Dynamics of molecular rotors in bulk superfluid helium

Molecules immersed in liquid helium are excellent probes of superfluidity. Their electronic, vibrational, and rotational dynamics provide valuable clues about the superfluid at the nanoscale. Here we report on the experimental study of the laser-induced rotation of helium dimers inside the superfluid 4He bath at variable temperature. The coherent rotational dynamics of He2∗ is initiated in a controlled way by ultrashort laser pulses and tracked by means of time-resolved laser-induced fluorescence. We detect the decay of rotational coherence on the nanosecond time scale and investigate the effects of temperature on the decoherence rate. The observed temperature dependence suggests a nonequilibrium evolution of the quantum bath, accompanied by the emission of the wave of second sound. The method offers ways of studying superfluidity with molecular nanoprobes under variable thermodynamic conditions.


INTRODUCTION
The enduring fascination with the superfluid phase of liquid helium (LHe), known as He II, stems in part from the fact that several important questions have yet to be understood in relation to the unique physical properties of this strongly interacting quantum system. Among them is the microscopic interpretation of the intrinsically macroscopic two-fluid model of superfluidity, which describes the system as a mixture of two interpenetrating components: A normal fluid that behaves like a classical liquid, and a superfluid that exhibits zero viscosity and can flow without resistance [for a recent review, see (1)]. According to Landau's theory (2), the normal component consists of collective elementary excitations, such as phonons and rotons, whose dispersion and scattering properties govern the behavior of the system as a whole. The two-fluid model also predicts the phenomenon of second sound-a temperature wave, which moves through the liquid via the periodic exchange between the normal and superfluid fractions (3,4). Understanding the microscopic origins of both the collective excitations and second sound in He II remains an active area of research.
Elementary excitations in superfluid helium have been studied predominantly with neutron scattering (5), as well as by observing the dynamics of embedded atoms and molecules (6). Because of the vanishingly small solubility of impurities in LHe, the use of molecular probes has been largely limited to studies in helium nanodroplets that can be doped by injection of foreign species in pick-up cells (7)(8)(9). A wealth of information has been extracted from such studies about the coupling between the molecular electronic, vibrational, and rotational degrees of freedom and the quantum bath, be it through frequency (10)(11)(12) or time domain (13) measurements. As the microscopic analog of the Andronikashvili experiment, which used a torsion balance to verify the phenomenological twofluid model of He II, molecular rotors have been most informative: The change in the moment of inertia and centrifugal distortion constant of an embedded molecule serves as a gauge of the dragged normal fraction, and nearly free rotation is taken as the signature of a frictionless superfluid bath (14)(15)(16)(17).
Despite their elegance, nanodroplets suffer from a serious limitation: Their thermodynamic state is fixed to a single point on the temperature-pressure (T, P) plane because of the evaporative cooling used in their production. Yet, to investigate the inherently macroscopic two-fluid model of He II, it is essential to carry out measurements as a function of thermodynamic variables. This can be accomplished by resorting to helium dimers in the lowest metastable triplet state (a 3 Σ þ u ), known as He � 2 excimers, as LHe's native molecular probe (18)(19)(20). With a lifetime on the order of seconds (21)(22)(23), helium excimers are ideally suited for time-resolved probing of the quantum environment.
Similar to solvated electrons, He � 2 excimers form in~14-Å-diameter cavities (or "bubbles") that expel the superfluid around the molecule (19,24,25). Electronic transitions of He � 2 have been used to drive damped bubble oscillations, whose dependence on temperature and pressure was shown to track the normal fraction, establishing that the two-fluid model extends down to the molecular scale (26). Rotational lines in the fluorescence spectra, albeit unresolved but with the envelope similar to that in the gas phase, indicated free rotation of He � 2 inside the bubble (19,20). However, large inhomogeneous broadening (27,28) due to bubble shape fluctuations (29) prohibited the spectroscopic analysis of the excimer's rotational dynamics. The observed slow time dependence of the broadened absorption lineshape indicated the characteristic time scale for the rotational cooling of a few milliseconds (27,30) but offered no information on the (potentially much faster) decay of rotational coherence and, therefore, on the finer details of the molecular interaction with He II. With no direct access to molecular rotation in bulk superfluid, the microscopic Andronikashvili experiment under controlled thermodynamic conditions remained unrealized.
