Laser-induced rotation of iodine molecules in He-nanodroplets: revivals and breaking-free

TL;DR

This study investigates how laser pulses induce and control the rotation of iodine molecules in helium nanodroplets, revealing conditions for coherent revivals and their breakdown, which advances understanding of solute-solvent quantum dynamics.

Contribution

It combines experimental laser alignment techniques with angulon theory to elucidate the rotational behavior of molecules in superfluid helium, highlighting the transition from collective to isolated-like rotation.

Findings

Coherent rotational revivals occur at low laser fluence.

High fluence causes revivals to disappear and rapid rotation to dominate.

Transient decoupling of molecule and helium shell explains observed dynamics.

Abstract

Rotation of molecules embedded in He nanodroplets is explored by a combination of fs laser-induced alignment experiments and angulon quasiparticle theory. We demonstrate that at low fluence of the fs alignment pulse, the molecule and its solvation shell can be set into coherent collective rotation lasting long enough to form revivals. With increasing fluence, however, the revivals disappear -- instead, rotational dynamics as rapid as for an isolated molecule is observed during the first few picoseconds. Classical calculations trace this phenomenon to transient decoupling of the molecule from its He shell. Our results open novel opportunities for studying non-equilibrium solute-solvent dynamics and quantum thermalization.

Tables

Table 1. Table 1: Classical calculation of the maximum angular velocity using Eq. ( 1 ) with θ 0 = 45   ° / subscript 𝜃 0 times 45 divide degree absent \theta_{0}=$45\text{\,}\mathrm{\SIUnitSymbolDegree}\text{/}$ for the nine different fluences used in the experiment. From ω 𝜔 \omega the linear speed, v He subscript 𝑣 He v_{\text{He}} , and the rotational energy, E rot ​ ( He ) subscript 𝐸 rot He E_{\text{rot}}(\text{He}) , of the He atoms at the ends of the molecules are calculated – see text.

$F_{\text{kick}}$, $J/\mathrm{cm}{}^2$	$\omega$, ${10}^{10}$ $\mathrm{Hz}\text{/}$	$v_{\text{He}}$, $m/s$	$E_{\text{rot}}\left(\text{He}\right)$, ${\mathrm{cm}}^{-1}$
$0.25$	$2.7$	0$13$	$0.029$
$0.50$	$5.5$	0$26$	$0.12$
$1.2$	$14$	0$65$	$0.71$
$2.5$	$27$	$130$	$2.8$
$3.7$	$41$	$195$	$6.4$
$5.0$	$54$	$260$	$11$
$6.4$	$70$	$338$	$19$
$7.4$	$81$	$390$	$26$
$8.7$	$95$	$454$	$35$

Equations

$$\omega =\frac{1}{2}\frac{\Delta \alpha F_{\text{kick}}\sin (2{\theta }_0)}{I_{\text{eff}}{\epsilon }_{0}c},$$

$$\begin{gathered} ⟨{\cos }^2{\hat{\theta }}_{2D}⟩(t) \\ =\sum _{j,j^{\prime },j_1,j_2,m}{c}_{j^{\prime }}^{\ast }{c}_j{e}^{it(E_{j_2}+E_{j^{\prime }}-E_{j_1}-E_j)/2}⟨j_{2}m|{\cos }^2{\hat{\theta }}_{2D}|j_{1}m⟩ \\ \times \int d{\Omega }_1\int d{\Omega }_2{Y}_{j^{\prime }{m}_0}^{\ast }({\Omega }_2)Y_{j_{2}m}({\Omega }_2)Y_{j{m}_0}({\Omega }_1)Y_{j_{1}m}^{\ast }({\Omega }_1) \\ \times e^{{\alpha }_1{t}^2\left[\frac{4\pi }{5}{\sum }_{\mu }{Y}_{2,\mu }({\Omega }_2)Y_{2,\mu }^{\ast }({\Omega }_1)-1\right]}. \end{gathered}$$

$$\hat{H}=-\frac{{\hslash }^2}{2M}{\nabla }_0^2+B{\hat{L}}^2-\frac{{\hslash }^2}{2m}\sum _i{\nabla }_i^2+\sum _{i}u({\mathbf{r}}_0,{\mathbf{r}}_i,\Omega )+\sum _{i<j}v(|{\mathbf{r}}_i-{\mathbf{r}}_j|)$$

$$\rho ({\mathbf{R}}_0,{\mathbf{R}}_M;\beta )=\int d{\mathbf{R}}_1\cdots {\mathbf{R}}_{M-1}\rho ({\mathbf{R}}_0,{\mathbf{R}}_1;\tau )\cdots \rho ({\mathbf{R}}_{M-1},{\mathbf{R}}_M;\tau ).$$

$${\rho }_B(\mathbf{R},\mathbf{R};\beta )=\frac{1}{N!}\sum _P\rho (\mathbf{R},P\mathbf{R};\beta )$$

$$F_{\ell }(t)=\frac{4\pi }{2\ell +1}\frac{1}{Z}\sum _m\mathrm{Tr}[Y_{\ell m}^{+}(t)Y_{\ell m}(0)e^{-\beta \hat{H}}]$$

$$F_{\ell }(t)=\frac{4\pi }{2\ell +1}\frac{3}{Z}\sum _{{\ell }_1,{\ell }_2}\frac{(2{\ell }_1+1)(2{\ell }_2+1)}{4\pi }{\left(\genfrac{}{}{0ex}{}{{\ell }_1}{0}\genfrac{}{}{0ex}{}{\ell }{0}\genfrac{}{}{0ex}{}{{\ell }_2}{0}\right)}^2{e}^{-(\beta -t)B{\ell }_2({\ell }_2+1)}{e}^{-tB{\ell }_1({\ell }_1+1)}$$

$$\hat{H}=B{\hat{\mathbf{J}}}^2+\sum _{k\lambda \mu }{\omega }_k{\hat{b}}_{k\lambda \mu }^{†}{\hat{b}}_{k\lambda \mu }+\sum _{k\lambda \mu }{U}_{\lambda }(k)\text{leavevmode}[Y_{\lambda \mu }^{\ast }(\hat{\theta },\hat{\varphi }){\hat{b}}_{k\lambda \mu }^{†}+Y_{\lambda \mu }(\hat{\theta },\hat{\varphi }){\hat{b}}_{k\lambda \mu }],$$

$$\hat{S}=e^{-i\hat{\varphi }\otimes {\hat{\Lambda }}_z}{e}^{-i\hat{\theta }\otimes {\hat{\Lambda }}_y}{e}^{-i\hat{\gamma }\otimes {\hat{\Lambda }}_z}.$$

$$\hat{\mathbf{\Lambda }}=\sum _{k\lambda \mu \nu }{\hat{b}}_{k\lambda \mu }^{†}{\mathbf{\sigma }}_{\mu \nu }^{\lambda }{\hat{b}}_{k\lambda \nu }$$

$$\hat{\mathcal{H}}\equiv {\hat{S}}^{-1}\hat{H}\hat{S}=B(\hat{\mathbf{L}}-\hat{\mathbf{\Lambda }}{)}^2+\sum _{k\lambda \mu }{\omega }_k{\hat{b}}_{k\lambda \mu }^{†}{\hat{b}}_{k\lambda \mu }+\sum _{k\lambda }{\tilde{U}}_{\lambda }(k)\left[{\hat{b}}_{k\lambda 0}^{†}+{\hat{b}}_{k\lambda 0}\right]$$

