Excitation of rotons in parahydrogen crystals: the laser-induced molecular alignment mechanism

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I. INTRODUCTION
Controlling the direction of molecular bonds by moderately intense laser pulses has been the topic of significant amount of experimental and theoretical activity. 1,2 The basic physics of intense-field alignment of gas phase molecules is now well understood. It is generally favorable to utilize short, nonresonant laser pulses instead of an adiabatic excitation with static electric field. Then, possible side effects due to electric field are suppressed as long as the ionization or other thresholds of sample breakdown processes are not crossed. In this dynamic, field-free regime, oscillating alignment results due to formation of a wave packet state. It exhibits a transient postpulse alignment due to rotational revival structure.
Spatial aspects of alignment depend on the polarization of the laser field that interacts with the anisotropic molecular polarizability. Molecules can become aligned with respect to a given space-fixed axis by choosing a linear or a circularly polarized field. Three-dimensional alignment, to force molecules into a plane or to suppress rotations completely, must use elliptical polarization or suitable combination of laser pulses. 2-4 Advanced control methods to manipulate rotations in 3D include multipulse schemes [5][6][7] and optimal control theory 8,9 with interesting outcomes such as molecular cogwheels 10 and stopwatches. 11 Having reached a relatively mature state, the concept of molecular alignment has already been manifested in variety of applications resulting in fascinating demonstrations of fundamental molecular physics in the gas phase. [12][13][14] In condensed phase the topic of quantum control has advanced only recently. 15,16 The motivation lies in the unique properties of rotationally broad wave packets that can be utilized as sensitive probes of the dissipative properties 17,18 and the energy level structure 19 of the environment. Rotational Raman transitions underlying the induced alignment process can be used themselves to study intermolecular forces. Molecular hydrogen is a suitable probe due to relatively weak interactions and its simple structure. 20,21 As a coherence-based control tool, efficient extension of the alignment mechanism to the condensed phase is posing challenges both experimentally and theoretically. [22][23][24][25] Therefore, parahydrogen (pH 2 ) as a minimally perturbing quantum condensed phase makes natural sense as the media to study.
Particularly appealing is the manifestation of long-lived condensed-phase coherence by relaxation of rotational excitation (roton). 26,27 In the presence of one J = 2 roton in solid pH 2 , there is one molecule in the J = 2 rather than in J = 0 state. The states in this J = 2 band form a traveling rotational excitation, i.e., the total angular momentum is located at a different molecule at different times. 28 The J = 2 band is split due to anisotropic intermolecular forces, and the main interaction originates from electric-quadrupole-quadrupole coupling (EQQ) between the relatively distant hydrogen molecules. 28 Thus elastic collisions or fluctuations of the EQQ interactions are the main source of dephasing, and apart from energy level splitting, pH 2 molecules behave as free rotors. These nearly free rotors possess a large zero point motion as the quantum solid property, and as a consequence are insignificantly affected by lattice vibrations (phonons).
A rotational coherence that lasts more than 12 ps was revealed by femtosecond pumpprobe 2D optical Kerr effect (OKE) spectroscopy in recent pH 2 experiment. 27 The coherent oscillation of ca. 94 fs period originates from the J = 2 ← 0 excitation S 0 (0) that can be categorized as a creation of delocalized rotons. The method relies on the pump laserinduced birefringence in the solid. The signal appears as stimulated Raman side bands of the probe pulse, while the Raman-active optical phonon mode exhibits ca. 900 fs oscillation period at the probe center. A closer examination with higher resolution revealed a splitting into |M| = 0, 1, 2-state substructure, where three components with 93.7, 94.2, and 94.8 fs oscillation periods can be resolved along with their mutual quantum beats. 29,30 The refined experiments also reported extended dephasing times T 2 on the order of 100 ps -a value which is controlled by the purity with respect to orthohydrogen concentration. 30,31 The extent of delocalization for the roton-type excitation in the solid parahydrogen is another interesting point accessible to evaluation by the impurity dependence. 30 Here, we report quantum simulations together with an illustrative experimental signal.