In the time-domain study presented here, we prepare coherent rotational wave packets in He � 2 and investigate their decoherence with femtosecond resolution in the superfluid quantum bath at variable temperature. After producing a-state excimers with intense pump pulses (22,23), we excite molecular rotation by a linearly polarized fs "kick" pulse (31)(32)(33) and then follow it in time with a delayed probe pulse (see Materials and Methods). Two-photon probe absorption promotes the molecule to a fluorescent d state (d 3 Σ þ u ), which decays to b 3 Π g by emitting a photon at ≈640 nm (22,28,34,35). Owing to the anisotropic absorption cross section, the difference between the laser-induced fluorescence (LIF) signals corresponding to two orthogonal probe polarizations (known, and hereafter referred to, as "linear dichroism" LD LIF ) reflects the ensemble-averaged alignment of molecular axes. As the latter rotate with respect to the probe polarization, the LD LIF signal becomes modulated at the frequency of molecular rotation, offering the direct measure of rotational coherence.

RESULTS
An example of the LD LIF (t) signal, recorded as a function of the kick-probe delay, is shown in Fig. 1A. The main oscillation frequency of (2.28 ± 0.02) THz corresponds to the energy difference ΔE 1,3 /h = 2.27 THz between the N = 1 and N = 3 rotational states of the ground vibrational level (v = 0) of the a 3 Σ þ u manifold. The observed oscillations are the result of the quantum coherence between the N = 1 and N = 3 states induced by the kick pulse (hence, labeled as LD 1,3 ). Owing to this coherence, the two-photon a → d absorption channels originated from these two states and sharing the same rotational level in the upper d 3 Σ þ u manifold, interfere as schematically illustrated by the diagram in Fig. 1. The interference leads to the time-dependent total absorption and hence the d → b fluorescence intensity, oscillating at the frequency ν 1,3 = ΔE 1,3 /h.
As indicated in the level diagram, each state with rotational quantum number N is split into three fine-structure components J = N, N + 1, N − 1 (in the order of increasing energy) due to the spin-spin and spin-rotational interactions (36,37). The splitting is of the order of a few gigahertz, which results in a slow modulation of the signal amplitude on the nanosecond scale, discussed later in the text. On the other hand, amplitude modulation on the 10-ps scale, clearly visible in Fig. 1A, is due to the frequency beating between multiple vibrational states with slightly different rotational constants. The Fourier transform of the LD LIF (t) signal is plotted in Fig. 1B, showing the rovibrational splitting of the LD 1,3 rotational line.
Since the frequency bandwidth of our pulses [≈14 THz full width at half maximum (FWHM)] is smaller than the excimer's vibrational frequency [54 THz (38)], the vibrational excitation is inherent in the energetic process of the He � 2 formation (39). The vibrational relaxation is far from complete 1 ms after the pump pulse (at the arrival time of the kick-probe pulse pair), in agreement with the previously determined vibrational decay time of order of 100 ms (27). Applying the known gas-phase molecular parameters (37,40) results in a good fit of the observed rovibrational spectrum (black dashed curve), indicating that within the experimental uncertainty of ≈10 GHz, the rotational constants in the three vibrational states are unaffected by the liquid environment.
Fourier transform of a delay scan with a lower frequency resolution but higher frequency range reveals the second excited rotational line in the LD LIF spectrum, corresponding to the laser-induced coherence between the N = 3 and N = 5 rotational levels [LD 3,5 in Fig. 1C]. Similar to LD 1,3 , the frequency of the second rotational peak ν 3,5 = (4.10 ± 0.02) THz agrees well with the energy difference between the N = 5 and N = 3 rotational levels of the a state in the gas phase (4.08 THz).