$$|{\psi }_{LM}⟩=e^{{\sum }_{k\lambda }\frac{{\tilde{U}}_{\lambda }(k)}{{\omega }_k}\left({\hat{b}}_{k\lambda 0}-{\hat{b}}_{k\lambda 0}^{†}\right)}|0⟩|LM⟩.$$

$$⟨{\hat{\mathbf{\Lambda }}}^2⟩\equiv ⟨{\psi }_{LM}|{\hat{\mathbf{\Lambda }}}^2|{\psi }_{LM}⟩=\sum _{k\lambda }\lambda (\lambda +1)\frac{{\tilde{U}}_{\lambda }^2(k)}{{\omega }_k^2},$$

$$B_{\text{eff}}=B{\left(1-\beta \right)}^2,$$

$$\beta ={\left(\frac{1}{2}\sum _{k,\lambda }\lambda (\lambda +1)\frac{{\tilde{U}}_{\lambda }(k{)}^2}{{\omega }_k^2}\right)}^{1/2}.$$

$$\hat{V}(t)=-\eta \delta (t){\cos }^2\hat{\theta }.$$

$$|\psi (t)⟩=e^{-i\hat{H}t}{e}^{\eta {\cos }^2\hat{\theta }}|{\psi }_0⟩.$$

$$|{\psi }_0⟩=e^{{\sum }_{k\lambda \mu }\frac{{\tilde{U}}_{\lambda }(k)}{{\omega }_k}\left({\hat{b}}_{k\lambda \mu }{Y}_{\lambda \mu }(\hat{\Omega })-{\hat{b}}_{k\lambda \mu }^{†}{Y}_{\lambda \mu }^{\ast }(\hat{\Omega })\right)}|j_0,m_0{⟩}_{\mathrm{mol}}\otimes |0{⟩}_{\mathrm{b}},$$

$$|\psi (t)⟩=e^{-i\hat{H}t}\sum _j{c}_j{e}^{{\sum }_{k\lambda \mu }\frac{{\tilde{U}}_{\lambda }(k)}{{\omega }_k}\left({\hat{b}}_{k\lambda \mu }{Y}_{\lambda \mu }(\hat{\Omega })-{\hat{b}}_{k\lambda \mu }^{†}{Y}_{\lambda \mu }^{\ast }(\hat{\Omega })\right)}|j,m_0{⟩}_{\mathrm{mol}}\otimes |0{⟩}_{\mathrm{b}},$$

$$⟨{\cos }^2{\theta }_{2D}⟩(t)\equiv ⟨\psi (t)|{\cos }^2{\hat{\theta }}_{2D}|\psi (t)⟩,$$

$$e^{-it\hat{H}}\approx e^{-itB{\hat{\mathbf{J}}}^2/2}{e}^{-it{\sum }_{k\lambda \mu }{U}_{\lambda }(k)\left[Y_{\lambda \mu }^{\ast }(\hat{\theta },\hat{\varphi }){\hat{b}}_{k\lambda \mu }^{†}+Y_{\lambda \mu }(\hat{\theta },\hat{\varphi }){\hat{b}}_{k\lambda \mu }\right]}{e}^{-itB{\hat{\mathbf{J}}}^2/2},$$

$$⟨{\cos }^2{\hat{\theta }}_{2D}⟩(t)=\sum _{j,j^{\prime },j_1,j_2,m}{c}_{j^{\prime }}^{\ast }{c}_j{e}^{it(E_{j_2}+E_{j^{\prime }}-E_{j_1}-E_j)/2}⟨j_{2}m|{\cos }^2{\hat{\theta }}_{2D}|j_{1}m⟩$$

$$\times \int d{\Omega }_1\int d{\Omega }_2{Y}_{j^{\prime }{m}_0}^{\ast }({\Omega }_2)Y_{j_{2}m}({\Omega }_2)Y_{j{m}_0}({\Omega }_1)Y_{j_{1}m}^{\ast }({\Omega }_1)e^{({\alpha }_1{t}^2+{\alpha }_2)\left[\frac{4\pi }{5}{\sum }_{\mu }{Y}_{2,\mu }({\Omega }_2)Y_{2,\mu }^{\ast }({\Omega }_1)-1\right]},$$

$${\alpha }_1\equiv \sum _k{U}_2^2(k),{\alpha }_2\equiv \sum _k\frac{U_2^2(k)}{{\omega }_k^2}.$$

$$S(t)={\left|\sum _j|c_j{|}^2{e}^{-it{E}_j}\right|}^2{e}^{-{\alpha }_1{t}^2}.$$

$$P_j=Z^{-1}{e}^{-B_{\text{eff}}j(j+1)/(k_{B}T)}.$$

$$P_j^{\mathrm{even}}$$

$$P_j^{\mathrm{odd}}$$

$$⟨{\cos }^2{\theta }_{2D}{⟩}_{\mathrm{even}(\mathrm{odd})}(t)=\sum _{j_0\in \mathrm{even}(\mathrm{odd}),m_0}⟨\psi (t)|{\cos }^2{\theta }_{(2D)}|\psi (t){⟩}^{(j_0,m_0)}\times P_j^{\mathrm{even}(\mathrm{odd})},$$

$$⟨{\cos }^2{\theta }_{2D}⟩(t)=\frac{15\times ⟨{\cos }^2{\theta }_{2D}{⟩}_{\mathrm{even}}(t)+21\times ⟨{\cos }^2{\theta }_{2D}{⟩}_{\mathrm{odd}}(t)}{15+21}.$$

Full text

Usually, molecules dissolved in a liquid are not rotating freely due to the intermolecular forces exerted by the surrounding solvent. An important exception is molecules embedded in liquid helium nanodroplets where high-resolution infrared Grebenev et al. (2000) and microwave Lehnig et al. (2009) spectroscopies display discrete rotational structure. These observations along with theoretical modelling has established a picture that molecules inside He nanodroplets can rotate frictionless although followed by a local solvation shell of He atoms. This shell increases the effective molecular moment of inertia compared to the gas-phase value Toennies and Vilesov (2004); Choi et al. (2006).

These unique properties build the expectation that it should be possible to induce frictionless rotation of molecules inside Helium droplets and follow it in real time. For isolated molecules versatile techniques based on moderately intense fs or ps laser pulses have been developed to control the rotational degrees of freedom Stapelfeldt and Seideman (2003); Ohshima and Hasegawa (2010); Korobenko et al. (2014). In particular, such methods have been extensively used to confine molecular axes to laboratory-fixed axes – methods referred to as alignment and orientation Stapelfeldt and Seideman (2003). Recently, the first time-resolved experiments of molecular rotation inside He droplets revealed that moderately intense laser pulses can induce alignment of molecules Pentlehner et al. (2013); Christiansen et al. (2015). The measurements showed, however, no sign of frictionless rotation. Notably, the transient alignment-recurrences (revivals) characteristic of freely rotating molecules in gas phase were absent. These observations seemed at odds with the prevailing conception of rotational structure obtained through spectroscopy Toennies and Vilesov (2004); Choi et al. (2006).