We have chosen to highlight the nonresonant alignment point-of-view as the underlying mechanism in OKE spectroscopy. This leads to extension of the alignment concept from the gas phase to condensed phase (molecular solids). Theoretical foundations of solid pH 2 are well known 28 and rotational Raman spectra of the solid have also been thoroughly considered. [32][33][34] Although the transverse optical phonon part of the signal is a very prominent feature in the experiments, 27,29,30 it is left out from the discussion by assuming negligible interplay of phonon and roton dynamics. These elementary excitations have a tenfold energy mismatch and a fivefold difference in dephasing time. 30 Phonon sidebands are only a very weak feature in rotational Raman spectra, and rotational excitations are strongly decoupled from intramolecular vibrations, too. 35 Furthermore, roton lifetimes are independent of temperature, which eliminates possible influence of lattice vibrations. 26 We use this information to incorporate the bulk solid effect into the alignment scheme, and resolve the labor of implementing the EQQ and the three reference frames: laboratory (laser fields), crystal, and molecule. The emphasis is put on the geometric aspects in light-molecule interaction.
We present the dependence of the roton excitation on the polarization direction of the field relative to the hexagonal direction of the crystal. In particular, the purpose of the paper is to computationally reproduce and explain the recent experimental spectra. 27,29 The present theoretical work provides insights that cannot easily be gained from experiments alone. In connection with the experiments, 27 we also felt the H 2 molecule itself deserved more attention. Therefore, we report a brief gas-phase survey, where the alignment process is worked out much in the same manner as in previous N 2 studies, 12,36-48 elaborating on the differences in the high-B (rotational constant), small-Δα (anisotropy of polarizability) molecule H 2 as compared to the N 2 .

II. EMPIRICAL BACKGROUND
Parahydrogen crystals grown in closed copper cells are large and transparent with regular shapes and known orientations of the hcp single-crystals. Although cubic centimeter scale single crystals with zero birefringence can be grown, we consider crystals grown in cylindrical cells, where the c-axis points radially from the copper wall towards the center of the cylinder. 27,29 The radial orientation of the c-axis with respect to the linearly polarized excitation laser field (Z) then depends on the location where the beam enters the cell, see Fig. 1 for definition of the angular coordinate system. In order to sample a unique c-axis direction by the excitation and probe laser pulses, the collinear beam propagation direction Y is aligned with the cylinder axis, which is perpendicular to c-axis and therefore to the ZX laboratory plane. This choice sets the Euler angle φ = 0. The long crystal cell dimensions (ca. 3 cm) are necessary to prevent damage of the windows on the cell due to use of focused ultrafast laser beams. Along the beam path within the crystal the angle χ can, in principle, be randomly distributed among single crystallites. However, the beams are focused such that only a volume of cubic millimeter scale is sampled, ensuring the signal is dominantly generated from one single crystal in each experiment. 27,29,30 The optical Kerr effect spectroscopy is a pump-probe polarization spectroscopy, also After R z (χ ), the crystal axes (x, y) are rotated by an angle χ relative to x , y (light gray).

III. THEORY
We consider the molecular rotational energy structure in the solid parahydrogen subject to the interaction with the aligning laser pulse, and the consequent coupling of the exciton (J = 2 roton) states via the electric-quadrupole-quadrupole interaction in the crystal. Ignoring dissipation effects in the rotational alignment, the Hamiltonian for the system reads where where γ labels the site (α or β) within the unit cell. The proper combination symmetry is fulfilled by the Bloch functions 54 It is essential to restrict to the k ≈ 0 wave vector region of the Brillouin zone, where the even and odd states do not mix. Furthermore, only transitions to even exciton states |ψ M (0) + are Raman active. For each value of M, the exciton has an equal probability to be found at any of the site locations R i . The kinetic energy operator H 0 + H c reduces to diagonal H cry where the energy E M (0) + represents the lattice sum of the anisotropic components of the intermolecular potential. In this treatment, other states than single J = 2 excitations coupled to J = 0 are disregarded. Double excitations of pairs of molecules, such as to form a |J 1 J 2 = |22 crystal state, are very weak (of the order of phonon side bands) in Raman spectra, 35 and appear at ca. 12B 0 , which is outside the laser pulse widths used in impulsive Raman processes of this work (see below). Moreover, ΔJ = 4 transitions are reported two orders of magnitude lower in intensity than ΔJ = 2, 55 and sequential J = 4 ← 2 pumping probability also suffers from the 14B 0 energy difference.
The basis set actually used in the calculation of the time-dependent alignment in the crystalline pH 2 consists of the |JM rotational eigenstates of isolated molecules. The equivalent in the coordinate space is the set of spherical harmonics Y JM (θ, φ). We thus abandon the crystal state description only needed to obtain the known level structure in Sec. III A.