Unlike the case of vibrational excitation, transferring the rotational population from the ground N = 1 to the excited N = 3 and N = 5 states requires two-photon Raman frequencies well within the bandwidth of our kick pulses. Therefore, one may wonder whether the LD lines originate from the rotationally hot molecules created by the pump pulse, which have not decayed yet to the ground rotational state, or whether they stem from the molecules coherently excited by the kick pulse. To answer this question, we measured the ratio of the second-to-first rotational peak amplitudes, LD 3,5 /LD 1,3 , as a function of the pulse energy. The results are shown by green squares in Fig. 2A. The quick drop in the relative amplitude of the second peak with decreasing pulse intensity indicates the degree of rotational excitation largely controlled by the kick pulse.
To further support this conclusion, we carried out numerical calculations of the expected ratio between the two LD peaks by solving the Schrödinger equation in the rigid-rotor approximation (see Materials and Methods). In Fig. 2A, we plot the ratio LD 3,5 / LD 1,3 calculated for the experimentally used kick energies. The fit provides us with the rotational population of N = 3 and N = 5 levels before the arrival of the kick pulse, which are respectively 0.5 and 0.05% (upper confidence limits of 5 and 0.2%). This suggests that the majority of He � 2 dimers have relaxed to the ground rotational N = 1 state 1 ms after their creation by the pump pulse, indicating a rotational decay constant much shorter than that found in earlier studies [≈15 ms (30)]. The numerical calculations also show the major redistribution of, and hence the possibility to control, the rotational population by the kick pulse. With the energy of the latter at 3.5 μJ (≈5 × 10 11 W/cm 2 ), more than 15% of molecules are occupying N = 3, and almost 2% are at N = 5 [in thermal equilibrium, the former (latter) would correspond to rotational temperatures of 43 K (64 K)].
While the analysis presented in Fig. 2A offers a method of determining the decay of rotational population, our approach also provides a way of measuring the decay of rotational coherence. Both peaks in the LD spectrum exhibit strong time dependence, shown for LD 1,3 (t) in Fig. 2B. Here, a fine scan from t to (t + 20 ps) has been carried out for calculating the amplitude of the LD 1,3 peak at each (coarse) value of t between 0 and 1.65 ns. The oscillatory behavior is a consequence of the spin-rotational and spin-spin interactions mentioned earlier (see level diagram in Fig. 1).
To verify this conclusion, we modeled the expected signal numerically as where ν k 1;3 are the frequencies of the five transitions allowed by the selection rules (see Materials and Methods), and calculated using the known accurate values for the spin-rotational and spin-spin coupling strength in the ground state of He � 2 (37,40). Being on the scale of ≈2 GHz, the splitting is significantly smaller than the kick bandwidth, justifying our assumption that all coherences are created with the same phase. On the other hand, coefficients c k 1;3 account for the differences in the two-photon J-dependent matrix elements between different absorption pathways. Here, we used these coefficients as free fitting parameters, leaving the comparison to their ab initio values to future theoretical analysis.
Our assumption of a single decay constant τ 1,3 in Eq. 1 is justified by the quality of the fit in Fig. 2B. From the fit, we extract the coherence lifetime τ 1,3 = (1.0 ± 0.5)ns, during which the molecule completes more than a thousand full rotations. The corresponding rotational linewidth of ≈0.3 GHz is significantly narrower than the scan-length limited lines in Fig. 1 (B and C). We note that v > 0 vibrational branches, not included in the fit, add fast oscillations around the plotted curve without changing the optimal fit parameters. At this time, we were unable to apply the same numerical analysis to the much weaker second rotational line (LD 3,5 ). Improving the signal quality and comparing the two decays is the objective of current investigation.