Here we experimentally demonstrate that a sufficiently weak fs pulse can initiate coherent rotation of iodine molecules together with their He solvation shell – lasting long enough to form revivals. Our observations are rationalized by a quantum theory based on the angulon quasiparticle Schmidt and Lemeshko (2015); Lemeshko and Schmidt (2017); Schmidt and Lemeshko (2016); Yakaboylu and Lemeshko (2017); Midya et al. (2016); Li et al. (2016). For strong alignment pulses the revivals disappear and, instead, strikingly fast rotational dynamics appears immediately after the pulse. Classical estimates indicate that, in this regime, He atoms of the solvation shell detach from the molecule due to the centrifugal force generated by the rapid rotation. This can be seen as a sudden decoupling of the molecule from its solvent and for a short time the rotational motion resembles that of a free molecule.

In our experiment, 10-nm-diameter helium droplets – each doped with at most one iodine (\ceI_2) molecule – are first irradiated by a $450\text{\,}\mathrm{f}\mathrm{s}$ linearly polarized laser pulse at $800\text{\,}\mathrm{n}\mathrm{m}$. The purpose of this kick pulse is to induce alignment of the molecules, i.e. confine their I–I internuclear axis along the polarization direction Stapelfeldt and Seideman (2003). Next, the molecules are Coulomb exploded by a delayed, intense probe pulse ($40\text{\,}\mathrm{f}\mathrm{s}$, $3.7\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$) which produces \ceIHe^+ ion fragments with recoil directions given by the angular distribution of the molecular axes at the instant of the probe pulse. By detecting the emission directions of the \ceIHe^+ ions with a 2D imaging detector at many different kick-probe delays, $t$, the time-dependent degree of alignment, $\langle\cos^{2}\theta_{\text{2D}}\rangle$, can be determined – $\theta_{\text{2D}}$ being the angle between the alignment pulse polarization and the projection of an \ceIHe^+ ion velocity vector on the detector Søndergaard et al. (2017). More details on the experimental setup are provided in the Supplemental Material sup .

Figure 1 shows $\langle\cos^{2}\theta_{\text{2D}}\rangle$ as a function of time for a series of different fluences of the kick pulse, $F_{\text{kick}}$. At low $F_{\text{kick}}$ there is a distinct maximum in $\langle\cos^{2}\theta_{\text{2D}}\rangle$ shortly after the kick pulse [Figure 1(a)-(d)]. The prompt peak grows in amplitude and appears earlier as $F_{\text{kick}}$ is increased [Figure 1(a2)-(d2)]. This behavior is the result of faster rotation and more efficient alignment induced by a stronger kick pulse and appears similar to previous measurements on \ceCH_3I molecules in He droplets Pentlehner et al. (2013). The current data exhibit, however, new, previously unobserved features. First, at $F_{\text{kick}}$ = 1.2 J/cm2 the prompt alignment peak is followed by pronounced yet decreasing oscillations out to $\sim$200\text{,}\mathrm{p}\mathrm{s}. Second, for $F_{\text{kick}}$ = 0.25, 0.50 and 1.2 J/cm2 an oscillatory structure is observed in the interval $550\text{\,}\mathrm{-}750\text{,}\mathrm{p}\mathrm{s}$. The structure is very similar for the three fluences with local maxima and minima at essentially the same times. Third, on average the$\langle\cos^{2}\theta_{\text{2D}}\rangle$curves are gradually decaying in the range$\sim100\text{\,}\mathrm{-}$$1500\text{\,}\mathrm{p}\mathrm{s}$$ for F_{\text{kick}}$ = 0.50, 1.2 and 2.5 J/cm2.

For $F_{\text{kick}}$ $\geq$ 2.5 J/cm2 the structure in the 550\text{\,}\mathrm{-}$$750\text{\,}\mathrm{p}\mathrm{s} interval disappears. Also, the oscillations after the main peak are strongly reduced for $F_{\text{kick}}$ = 2.5 J/cm2 and essentially absent at larger fluences. Instead a substructure in the prompt alignment peak starts to appear at $F_{\text{kick}}$ = 3.7 J/cm2 [Figure 1(e2)]. As the fluence is increased the substructure grows to a prominent sharp peak ending with a maximum already at $t\sim$1.3\text{,}\mathrm{p}\mathrm{s}$$ for $F_{\text{kick}}$ = 8.7 J/cm2 [Figs. 1(i) and 4].

We interpret the oscillations after the prompt peak and the 550\text{\,}\mathrm{-}$$750\text{\,}\mathrm{p}\mathrm{s} structure as manifestations of coherent rotation of the molecules and their local He solvation shell –- hereafter termed He-dressed molecules. To substantiate this interpretation we first model He-dressed molecules as classical rigid rotors driven by the polarizability interaction with the kick pulse. A He-dressed molecule initially at an angle $\theta_{0}$ to the kick pulse polarization [Fig. 2(a3)] gains an angular velocity, $\omega$, of Leibscher et al. (2004):

[TABLE]

where $\Delta\alpha$ is the polarizability anisotropy of \ceI_2 and $I_{\text{eff}}$ is the effective moment of inertia of \ceI_2 in the droplets. No experimental value exists for $I_{\text{eff}}$ so we determined it by a path integral Monte Carlo calculation Zillich et al. (2005), which gave $I_{\text{eff}}=1.7\times I_{0}$ where $I_{0}$ is the moment of inertia of the bare \ceI_2 molecule. The calculated He density around the \ceI_2 molecule is shown in Fig. 3. In our classical calculations a He-dressed molecule is treated as an \ceI_2 molecule rigidly attached to eight He atoms placed in the minima of the \ceI_2–He potential Grebenev et al. (2000); García-Gutierrez et al. (2009) (six He atoms in the central ring around the molecule and two at the ends), see Fig. 2. The value of $I_{\text{eff}}$ determined from this structure (Fig. 2) is essentially equal to the Monte Carlo one.

Equation (1) predicts that the He-dressed molecules are set into end-over-end rotation (Fig. 2) leading to a prompt alignment peak as observed experimentally. Continued rotation for extended times requires that superfluidity of the droplets is undistorted. A simple classical criterion for this is that the linear speed of the outer components of the He-dressed molecules should not exceed the Landau velocity, $v_{L}=56$ m/s Brauer et al. (2013). The highest linear speed is calculated as $v_{\text{He}}=\omega r_{\text{He}}$, where $r_{\text{He}}$ is the distance from a He atom at the ends to the axis of rotation (see Fig. 2). Table 1 displays the values of $\omega$ and $v_{\text{He}}$ for the nine different fluences used in the experiment. For $F_{\text{kick}}$ = 0.25 and 0.50 J/cm2 $v_{\text{He}}$ $<$ $v_{L}$, whereas at $F_{\text{kick}}$ = 1.2 J/cm2 $v_{\text{He}}$ is just above $v_{L}$. At higher fluences, $v_{\text{He}}\gg v_{L}$ for almost all of the He-dressed molecules – independent of their initial orientation. These simple classical considerations indicate that long-time coherent rotational dynamics of the He-dressed molecules is only possible for the three lowest fluences – in accordance with the observations – and illustrated by panels (a1)-(a3) of Fig. 2.