The case is analogous to a matrix-isolated impurity molecule, and the EQQ interactions are pairwise summed to yield a quantum crystal-field effect. Technically, we consider a localized case instead of the traveling exciton, but we retain the term roton for the wave packet. The probe pulse can be thought to propagate along with the roton wave throughout the sample and thus be affected in each region of the crystal in an uniform way. 27 The induced Hamiltonian (derived in Sec. III B) is dependent on the relative orientation of the crystal with respect to the laser polarization, which in turn dictates the M-level composition. The quantum state is expanded as The numerical task then is to integrate the time-dependent Schrödinger equation in the rotational basis of Eq. (6) using an explicit Runge-Kutta formula. Based on the moderately low intensity of the laser field, we can restrict to J max = 2 here. The corresponding 6 × 6 Hamiltonian matrices are evaluated analytically and implemented in the numerical wave packet propagation code. While expanding the basis to J = 4, and computing the matrix elements of H ind and the expectation values related to the signal field, would be rather straightforward, developing the corresponding crystal field theory would increase the workload significantly.
In the gas-phase case, the angle θ is between the molecular symmetry axis and polarization direction of the laser (laboratory instead of a crystal frame), and becomes the sole angular variable in the expression for the induced Hamiltonian. For the gaseous pH 2 and N 2 molecules, we propagate in Sec. IV A the density matrix ρ rot (t) = |Ψ rot (t) Ψ rot (t)| directly in the Liouville-von Neumann formalism: 18 to account for the distribution of initial states at 80 K. Basis set size can be set to a low value (J max ∼ 10), since J = 4 state amplitude is practically zero for pH 2 with the present pulse durations and intensities. We report the time evolution without any dissipation effects.
The expectation value of cos 2 θ is used as the time-dependent degree of alignment, and is obtained as: The trace can further be divided into population and coherence parts: cos 2 θ = cos 2 θ p + cos 2 θ c , which account for the time-evolution of the diagonal and off-diagonal elements of the density matrix, respectively. For a random ensemble, the average of the alignment degree equals the isotropic value of 1/3. The offset from this value is contained in the diagonal part of the density matrix, while the off-diagonal elements represent coherences between molecular eigenstates and show oscillation with respect to the population part. The partitioning opens a direct way to separable measurements of the population decay and the loss of phase coherence. 17,56

A. Crystal field effect
We employ the spherical tensor formulation for the electrostatic interaction between the molecules. Starting from the spherical harmonic addition theorem 57 one finds the following expansion in a global coordinate system 58 where C lm = 4π/(2l + 1)Y l,m are renormalized spherical harmonics,Q lm are multipole moment operators, and the Wigner 3j symbols are defined by ⎛ The Coulomb interactions are additive, so one can sum over the distinct i, j pairs separated by distances R ij with relative directionsR ij . The multipole moment operators can be expressed in local coordinates by substitutinĝ and replacing operators by expectation values. The electric quadrupole moments are the main contribution to anisotropic forces in solid hydrogen, so one restricts to l 1 = l 2 = 2 terms. Moreover, the linear molecule has Cartesian quadrupole moment components Θ xx = Θ yy = −Θ zz /2 so that the only nonzero spherical component is Q 20 = Θ zz ≡ Q, and k = 0. Noting that the Wigner function for rotation is related to the spherical harmonics by D

the Hamiltonian term becomes
where Choosing a central molecule i with R i = 0, the summation is carried over the hexagonal lattice to give the energy levels of the J = 2 roton band split by E M (0) + as 54 where R e = 3.783Å is the nearest neighbor distance 59 and Q 0 = 0.4858 ea 2 0 is the quadrupole moment. 28 The S 0 (0) Raman triplet splitting to M = ±1, ±2, 0 follows predominantly from the quadrupolar interaction, and the local van der Waals effects in the crystalline field can

B. Laser field interaction