One of the main advantages of studying molecular dynamics in bulk LHe is the ability to vary the temperature and pressure of the superfluid, probing the macroscopic nature of superfluidity. Here, we explored the temperature dependence of the rotational coherence between N = 1 and N = 3 rotational levels, reflected by the amplitude of the LD 1,3 peak in the dichroism spectrum. The experimental result, measured at a fixed kick-probe delay of 850 ps, is shown by red circles in Fig. 3. A clear decrease of LD 1,3 with temperature increasing toward the lambda point is a signature of the apparent interaction between the liquid and the laser-induced coherent rotation of helium dimers [unfortunately, taking data at T >  T λ proved impossible due to thermal instabilities in the liquid above the phase transition (22)].

DISCUSSION
Unlike the T-dependent change in the total fluorescence signal, the observed rotational decoherence cannot be attributed to bimolecular collisions. From the known diffusion constant of the He � 2 molecules in our temperature range [≲10 −3 cm 2 s −1 (41)], their average displacement on the time scale of our experiment is about 10 nm, which is significantly smaller than the intermolecular separation of >300 nm for the experimentally determined molecular density of 2 × 10 13 cm −3 (see Materials and Methods).
On the other hand, scattering of thermal quasiparticles (i.e., the normal component of the liquid) on the molecular rotor could be responsible for the observed temperature dependence of LD LIF . In a simple kinematic picture, where the (quasi)stationary dimers are colliding with He atoms moving with the velocity of first sound u 1 (T ), one can write LD 1;3 ðt; TÞ ¼ LD where γ 1,3 is the decoherence rate, N eq n is the equilibrium atom number density of the normal fluid, and σ 1,3 is the scattering cross section. Given the unknown T dependence of σ 1,3 , we used it as a temperature-independent single fitting parameter. The best fit, shown with the thick blue dashed curve in Fig. 3, captures the overall trend in the data but fails to reproduce the nonexponential flattening of the curve at lower temperatures. It also results in the decoherence rate (thin blue dashed curve), which is significantly lower than γ 1,3 ≈ 1 GHz observed in our scans of the kick-probe delay at T = 1.95 K discussed earlier (Fig. 2).
While further theoretical investigation of σ 1,3 (T ) may reconcile our data with the collisional model of Eq. 3, we note that it assumes thermal equilibrium between the molecular rotor and the surrounding liquid, implicit in the use of the time-independent normal fraction N eq n ðTÞ. However, the impulsive excitation of He � 2 by the kick pulse transiently promotes the dimer to the electronic d state, whose interaction potential with the closest He atoms in the liquid (7 Å away) is about 100 K higher than that of the a state (42). This instantaneous injection of energy creates a nonequilibrium state, in which molecular rotors find themselves surrounded by a microscopic local volume of "hot" liquid. The sudden imbalance of entropy may trigger a coherent pulse of second sound, initiating a flow of the normal component away from the molecule, and a corresponding counterflow of superfluid toward it. Consider, for simplicity, a Gaussian pulse of width w, traveling with the speed of second sound u 2 (T ), and describing the nonequilibrium density of the normal fraction at the location of the molecule where N is the total density of the liquid-entirely normal at time zero. Substituting N eq n ðTÞ in Eq. 3 by this time-dependent N neq n ðT; tÞ, and fitting it to the data in Fig. 3 using σ 1,3 and w as free parameters, results in the thick red solid line. In contrast to the equilibrium model, the decoherence rate (thin red solid line) mediated by the wave of second sound is more consistent with our findings from the delay scan at 1.95 K.