To elucidate the quantum dynamics of the system, we apply the recently-developed angulon theory Lemeshko (2017); Schmidt and Lemeshko (2015); Lemeshko and Schmidt (2017); Schmidt and Lemeshko (2016); Yakaboylu and Lemeshko (2017); Midya et al. (2016); Li et al. (2016); Redchenko and Lemeshko (2016). The angulon represents a quasiparticle consisting of a molecular rotor dressed by a many-body field of superfluid excitations, and can be thought of as a quantum formulation of the He-dressed molecule. Recently it was shown that molecules in superfluid helium form angulons Lemeshko (2017). The case of I2 in helium belongs to the strong-coupling regime, where the molecular kinetic energy is small compared to the molecule-helium interactions Lemeshko (2017); Schmidt and Lemeshko (2016). In this regime, the angulon theory furnishes a closed-form expression for the alignment cosine:

[TABLE]

(in units of $\hbar\equiv 1$). Here $E_{j}=B_{\rm eff}j(j+1)$ are the molecular rotational energies, with $B_{\rm eff}$ the effective rotational constant of \ceI_2. $\alpha_{1}$ parametrizes the anizotropic molecule-helium interactions, with the strong-coupling regime defined by $B_{\rm eff}\ll\sqrt{\alpha_{1}}$. The coefficients, $c_{j}=\langle j,m_{0}|\,\exp(\eta\cos^{2}\hat{\theta})\,|j_{0},m_{0}\rangle$, describe the rotational wavepacket created from the initial molecular state $\,|j_{0},m_{0}\rangle$ by a short laser pulse with a dimensionless intensity $\eta$. In order to compare the theory to experiment, the results of Eq. (2) were averaged over the thermal distribution of the initial states and the finite width of rotational lines due to dephasing was accounted for. More details on the theoretical approach are provided in the Supplemental Material sup .

The strong-coupling angulon theory is straightforward to apply if the molecule-laser interaction energy $\eta\lesssim\sqrt{\alpha_{1}}$. This is the case in Figure 1 (a) and (b), where $\eta/\sqrt{\alpha_{1}}\approx 1.4$ and $2.7$, respectively. Both calculated $\langle\cos^{2}\theta_{\text{2D}}\rangle$ curves (red) are dominated by a prominent peak at early times. For $F_{\text{kick}}$ = 0.25 and 0.50 J/cm2 the prompt alignment peak agrees with the experimental curves although the peak amplitude is somewhat higher for the calculated curves. We ascribe this to an underestimation of the measured degree of alignment due to non-axial recoil effects in the Coulomb explosion process, caused by the He environment Christensen et al. (2016). The fluence of $F_{\text{kick}}$ = 1.2 J/cm2 corresponds to $\eta/\sqrt{\alpha_{1}}\approx 7$ and therefore lies beyond the reach of the strong-coupling angulon theory which predicts a faster initial dynamics compared to the experiment. Nevertheless, the long-time decay of alignment observed experimentally is reproduced. For higher fluences, $F_{\text{kick}}$ = 2.5 J/cm2 and $F_{\text{kick}}$ = 3.7 J/cm2, the molecule-laser interactions dominate over molecule-helium interactions, since $\eta/\sqrt{\alpha_{1}}\approx 14$ and $21$, respectively.

For $F_{\text{kick}}$ = 0.25 and 0.50 J/cm2 low-amplitude structures just before $400\text{\,}\mathrm{a}$nd $800\text{\,}\mathrm{p}\mathrm{s}$ are visible. The angulon model identifies these as the half- and full-revival of the He-dressed molecule, marked by red arrows for the theoretical curves. The locations of the revival structures match $h/(4B_{\text{eff}})$ and $h/(2B_{\text{eff}})$, with $B_{\text{eff}}=\hbar^{2}/(2I_{\text{eff}})$, similar to the well-studied case of isolated molecules. The magnitude of the revivals decreases for larger fluences, and they are no longer visible for F${}_{\text{kick}}\geq 2.5$ J/cm2.

Importantly, the angulon theory predicts that rotational revivals are possible for molecules strongly interacting with superfluid helium. Therefore, we interpret the observed oscillatory structure in the 550\text{\,}\mathrm{-}$$750\text{\,}\mathrm{p}\mathrm{s} interval as a full rotational revival of the He-dressed molecule. Furthermore, we note that the model captures the overall decay of $\langle\cos^{2}\theta_{\text{2D}}\rangle$ observed most clearly for $F_{\text{kick}}$ = 0.50 and 1.2 J/cm2. We quantify this decay by the survival probability (closely related to the Loschmidt echo Zangara et al. (2012)), $S(t)=|\langle\psi(t)|\,\psi_{0}\rangle|^{2}\equiv|\langle\psi_{0}|\,e^{iHt}\,|\psi_{0}\rangle|^{2}$, of the state $\,|\psi_{0}\rangle$ immediately after the kick pulse excitation during time evolution under the angulon Hamiltonian $H$ sup . The time-dependence of the survival probability is shown in Figure 1j. While the gas-phase survival probability (dashed line) exhibits revivals similar to the gas-phase molecular alignment, for I2 in helium we predict a Gaussian decay $S(t)\sim\exp(-\alpha_{1}t^{2})$ sup . The latter comes from redistribution of the angular momentum between the molecule and the superfluid. Note that this decay occurs on a faster timescale compared to the exponential decay common for Markovian reservoirs Ramakrishna and Seideman (2005).

In the high-fluence regime, $\eta/\sqrt{\alpha_{1}}\gg 1$, the strong-coupling angulon theory is not applicable. However, classically, a high-fluence pulse can induce such a fast rotation of the He-dressed molecule that helium atoms detach due to the centrifugal force – a mechanism which, we believe, is responsible for the sharp structure appearing in the prompt alignment peak. A simple criterion for detaching one He atom is: $E_{\text{rot}}(\text{He})>E_{\text{binding}}(\text{He})$, where $E_{\text{rot}}(\text{He})=\tfrac{1}{2}m_{\text{He}}$ $r_{\text{He}}^{2}\omega^{2}$ is the rotational energy of a He atom and $E_{\text{binding}}(\text{He})\approx 16$ cm*-1* is the ground-state binding energy of the \ceHeI_2 complex  García-Gutierrez et al. (2009); Ray et al. (2006). Table 1 displays $E_{\text{rot}}(\text{He})$ calculated for the different fluences. At F${}_{\text{kick}}\gtrsim$ 6 J/cm2 the criterion is met implying that one or indeed several He atoms detach from the molecule (lower panels in Fig. 2) since the binding energies of the first few He atoms are similar Paesani and Whaley (2004).

Fig. 4 compares the short-time alignment dynamics of molecules in He droplets to that of isolated molecules at $F_{\text{kick}}$ = 8.7 J/cm2. In droplets, $\langle\cos^{2}\theta_{\text{2D}}\rangle$ evolves almost as fast as $\langle\cos^{2}\theta_{\text{2D}}\rangle$ of isolated molecules during the first \sim$$2\text{\,}\mathrm{p}\mathrm{s}. In classical terms, this indicates that \ceI_2 rotates almost freely, detached from the He atoms. We observed the same rapid short-time alignment dynamics for \ceOCS and \ceCS_2 molecules in He droplets. At t $>$ $2\text{\,}\mathrm{p}\mathrm{s}$ the free rotation is quenched, which indicates a dynamical re-formation of the He-dressed molecule.

