E_{\rm coll}\sim$ 5 K, which unlike the 0.1 K resonance is slightly enhanced by this external preparation.

While the $\beta=90^{\circ}$ and $\beta=$ mag preparations have a minor effect on the integral cross section, the polarization of $\bm{j}_{\rm HD}$ has a dramatic effect on the shape of the DCS. Fig. 3 (panels (c)–(f)) shows the effect of $\beta=90^{\circ}$ and $\beta=$ mag and $\alpha=0^{\circ},180^{\circ}$ preparations on the DCS. The shape and magnitude of the DCS for all these cases differ from each other and from the isotropic case. Moreover, all of them show distinct features at the resonance. For $\beta=90^{\circ}$, $\alpha=0,90^{\circ}$, the DCS at the resonance has two prominent peaks at around $30^{\circ}$ and $150^{\circ}$, while for $\beta=$ mag and $\alpha=0^{\circ}$ there is a strong enhancement of forward scattering at the resonance. While for all non-zero $\beta$ values the resonance at 0.1 K is present, its angular distribution is strongly sensitive to $\beta$, showing that the resonance can be used to control not just the magnitude of the cross section, but also the scattering direction. Heid et al. This provides a powerful tool to elucidate stereodynamics of resonance-mediated collisions and fine-tune calculated interaction potentials against controlled experiments.

To gain further insight into the reaction mechanism we analyze the remaining polarization parameters besides s${}^{\{2\}}_{0}$. For initial $j=2$ there are 12 independent parameters of which 8 contribute to the alignment of the internuclear axis distribution. To aid the interpretation of these parameters we use “stereodynamical portraits” de Miranda and Aoiz (2004); de Miranda et al. (2004), which show the spatial distribution of the angular momentum on the internuclear axis for a given polarization of the rotational angular momentum, with the $z$ axis parallel to $\bm{k}$. Top panel of figure 4 presents the stereodynamical portraits associated with the internuclear axis of HD for $\Delta j=-1$. At the lowest energies considered here, the HD internuclear axis is contained in the scattering plane, although it does not show a significant preference towards head-on or side-on encounters. Just below the resonance, however, it starts to show a strong preference towards head-on collisions (typically associated with small impact parameters). A sudden change of the mechanism occurs at the resonance, with a clear preference for side-on encounters (internuclear axis perpendicular to $z$). Just above the resonance the internuclear axis remains perpendicular to the approach direction, but preferentially contained in the $xy$ plane. With increasing collision energy, the internuclear axis is no longer aligned along or perpendicular to $z$. For $\Delta j=-2$, the stereodynamical portraits are similar except at the resonance found for $\Delta j=-1$ where, in contrast, the head-on encounters are preferred, evincing the dramatic change of the reaction mechanism caused by the resonance.

Altogether, these results demonstrate that, in the cold-energy regime, inelastic collisions between HD($v$=1,$j$=2) and p-H2 are controlled by a resonance at 0.1 K that causes profound changes to the reaction mechanism that favors side-on collisions, typically associated with large impact parameters, over head-on collisions that would have been preferred if the resonance were absent. This sudden change in mechanism permits exquisite control of the collision outcome by using different preparations of the HD internuclear axis, and makes it possible to switch-off the resonance altogether. The effect of the initial HD alignment becomes most evident in the DCS, which changes dramatically for the alternative preparations investigated. Energy resolved measurements of angular distribution of state-prepared HD in collisions with H${}_{2}(v=0,j=0)$ would be desirable to validate these predictions.

P.G.J. acknowledges funding by the Fundación Salamanca city of culture and knowledge (programme for attracting scientific talent to Salamanca). J.F.E.C. gratefully acknowledges support from the Dodd-Walls Centre for Photonic and Quantum Technologies. H. G. acknowledges US Department of energy (DE-SC0015997). M.B. thanks support of the UK EPSRC (to M.B. via Programme Grant EP/L005913/1). N.B. acknowledges support from the US National Science Foundation, Grant No. PHY-1806334. P.G.J. and F.J.A. acknowledges funding from the Spanish Ministry of Science and Innovation (Grants No. CTQ2015-65033-P and PGC2018-09644-B-100).