The nonequilibrium picture also explains the flattening of the LD 1,3 (T ) data between 1.4 and 1.8 K. The local minimum of γ 1,3 in this temperature window stems from the corresponding local maximum in the speed of second sound (1). The faster the entropy pulse, the faster the counterflow of the frictionless superfluid component toward the molecular rotor, the slower its rotational decoherence. As the speed of second sound decreases, with T increasing beyond 1.8 K, the heat wave created by the kick pulse travels a shorter distance away from the molecular rotor in a given amount of time. The correspondingly slower influx of the superfluid component results in faster decoherence and a lower signal amplitude. We note that thermal diffusion, which becomes faster with increasing T, would result in the opposite dependence of the signal on temperature, making it inadequate for explaining the experimental data.
The decoherence cross section σ 1,3 =2.5 × 10 −2 Å 2 , extracted from the fit to the nonequilibrium model, appears to be four orders of magnitude smaller than the size of the He � 2 bubble. This indicates very weak coupling between the normal fluid and the spherical a 3 Σ þ u state and explains why, within our experimental uncertainty, the rotational constants of the excimer in different vibrational and electronic spin states seem to be unaffected by the surrounding superfluid. The 22-nm width of the second sound pulse, provided by the fit, is larger than the distance of 17 nm covered by the pulse in 850 ps, which justifies the proposed far-from-equilibrium scenario. The latter could also explain the relatively large scatter of experimental data in Fig. 3, which we could not trace to any source of instrumental noise.
In summary, we report the first experimental observation of the laser-induced coherent molecular rotation in bulk superfluid LHe. Our time-resolved method enables us to detect and study various rotational dynamics in three different time windows: (i) we characterize the degree of rotational cooling on the millisecond time scale; (ii) probe the rovibrational, spin-rotational, and spin-spin dynamics on the picosecond time scale; and (iii) investigate the decay of rotational coherence on the nanosecond time scale.
By measuring the temperature dependence of the coherent rotational signal, we identify two possible decoherence mechanisms of qualitatively different nature: one mediated by the normal component of the helium bath in thermal equilibrium with the rotating molecule and another one based on nonequilibrium dynamics of the superfluid, governed by the wave of second sound. We note that such nonequilibrium response of He II to the sudden injection of energy by an intense ultrashort laser pulse has recently been observed in the ultrafast dynamics of rotons (43). Since rotons are predominant collective excitations of the normal component at T ≳ 1 K (41), one may also expect the latter to interact with a suddenly initiated rotation of a molecular probe in a nonequilibrium fashion. Work is underway to further investigate the molecule-superfluid interaction under variable temperature and pressure, to better differentiate between the proposed equilibrium and nonequilibrium models.
We also demonstrate the ability to vary the degree of rotational excitation, which offers a method of measuring the anisotropic polarizability of the helium dimer. Last, information about the rotational relaxation of He � 2 in He II may also help improve the methods of LIF-based molecular tagging (28,41,44) in the studies of the counterflow (45) and quantum turbulence (34,35,46) in superfluids, as well as open avenues for studying the microscopic implications of superfluidity with molecular nanoprobes.

MATERIALS AND METHODS
Our experiments are performed in a custom-built helium cryostat (Fig. 4). By pumping on the helium, the temperature of the liquid can be varied between ≈1.4 and 4.2 K, while the pressure above the surface is at the saturated vapor pressure. Three laser pulsespump, kick, and probe-are delivered to the cryostat at the repetition rate of 1 KHz and are focused in LHe with a 250-mm-focal length lens. Extracted from the same ultrafast Ti:Sapph laser system, they share the same central wavelength of 798 nm and bandwidth of 30 nm (FWHM) but differ in pulse length, energy, and the time of arrival. The kick-probe pulse pair is delayed from the pump by ≈1 ms, whereas the delay within the pair can be scanned up to 1.2 ns with fs accuracy.