Our results demonstrate that for molecules embedded in He droplets a moderately intense laser pulse can induce coherent collective rotation of a molecule and its solvation shell for times long enough to form rotational revivals. These findings reconcile femtosecond laser-induced molecular alignment and high-resolution infrared and microwave spectroscopy. Our angulon quasiparticle theory rationalizes the observations for the low-fluence experimental results. Future generalization of the theory may lead to a quantitative agreement with experiments in a broad range of laser fluences and is expected to provide new insights into the superfluid behavior of He droplets. Finally, the observed decoupling of the molecule from its He-solvation shell at high fluences draws parallels to the nonlinear response in the solute-solvent interaction of rapidly rotating \ceCN molecules dissolved in ethanol Moskun et al. (2006); Tao and Stratt (2006). Our results open unique opportunities for real-time studies of non-equilibrium solute-solvent dynamics, for instance, by gradually modifying the solvation shell through insertion of other noble gas atoms or even water molecules Choi et al. (2006). Furthermore, experiments on molecules in small helium droplets might yield insight into quantum thermalization of finite many-particle systems Rigol et al. (2008); Polkovnikov et al. (2011).

We thank Richard Schmidt for insightful discussions. JK acknowledges support from People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement No. [291734]. REZ acknowledges support from the Austrian Science Fund (FWF) under grant No. P23535-N20. ML acknowledges support from the Austrian Science Foundation (FWF), under grant No. P29902-N27. HS acknowledges support from the European Research Council-AdG (Project No. 320459, DropletControl) and the Villum Foundation.

S1 Supplemental Material

Appendix A Experimental setup and method

A schematic diagram of the experimental setup used for laser-induced alignment experiments of iodine molecules, both solvated inside helium droplets and isolated in a supersonic beam, is shown in Figure S1. Helium droplets are produced using a continuous helium droplet source with stagnation conditions of $14\text{\,}\mathrm{K}$ and $25\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$, giving \sim$$10\text{\,}\mathrm{n}\mathrm{m} diameter helium droplets Toennies and Vilesov (2004). Shortly after the exit of the continuous source the droplet beam passes through a skimmer with a $1\text{\,}\mathrm{m}\mathrm{m}$ diameter opening and enters a pickup cell containing iodine vapor. The partial pressure of the iodine vapor was kept sufficiently low to ensure the pickup of at most a single iodine molecule. Hereafter the doped droplets pass through a liquid nitrogen trap that captures the majority of the effusive iodine molecules that are not picked up by the droplets. In order to further reduce the contribution from effusive molecules the doped droplets pass through a second skimmer with a $2\text{\,}\mathrm{m}\mathrm{m}$ diameter opening followed by a second liquid nitrogen trap. Finally, the doped droplets enter the interaction region of the target chamber. In this region, the doped helium droplet beam is crossed perpendicularly by two collinear $800\text{\,}\mathrm{n}\mathrm{m}$ pulsed laser beams. The doped droplets are first irradiated with a linearly polarized kick pulse that is used to induce alignment. For the measurements up to $F_{\text{kick}}$ = 5.0 J/cm2 the duration of the kick pulse is $450\text{\,}\mathrm{f}\mathrm{s}$. At this duration the fluence cannot be increased further because the intensity becomes so high that the iodine molecules starts to be ionized by the kick pulse alone. The measurements with $F_{\text{kick}}$ = 6.4, 7.4 and 8.7 J/cm2 are, therefore, recorded with a kick pulse duration of $1300\text{\,}\mathrm{f}\mathrm{s}$. This is still much shorter than the rotational time of the iodine molecule (446 ps in gas phase) and thus keeps the experiment in the strictly non-adiabatic limit of alignment Stapelfeldt and Seideman (2003); Seideman and Hamilton (2005). For consistency, we recorded the alignment experiment at $F_{\text{kick}}$ = 5.0 J/cm2 with the $1300\text{\,}\mathrm{f}\mathrm{s}$ kick pulse and got results essentially identical to those recorded with the $450\text{\,}\mathrm{f}\mathrm{s}$ kick pulse (Fig. 1f in the main text).

After the kick pulse the molecules are Coulomb exploded by a delayed, intense probe pulse (40 fs, $3.7\text{\times}{10}^{14}\text{\,}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$), which produces \ceI^+ or \ceIHe^+ ion fragments. The recoil directions of either ion species are given by the angular distribution of the molecular axes at the instant of the probe pulse. For the measurements reported here the \ceIHe^+ signal was chosen as the observable because these ions can only be produced from molecules inside He droplet  Pentlehner et al. (2013, 2013b). In the case of \ceI^+ ions there is a contribution from those iodine molecules that manage to effuse from the pick-up cell to the interaction region in the target chamber. This contribution is, however, at most a few percent and recording of \ceI^+ images could, therefore, also have been carried out and should give the same rotational dynamics as that obtained from the \ceIHe^+ images.

By detecting the emission directions of the \ceIHe^+ ions with a 2D imaging detector at many different kick-probe delays, $t$, the time-dependent degree of alignment, $\langle\cos^{2}\theta_{2\mathrm{D}}\rangle$, can be determined, where $\theta_{2\mathrm{D}}$ is the angle between the kick pulse polarization and the projection of an \ceIHe^+ ion velocity vector on the detector. The experimental setup is equipped with a pulsed Even-Lavie valve located beneath the target chamber and allows for a molecular beam of isolated iodine molecules to be sent into the interaction region. The alignment dynamics for isolated molecules was recorded under the same laser conditions as those used for the helium droplet experiments. Here \ceI^+ ions were used as observables.

Appendix B Path integral Monte Carlo

Quantum many-body systems of $N$ particles in equilibrium can be mapped to a classical system of polymer chains Feynman and Hibbs (1965); Chandler and Wolynes (1981). The path integral Monte Carlo (PIMC) method exploits this isomorphism. For Bose systems like 4He droplets, finite-temperature results obtained by PIMC can be considered virtually exact, given sufficient simulation time.