The nonresonant laser field causing the exciton formation interacts with the molecules via the polarizability of the medium 2 where the polarizability is equal to the sum of the polarizabilities of the separate molecules, In terms of spherical tensors, the dipole-dipole polarizability can be written whereμ m is the m = 1, 0, −1 component of the dipole moment operator, and the states |n i = |ξ i ν i and |n = |ξν stand for the ground and excited vibronic state manifolds, respectively, with electronic states labeled as ξ and the corresponding vibrational states as ν. It is convenient to construct a tensor product from α m,m , which transforms according to angular momentum labels L and M: 57 where 1m1m |LM ≡ C(11L, mm ). For the linear molecules only α 0 0 and α 2 0 components exist. These can be denoted by the symbols α = −α 0 0 / √ 3 and Δα = 3/2α 2 0 for the mean polarizability and anisotropy, respectively, which are given by relations α = (α + 2α ⊥ )/3 = (α zz + α xx + α yy )/3 and Δα = α − α ⊥ = α zz − (α xx + α yy )/2. With these definitions, the field-induced part of the Hamiltonian becomes For the isotropic, L = 0 part we have For the L = 2 part we obtain So far everything is defined in the crystal frame. We want to define the electric field of the laser in the laboratory coordinates and use the polarizability components known for the molecule in the body-fixed frame. Referring to the molecule, we write the anisotropic polarizability in terms of the Euler angles (φ, θ, χ) as where the latter equality results since only the α 2 0 exists in the body-fixed frame. Using D (2) * m+m ,0 (φ, θ, χ) = 4π/5Y 2,m+m (θ, φ), we obtain the following form for the interaction: For the field terms we rotate the laboratory frame Cartesian components to the crystal frame, thus specifying the angles (φ , θ , χ ), and apply the Cartesian to spherical tensor transformation (xyz) → (1, 0, −1). We denote the former by a rotation matrix Φ and the latter by a transformation matrix U: 57 and write With linearly polarized pump field along the laboratory coordinate Z, only the column is needed, and the field is defined as The field product term in Eq. (21) then obtains the following form where the oscillatory terms are neglected and the intensity envelope is contained implicitly, i.e., ε Z = ε(t).
In the coordinate system depicted in Fig. 1, the field-molecule interaction Hamiltonian H ind takes the following five-term expansion: where the θ , χ -dependence is explicit. The matrix elements are readily evaluated in spherical harmonics basis. 57 We have used a short-hand notation Y 2,M ≡ Y 2,M (θ, φ). By setting θ = χ = 0 in Eq. (25), i.e., r = s = z and U mz = (0, 1, 0) T [Eq. (22)], the laboratory and crystal frames coincide, and Eq. (28) reduces to the familiar (gas-phase) form: where θ is again the angle between the molecular axis and the polarization direction of the laser. The prefactor term ε(t) 2 Δα/4 offers a control for the extent of the rotational excitation via the pulse duration and intensity. The employed laser pulse ε(t) is a Gaussian of the form where ε a is the laser field amplitude, the intensity I (TW/cm 2 ) follows from the relation ε 2 a = 2I/ε 0 c, and σ is the pulse duration (FWHM of pulse intensity) in ps. The anisotropy of polarizability Δα was set to a constant value 60 of 0.30Å 3 , which is by a factor of three lower than that for N 2 (0.89Å 3 ). 61
When treating the OKE spectroscopy as a four-wave mixing technique, the system interacts twice with the pump pulse (k 1 = k 2 ) and is nonintrusively interrogated with the probe pulse (k 3 ) at delay time τ after the pump. The pump-induced rotational density has only coherence terms |2M 00| and |00 2M|, which can be assigned to two different 3rd-order responses R ijkl . 62 The coherent Raman scattering signal is detected in the direction of the probe: k s = k 1 − k 2 + k 3 . Although the interaction strength with intense pump pulses goes beyond the perturbative regime, the theoretical framework relies on expanding in powers of the electric field. Consideration of the J = J = 2 elements, such as |22 22| population or |22 21| coherence, involves a 5th-order process (six-wave mixing), where the signal still propagates along the probe as the pump accounts for the first four directionally canceling interactions; twice from the ket and twice from the bra side of the Feynman diagram, before the probe interaction. Adapting from Mukamel, 62-64 we write the density operator in a form We concentrate on times after the pump-induced creation of the ρ rot (being part of ρ gn,gn (t) state space), and consider this nonequilibrium density matrix as the initial state ρ(t 0 = 0) interrogated by the well separated probe field (acting to ket from the left) The material response to applied laser fields is imprinted in the optical polarization P (t; τ ) which is the expectation value of the dipole operatorμ dictated by the density matrix: The time propagation of the density matrix is here brought by the dipole operatorμ in the interaction picture: where μ mn = em|μ|gn . Successive operations to ρ gn,gn (0) yield the signal source as mediated by the vibronic polarizability tensor: where instantaneous decay (γ eg → ∞) is used and the complex conjugate is not written for the intermediate steps to Eq. (36). The result includes the Kramers-Heisenberg-Dirac expression for the dynamic polarizability α(ω p ) similar to Eq. (15). Due to the initial state weighting by coefficients c JM c * J M , we can restrict to J = J so that the polarization has Raman shifts ω p ± ω nn where plus and minus mean |2M 00| (anti-Stokes) and |00 2M | (Stokes) branches, respectively. The trace is over the rotational manifold.