Pump pulses, stretched to ≈2 ps and carrying 80 μJ per pulse, are used to create helium excimers. Their peak intensity of 4 × 10 11 W/ cm 2 is significantly below the breakdown threshold I break ≈ 5 × 10 13 W/cm 2 , determined in previous studies (22,35). We note, however, that subbreakdown intensities do not guarantee the production of the desired "bubble phase," i.e., an ensemble of isolated He � 2 molecules, each solvated in the liquid in its own bubble. To illustrate this, Fig. 5A shows the observed fluorescence spectrum corresponding to pump intensities of 1.7 × 10 13 W/cm 2 (lower red curve) and 6.8 × 10 12 W/cm 2 (upper blue curve). Although both intensities are below I break , the gas-phase-like narrow rotational lines in the lower trace indicate that the molecules are created in macroscopic gas pockets (47). We found the transition between the bubble and gas phases to occur at I gas ≈ 5 × 10 12 W/cm 2 , which dictated our choice of all pulse intensities well below this threshold value.
To further verify the important aspect of preparing the excimers in the solvated bubble state, we investigated the influence of the LHe temperature on the total LIF signal. The latter is proportional to the time-dependent He � 2 number density N(t). The time dependence is governed by the bimolecular annihilation reaction, He � 2 þ He � 2 ! He �� 2 þ 2He (21,22) NðtÞ The dependence on temperature enters through the reaction rate K(T ). In the bubble phase, K(T ) is determined by the diffusion of He � 2 molecules in the liquid due to their scattering on thermal rotons (39,48). As the roton energy Δ(T ) decreases with increasing temperature (49), their equilibrium density grows proportionally to ffi ffi ffiffi T p exp½À ΔðTÞ=T� (50), causing the scattering length and the diffusion coefficient to decrease. Slower diffusion at higher T results in a lower annihilation rate and correspondingly larger number of excimers at any given time.
Dark dots in Fig. 5B show our measured temperature dependence of the fluorescence intensity, induced by a fs probe pulse (pulse length of 70-fs FWHM, intensity of 1.3 × 10 12 W/cm 2 ) following the ps pump pulse after a fixed delay of ≈1 ms. The red solid line is a fit to Eq. 5, with the known roton energy Δ(T ) from the neutron scattering experiments (49). An excellent fit to the rotonmediated diffusion model confirms the production of He � 2 molecules properly solvated in LHe. Note that in the gas phase, one would expect a very different rate of bimolecular decay, owing to the increasing gas pressure, and hence higher annihilation rate, with increasing temperature. From the fit in Fig. 5B, the initial molecular number density (N 0 , the only fitting parameter) is (1.9 ± 0.1) × 10 13 cm −3 .
To excite the rotation of helium excimers, we send the linearly polarized femtosecond kick pulse before the probe (upper red in Fig. 4). Using the 1-KHz repetition rate of our laser system, we make the kick-probe pair trail the pump pulse by about 1 ms. At that time, the amount of He � 2 molecules which survived the bimolecular reaction is still quite high, yet given the experimentally observed decay of the pump-induced fluorescence on the time scale of a few tens of nanoseconds [consistent with the earlier study of Benderskii et al. (22)], all these molecules have already decayed to their lowest metastable electronic state a 3 Σ þ u . On the other hand, the rotational relaxation is expected to occur on the scale of a few milliseconds (30). As we demonstrate in this work, even after 1 ms, most of the molecules have also relaxed to the ground rotational state corresponding to the angular momentum (excluding electronic spin) N = 1 [due to the nuclear spin statistics, only odd values of N are allowed in a 3 Σ þ u (37)]. Our method of detecting molecular rotation is based on the anisotropic absorption cross section, common to linear molecules (51). Two probe photons with a wavelength of 800 nm promote the excimer from the ground a to the excited d state (26,28) with an absorption rate dependent on the angle between the molecular axis and the vector of the probe polarization. The difference in the absorption of two orthogonally polarized probe pulses (known as LD) corresponds to the anisotropy of the ensemble-averaged distribution of molecular axes, whereas its time dependence reflects the rotational dynamics of the molecules. The approach is similar to the polarization-based studies of the laser-induced rotation of gasphase molecules (52). Since we detect this LD via the induced fluorescence on the d → b transition, we refer to it as LD LIF .