The PIMC method calculates equilibrium properties in the canonical Ceperley (1995) or grand canonical Boninsegni et al. (2006) ensemble. In the present work we use the canonical ensemble, thus expectation values of an operator $\hat{A}$ are obtained as $\langle\hat{A}\rangle={1\over Z}{\rm Tr}[e^{-\beta\hat{H}}A]$, where $Z={\rm Tr}[e^{-\beta\hat{H}}]$ is the partition function, $\beta=1/k_{\text{B}}T$, and $\hat{H}$ is the many-body Hamiltonian. In our case of a linear molecule with bare rotational constant $B$ and mass $M$ and $N$ 4He atoms of mass $m$ $\hat{H}$ is

[TABLE]

$\hat{L}$ is the angular momentum operator, $\Omega$ are the two Euler angles of a linear molecule, ${\bf r}_{0}$ is the center of mass coordinate of the molecule and ${\bf r}_{i}$ are the coordinates of the 4He atoms. The interactions are modeled as pair-wise interactions between 4He atoms, $v$, and between 4He atoms and the molecule, $u$. We use the potential by Aziz et al. Aziz et al. (1987) for $v$ and the ab initio potential by Garcia-Gutierrez et al. García-Gutierrez et al. (2009) for $u$. The latter depends not only on the distance $|{\bf r}_{0}-{\bf r}_{i}|$ between 4He atom and molecule, but also on the angle $\theta_{i}$ between the distance vector ${\bf r}_{0}-{\bf r}_{i}$ and the axis of the molecule defined by $\Omega$. It is the dependence on $\theta_{i}$ which leads to the coupling of the rotational dynamics of the molecule to the helium droplet. We neglect the vibrational degree of freedom of the molecule, which for a diatomic molecule like $I_{2}$ is the distance between the two iodine atoms. The vibrational excitation energies are orders of magnitude larger than typical rotational energies and excitation energies in helium. Therefore, the coupling between vibrations and helium are negligible in the study of rotational dynamics. We assume $I_{2}$ to be a rigid rotor, with the two iodine atoms separated by their equilibrium distance of $2.666$ Å.

For PIMC simulations, it is convenient to work in coordinate space. Thus, for calculating expectation values $\langle\hat{A}\rangle$ the many-body density matrix in configuration space, $\rho({\bf R},{\bf R}^{\prime};\beta)=\langle{\bf R}|e^{-\beta\hat{H}}|{\bf R}^{\prime}\rangle$, is sampled using the Metropolis algorithm Metropolis et al. (1953). Here ${\bf R}$ denotes all coordinates of the many-body system, ${\bf R}=(\Omega,{\bf r}_{0},{\bf r}_{1},\dots,{\bf r}_{N})$. A numerical evaluation of $\rho({\bf R},{\bf R}^{\prime};\beta)$ is complicated by the fact that in general the exponential of the many-body Hamiltonian $\hat{H}$ cannot be calculated. Therefore, we split the “imaginary time” interval $\beta$ into small “time steps” $\tau=\beta/M$. This necessitates the introduction of new coordinates at intermediate time slices,

[TABLE]

$({\bf R}_{0},\dots,{\bf R}_{M})$ can be regarded as a discretized path in imaginary time. This way each particle coordinate ${\bf r}_{i}$ is replaced by a whole path of coordinates (“beads”) ${\bf r}_{i,j}$ where the new index $j=0,\dots,M$ labels the discretized imaginary time.

For completing the isomorphism between a quantum system and classical polymers of beads, we choose $\tau$ sufficiently small, such that $\rho({\bf R}_{0},{\bf R}_{1};\tau)$ can be approximated. We use the pair density approximation Ceperley (1995) for the He-He interaction, and the Trotter approximation for the He-molecule interaction, $e^{-\tau(\hat{T}+\hat{V})}=e^{-\tau\hat{V}/2}e^{-\tau\hat{T}}e^{-\tau\hat{V}/2}+{\rm O}(\tau^{3})$, where $\hat{T}$ and $\hat{V}$ are the (non-commuting) kinetic and interaction terms of the Hamiltonian $\hat{H}=\hat{T}+\hat{V}$, respectively. The Trotter approximation requires to use a relatively small time step $\tau=1/80$ K, which determines the number of beads as $M={\beta\over\tau}$ for a given inverse temperature $\beta$. If no off-diagonal operators such as the one-body density matrix need to be averaged, we can set ${\bf R}_{M}={\bf R}_{0}$. Thus all polymers are closed loops, one for each quantum particle.

We need to account for the indistinguishability of quantum particles. Bose statistics is implemented by symmetrization of the density matrix

[TABLE]

where the sum is over all permutations $P$. As can be seen from the right hand side of Eq. (S2), the symmetrization corresponds to reconnecting the imaginary time paths to form larger polymers. For a detailed review of the PIMC method for bosons see Ref. Ceperley (1995), for the application to dopants in 4He clusters see Refs. Blinov et al. (2004); Zillich et al. (2005).

In addition to static quantities, PIMC allows in principle also to calculate dynamical properties. It is straightforward to calculate correlation functions $\langle\hat{A}(t)\hat{A}(0)\rangle$, that are related to measurable spectra via the fluctuation-dissipation theorem. However, PIMC provides these correlation functions only in imaginary time. The analytic continuation of imaginary time data with statistical noise due to finite sampling time to real time is an ill-posed problem. In the present case, we are interested in the rotational spectrum of a linear molecule, which can be obtained from

[TABLE]

where $Y_{\ell m}$ are spherical harmonics. The spectrum $S_{\ell}(\omega)$ of rotational excitations $J\to J+\ell$ (where $\ell=2$ in the case of a homo-nuclear linear rotor) can be obtained by inverting the Laplace transform, $F_{\ell}(t)=\int_{-\infty}^{\infty}d\omega e^{-t\omega}S_{\ell}(\omega)$. Since the inversion is an ill-posed problem, it can only be done approximately and only if the error bars of $F_{\ell}(t)$ are very small. Therefore, we opted to simply fit the known solution $F^{0}_{\ell}(t)$ for a free linear rotor,

[TABLE]

to correlation function $F_{\ell}(t)$ obtained with PIMC for $I_{2}$ in helium with $B$ acting as fit parameter. This fit yields an effective $B=B_{\rm eff}$, under the assumption that the rotational spectrum of $I_{2}$ in helium is essentially that of a linear rotor with a renormalized rotational constant. For heavy rotors this assumption has been validated by experiments. Although the effective distortion constant $D_{\rm eff}$ usually increases by orders of magnitude in helium compared to the gas phase value, $D_{\rm eff}$ is still small and, therefore, we cannot determine it from an improved fit to a free linear rotor with distortion constant $D$ as second parameter.

PIMC simulations of droplets of $10^{3}$ or $10^{4}$ 4He atoms, as produced in the experiments, would be very demanding. Instead we performed two kinds of simulations of $I_{2}$ in helium, that bracket the experimental situation from both sides: $I_{2}$ in 4He clusters of $N=150$ atoms, much smaller than in experiment; and $I_{2}$ in bulk 4He. Bulk simulations are realized with periodic boundary conditions. 511 4He atoms and one $I_{2}$ are put in a simulation box of side length $L=28.6$Å. $L$ is determined by the condition that the 4He density in the molecule frame of reference, $\rho(r,\theta)$, approaches the equilibrium density of bulk 4He, $\rho_{eq}=0.02186\,$Å*-3*, for large distance $r$ between 4He atoms and molecule. The largest distance compatible with periodic boundaries is $L/2$, which is large enough to obtain a 4He density $\rho(r,\theta)$ that fluctuates only slightly around $\rho_{\text{eq}}$. Therefore, we believe that simulations for this size provide a good approximation to the bulk limit. Simulations of $I_{2}$ in 4He cluster were done at a temperature of $T=0.31$K, typical for 4He droplets in equilibrium. For simulations of $I_{2}$ in bulk helium, approximated by 511 4He atoms, we doubled the temperature to $T=0.62$K, in order to reduce the computational demands by cutting the number of beads in half. From the bulk simulations we obtained a ratio of $B_{\rm eff}/B=0.60$, while simulations of I2 in a cluster of $N=150$ 4He atoms gave a very similar value of $B_{\rm eff}/B=0.58$. We note that for I2 the ratio $B_{\rm eff}/B$ is large compared to values found for other heavy linear rotors in helium. This smaller relative reduction is due to the particularly large moment of inertia $I$ of I2 already in the gas phase. A significant relative increase of $I$, and thus significant relative reduction of $B$ requires, therefore, a much larger effect of the helium environment than, e.g., for the well-studied OCS molecule in helium, with a moment of inertia more than five times smaller.

Appendix C The angulon theory

C.0.1 The angulon Hamiltonian

The theoretical approach used here is based on the recently-developed angulon theory Schmidt and Lemeshko (2015, 2016); Lemeshko (2017); Lemeshko and Schmidt (2017); Midya et al. (2016); Yakaboylu and Lemeshko (2017); Redchenko and Lemeshko (2016); Li et al. (2016). We start from the angulon Hamiltonian, which describes a rotating molecule coupled to a bosonic bath Schmidt and Lemeshko (2015):

[TABLE]

where we used the notation $\sum_{k}\equiv\int dk$, and set $\hbar\equiv 1$. The first term of Eq. (S3) corresponds to the rotational kinetic energy of a linear-rotor molecule, with $\hat{\mathbf{J}}$ the angular momentum operator and $B=1/(2I)$ the molecular rotational constant, where $I$ is the molecular moment of inertia. The bare eigenstates of the molecular impurity are given by the $(2L+1)$-fold degenerate states, $|L,M\rangle$, with energies $E_{L}=BL(L+1)$. Here $L$ is the angular momentum quantum number, and $M$ is its projection on the laboratory-frame $z$-axis.

The second term of Eq. (S3) gives the kinetic energy of the superfluid excitations, such as phonons and rotons, whose spectrum is described by the dispersion relation $\omega_{k}$. Here, the operators $\hat{b}^{\dagger}_{k\lambda\mu}$ ($\hat{b}_{k\lambda\mu}$) are creating (annihilating) a superfluid excitation with linear momentum $k=|\mathbf{k}|$, the angular momentum $\lambda$, and its projection, $\mu$, onto the $z$-axis. These operators can be obtained using the spherical-harmonic expansion of the usual creation/annihilation operators in Cartesian space, $\hat{b}^{\dagger}_{\mathbf{k}}$ and $\hat{b}_{\mathbf{k}}$, see Refs. Schmidt and Lemeshko (2015, 2016); Lemeshko and Schmidt (2017) for details.

The last term of the Hamiltonian (S3) describes the angular-momentum exchange between the molecular impurity and the superfluid, where the coupling constants $U_{\lambda}(k)$ are proportional to the Legendre moments of the molecule–Helium potential energy surface in Fourier space. Here $Y_{\lambda\mu}(\hat{\theta},\hat{\phi})$ are spherical harmonics Varshalovich et al. (1988), which depend on the molecular angle operators in the laboratory frame, $(\hat{\theta},\hat{\phi})$. This type of coupling, explicitly dependent on the three-dimensional impurity orientation, makes Eq. (S3) substantially different from other impurity problems such as, e.g., the Bose-polaron Devreese (2015) or the spin-boson Leggett et al. (1987) models.

Originally, the Hamiltonian (S3) was derived to describe an ultracold molecule interacting with a dilute BEC, where the coupling constants $U_{\lambda}(k)$ assume a simple analytic form Schmidt and Lemeshko (2015); Lemeshko and Schmidt (2017). Helium, on the other hand, represents a dense, strongly-interacting superfluid, which makes it quite challenging to derive the coupling constants from first principles.

However, by analogy with effective field theories of nuclear Machleidt and Entem (2011) and condensed matter Fradkin (2013) physics, the angulon Hamiltonian (S3) can be approached from a phenomenological perspective, where the effective low-energy constants are extracted from experiment or ab initio calculations. As an example, recently it was shown that the effective rotational constants of 25 different molecules in superfluid helium can be obtained from the angulon theory in good agreement with experiment, based on only two phenomenological parameters Lemeshko (2017). Here we pursue a similar approach to calculate the dynamical properties of the I2 molecule in helium.

C.0.2 Effective rotational constants

Interactions of a heavy molecule, such as I2, with helium can be most naturally understood if one rewrites the Hamiltonian (S3) in the rotating molecular frame Lemeshko (2017). This is achieved by applying a canonical transformation recently introduced by Schmidt and Lemeshko Schmidt and Lemeshko (2016):

[TABLE]

Here $(\hat{\phi},\hat{\theta},\hat{\gamma})$ are the angle operators which act in the Hilbert space of the molecular rotor, and

[TABLE]

is the total angular momentum operator of the superfluid excitations, acting in their corresponding Hilbert space. The angular momentum matrices, $\bm{\sigma}^{\lambda}\equiv\{\sigma^{\lambda}_{-1},\sigma^{\lambda}_{0},\sigma^{\lambda}_{+1}\}$, fulfill the $SO(3)$ algebra in the representation of angular momentum $\lambda$. Thus, the transformation of Eq. (S4) transfers the superfluid degrees of freedom into the rotating molecular frame.

The transformed Hamiltonian assumes the following form:

[TABLE]

where $\tilde{U}_{\lambda}(k)=\sqrt{(2\lambda+1)/(4\pi)}U_{\lambda}(k)$. The operator $\hat{\mathbf{L}}\equiv\hat{\mathbf{J}}+\hat{\mathbf{\Lambda}}$ is the total angular momentum operator, which acts in the molecular Hilbert space. The components of $\hat{\mathbf{L}}$ define projections of total angular momentum in the rotating molecular frame and, therefore, obey anomalous commutation relations Lefebvre-Brion and Field (2004); Biedenharn and Louck (1981). In the absence of external fields, total angular momentum $\hat{\mathbf{L}}$ is conserved, which allows to solve the problem for each value of $L$ separately.

Another advantage of the transformed Hamiltonian (S6) is that it can be diagonalized exactly in the limit of a slowly rotating molecule, $B\to 0$. There, for each total angular momentum state, $\,|LM\rangle$, the ground state is given by:

[TABLE]

Note that the bosonic coherent state of Eq. (S7) involves an infinite number of superfluid excitations and, therefore, describes a collective anizotropic displacement of helium atoms. Such a deformation can be thought of as a microscopic formulation of the ‘nonsuperfluid helium shell’ which rotates along with the molecule Grebenev et al. (2000); Toennies and Vilesov (2004).

It is important to note that the angulon theory based on the transformed Hamiltonian, Eq. (S6), provides a simple physical explanation for renormalization of molecular rotational constants in superfluid helium. The rotational energy of the molecular impurity is defined by the first term of Eq. (S6), while the rest of terms ultimately determine how many phonons are excited due to the molecule-helium interactions. In the absence of helium, the total angular momentum is given by that of a free molecule, $\hat{\mathbf{L}}\equiv\hat{\mathbf{J}}$. In the presence of helium, $\hat{\mathbf{L}}$ is still a conserved quantity, however, the stronger the molecule-helium interactions the larger is the angular momentum of the superfluid, $\hat{\mathbf{\Lambda}}$. Thus, for a given total angular momentum $L$, the rotational energy is lower in the presence of helium ($\hat{\mathbf{\Lambda}}\neq 0$) compared to a gas-phase molecule ($\hat{\mathbf{\Lambda}}=0$), which leads to renormalization of the molecular rotational constant.