We keep the probe polarization constant and modulate the polarization direction of the kick pulses between 0 β and 90 β with a Pockels cell (PC in Fig. 4). The LD LIF signal is defined as where I k;?
LIF is the fluorescence intensity recorded (by a photomultiplier tube) with the kick polarization respectively parallel or perpendicular to the fixed probe polarization. We use a Boxcar integrator to gate the fluorescence signal around the arrival time of the kickprobe pulse pair. A lock-in amplifier is used to retrieve the dichroism from the LIF intensity as the signal component at the polarization modulation frequency ω m ≈ 200 Hz. To eliminate possible instrumental artifacts due to our detection geometry, we use a half-wave plate (λ/2 in Fig. 4) to rotate all polarization vectors by 45 β with respect to the excitation-observation plane. The angle of the plate is fine-tuned to bring the LD LIF signal to zero when the probe pulse is blocked. We note that due to the undesired kickinduced fluorescence, inadvertently entering the denominator in Eq. 6, we are currently unable to extract reliable absolute values of LD LIF .
Consider a rotational state ψ J,M (t) = ∑ J,M c J,M exp (−iE J t/ħ)|J,M⟩ interacting with a laser field E kick (t). Here, J and M are the molecular total angular momentum (including electronic spin) and its projection on the vector of kick polarization, whereas c J,M and E J are the amplitude and energy of the corresponding eigenstate. The interaction potential is given by (53) VðtÞ ¼ À 1 4 Δαcos 2 ðθÞE kick ðtÞ ð7Þ where Δα = 35.1 Å 3 is the difference between the molecular polarizability along and perpendicular to the molecular axis [calculated using the method of coupled cluster with single and double substitutions (42) and the basis set from (24)], and θ is the angle between the molecular axis and the laser field polarization. We numerically solved the Schrödinger equation in the rigid-rotor approximation, assuming that the kick length is much shorter than the period of the molecular rotation. For simplicity, here, we also neglect the effect of the electronic spin (which is discussed below) and take J ≡ N and E J ≡ E N . Linearly polarized kick field leaves the molecule in a coherent superposition of states with ΔN = ± 2 and ΔM = 0. The numerical solution of the Schrödinger equation provides us with the complex amplitudes c N,M of those states right after the kick. The observed LD N,N+2 signal, oscillating at the frequency ΔE N,N+2 , is proportional to the real part of the product c N;M c � Nþ2;M , summed over all independent M channels.
Consider the first rotational line LD LIF1,3 , corresponding to the coherent superposition of N = 1 (split into J = 1,2,0) and N′′ = 3 (split into J′′ = 3,4,2), created by the kick pulse (J values are listed in the order of increasing energy). The absorption of two probe photons on the a → d; (J, J′′) → J′ transitions must obey the following selection rules: |J′ − J| = 0,2 and |J′ − J′′| = 0,2. Therefore, there are five pairs (J, J′′ = J, J ± 2, J ± 4) producing the LD signal at the approximate frequency ν k 1;3 � 2:27 THz: (0,2), (0,4), (1,3), (2,2) and (2,4). Interference between two absorption channels J = 1 → J′ = 1 and J′′ = 3 → J′ = 1 is shown as an example in the inset to Fig. 1. Beating of these five frequencies results in the observed oscillations of LD LIF1,3 with a minimum around 500 ps (see Fig. 2). 2 at T = 2 K with P, Q, and R rotational branches labeled. The broadening of rotational lines with decreasing pump intensity demonstrates the transition between the molecules in macroscopic gas pockets ("gas phase," lower red line) and solvated molecules (bubble phase, upper blue line) and illustrates the effect of the superfluid on the molecular rotation. (B) Intensity of the probe-induced fluorescence as a function of temperature (dots) and its fit to the expected bimolecular decay (red solid line; see text for details).