In our approach, we calculate the amount of the angular momentum transferred to the superfluid for the state of Eq. (S7):

[TABLE]

and replace the boson angular momentum operator in Eq. (S6) by its expectation value, $\hat{\mathbf{\Lambda}}\to\langle\Lambda\rangle\equiv\langle\hat{\mathbf{\Lambda}}^{2}\rangle^{1/2}$. Then, assuming that $\hat{\mathbf{\Lambda}}$ points along the total angular momentum, $\hat{\mathbf{\Lambda}}\sim\hat{\mathbf{L}}$, we can evaluate the effective rotational constant as:

[TABLE]

where

[TABLE]

Within our approach, we treat $\beta$ as a phenomenological parameter and set it to $\beta=0.23$, which reproduces the results of quantum Monte Carlo calculations giving $B_{\text{eff}}=0.6B$. In addition, Eq. (S10) enables us to calculate the $\alpha_{2}$ parameter as discussed below.

C.0.3 Dynamics of I2 in helium

We perform the calculations of the time evolution in the laboratory frame, as given by the Hamiltonian (S3), with $B$ replaced by $B_{\text{eff}}=0.6\,B_{0}$, as discussed above. Since the pulse is very short, $\tau\ll B^{-1},U_{\lambda}(k)^{-1},\omega_{k}^{-1}$, the state after the pulse can be found within the impulsive approximation Henriksen (1999). To this end, we assume that the interaction with the laser pulse is described by the potential

[TABLE]

Here, $\eta$ is a dimensionless parameter related to the fluence of the kick pulse. The laser fluences presented in Fig. 1(a)–(e) correspond to the values of $\eta=2.6,5.2,13,26$, and $39$, respectively. The subsequent time evolution of the wave function is given by:

[TABLE]

Before the laser pulse, the He-dressed molecular states in the laboratory frame are obtained by applying an inverse transformation of Eq. (S4) to Eq. (S7), which results in:

[TABLE]

where $\hat{\Omega}\equiv(\hat{\theta},\hat{\phi})$ and $|0\rangle_{\rm b}$ is the bosonic vacuum state. Since after the pulse the wave function represents a superposition of rotational energy levels, the time-evolution of the angulon state is given by:

[TABLE]

where the coefficients $c_{j}$ depend on $j_{0}$, $m_{0}$, and $\eta$. For the wave function (S14), we calculate the evolution of the molecular alignment as

[TABLE]

where $\cos^{2}\hat{\theta}_{\rm 2D}\equiv\cos^{2}{\hat{\theta}}/(\cos^{2}{\hat{\theta}}+\sin^{2}{\hat{\theta}}\sin^{2}{\hat{\phi}})$ is a two-dimensional projection of the three-dimensional alignment cosine, $\cos^{2}\hat{\theta}$, as measured in the experiments.

Performing the time evolution of the considered wave function using Eqs. (S3) and (S14) is an involved many-particle problem because the interaction term of the Hamiltonian does not commute with the rest of the terms. First of all, we note that the boson kinetic energy can be eliminated by rewriting the Hamiltonian in the rotating frame, which corresponds to the replacements $\hat{b}^{\dagger}_{k\lambda\mu}\to\hat{b}^{\dagger}_{k\lambda\mu}e^{-i\omega_{k}t}$, $\hat{b}_{k\lambda\mu}\to\hat{b}_{k\lambda\mu}e^{i\omega_{k}t}$. Since in the final expression for the alignment cosine the oscillating exponents cancel with their corresponding complex conjugates, the boson kinetic energy contributes only through the initial state, Eq. (S13). In order to account for the rest of the Hamiltonian, we apply an expansion of the Suzuki-Trotter type:

[TABLE]

which becomes exact in the limit of small $t$. Furthermore, in order to decrease the number of free parameters of the model, we take into account only the leading anisotropic term, $\lambda=2$, of the He–I2 PES García-Gutierrez et al. (2009); Lemeshko (2017). The resulting alignment cosine can be derived in closed form:

[TABLE]

where $\int d\Omega\equiv\int\sin{\theta}\,d\theta\,d\phi$ and we have defined

[TABLE]

The $\alpha_{1}$ parameter determines the decay rate of the alignment. We take $\alpha_{1}=10$ (in units of $1/B_{\text{eff}}^{2}$), which reproduces the decay rate observed in experiment for $F_{\text{kick}}$ = 1.2 J/cm2. The $\alpha_{2}$ parameter can be determined using Eq.(S8) and the value of $B_{\text{eff}}=0.6B_{0}$ and turns out to be very small $\alpha_{2}\approx 0.04$. Previous experiments on microwave spectroscopy of molecules in helium droplets have shown a broadening of the molecular rotational lines, which amounts to $\sim 100$ MHz in the case of OCS Lehnig et al. (2009). The line shapes are attributed to inhomogeneous broadening, and at least for CO in helium this was confirmed by calculations Zillich et al. (2008). In order to account for this effect, we introduce a Gaussian broadening of $B_{\text{eff}}$ values with a standard deviation $\gamma=0.05\leavevmode\nobreak\ B_{\text{eff}}$, which corresponds to half width at half maximum of $66$ MHz for I2. Explicitly, we use the energy levels $E_{j}\equiv(B_{\text{eff}}+\gamma x)j(j+1)$, where $x$ is drawn from a normal distribution and we integrate the result for $\langle\cos^{2}{\theta_{\rm 2D}}\rangle(t)$ over $x$.

In addition, we calculate the survival probability of the initial state as:

[TABLE]

C.0.4 Ensemble averaging: temperature and symmetry

Since the temperature of He droplets, $T=0.38\;{\rm K}$, is larger than the rotational constant $B_{\text{eff}}=0.032\;{\rm K}$, one needs to account for the thermal distribution of population over several rotational states, as given by the Boltzmann distribution:

[TABLE]

where $k_{B}$ is the Boltzmann constant and $Z\equiv\sum_{j}P_{j}$ is the partition function. For the temperature stated above, it is sufficient to truncate the sum at $j=6$, which corresponds to $28$ lowest $\,|jm\rangle$-levels.

In addition to the thermal distribution, we account for the $21:15$ ortho-to-para ratio of I2. This corresponds to an averaging over molecules in even and odd rotational states:

[TABLE]

where the reference energy for odd states is $2B_{\text{eff}}$ – the rotational energy of the lowest odd state with $j=1$. The corresponding partition functions are given by $Z_{\rm even(odd)}\equiv\sum_{j\in{\rm even(odd)}}P_{j}^{\rm even(odd)}$. The thermally-averaged result for the alignment, $a(t)$, of the odd and even states is given by:

[TABLE]

where the ${(j_{0},m_{0})}$ superscript denotes alignment with the starting $|j_{0},m_{0}\rangle$ state of the molecule. Finally, the ensemble-averaged result is obtained as

[TABLE]

Eq. (S24) was the one we used to compare the theory to experiment.