The system response is detected by a square-law-detector after mixing the signal field E s = i(ω p l/c)P (t; τ ) with a local oscillator E lo , resulting in a heterodyne-detected signal.
The nonlinear refraction is dependent on the pump intensity I as n(I) = n 0 + n 2 (I), which introduces the multiplicative term where l is the optical length of the focus volume and c the speed of light. Self-heterodyning is achieved experimentally by turning the probe pulse polarization by an angle ϑ ≈ 1 • from exactly crossed polarizer-analyzer geometry. A portion of the probe field amplitude then penetrates the analyzer even when no birefringence is present in the medium: E lo = sin ϑ E p (t) exp(iϕ lo ). Since the total intensity is I = |E s +E lo | 2 , we have I = |E s | 2 + 2Re{E s E * lo } + |E lo | 2 = I hom + I het + I lo , respectively. The time-integrated homodyne signal is written as The homodyne part is a positive quantity as it measures only the deviation from the isotropy.
The heterodyne part is: The heterodyne-detected signal can have positive and negative values, and it carries the phase information of the signal field. The local oscillator intensity I lo is time-independent and only brings a background to the signal: Experimentally, the pump pulse is polarized in the Z-direction, the probe pulse is oriented at (45+ϑ) degrees with respect to the pump pulse:ˆ p ≈ (−1, 0, 1) T / √ 2 in the X, Y, Zframe, and the analyzer is oriented nearly perpendicular to probe's initial polarization: The angle θ is the angle between laser polarization direction and the molecular axis and φ is the azimuthal angle. The expression is the same as derived by Renard et al., 65 except for cos 2 φ instead of sin 2 φ. However, the trace operation removes the φ-dependence in the gas phase, and one obtains for the trace. Thus the usual measure for alignment degree is very suitable for characterization of the birefringence-dependent OKE signal.
In the case of solid parahydrogen, we denote the angles by capital letters to distinguish from the crystal frame coordinates. We denote the solid-state alignment degree function by A γ (t): where the decay function with 1/γ = 90 ps is added to account for the loss of rotational coherence. It is convenient to rotate the operator Y 2,0 (Θ, Φ) to make the crystal and laboratory frames to coincide: We have used the arguments mentioned in Sec. III B to set Euler angle φ = 0 and used the explicit form of the Wigner D-matrices. 57 To obtain the result for the gas phase alignment, the operator Y 2,0 (Θ, 0) is replaced with Y 2,0 (θ, φ). Similarly, we have where To reiterate, the final expression for solid-state alignment is implemented as We scale the OKE signal intensity by [ω p l/c] −2 to obtain the final implemented form: The local oscillator amplitude is defined through the value of C = sin ϑ c/ω p l. When C = 0, the signal is homodyne-detected and it beats at the inverse of difference (and sum) frequencies ω MM . When C is large, we observe the transition frequencies ω M . We set C suitably in-between the above limits.
In order to simulate the experiments 27,29 as a whole, we also produce spectrally dispersed signals. The intensity collected by a spectrometer is which we implement in the form The two terms in Eq. (49) are the homodyne (left) and the heterodyne (right) parts of the signal, and the local oscillator intensity is neglected. The probe carrier frequency is also eliminated in the Fourier transforms. The shape of the local oscillator is modified from that of the probe due to the wavelength dependence of the polarizer-analyzer extinction ratio. In particular, the probe center wavelength is suppressed the most leading to a double peak shape of the local oscillator. 29 We simulate this effect by implementing the field in a self-phase-modulated form 66

A. Gas-phase alignment for pH 2 and N 2
The rotational excitation occurs via the stimulated nonresonant Raman process in the time-resolved femtosecond pump-probe OKE spectroscopy. As the present pulses have durations in the fs scale, below the rotational period τ rot , the study falls in the category of nonresonant, field-free alignment. In this regime, a short pulse generally excites a broad wave packet in the angular space, which then spreads and at certain times, called revival times, the spreading fully reverses and molecular alignment is observed, see is rather simple for pH 2 , as the J = 2 state is not initially populated at low temperatures and the field intensities considered here do not extend the rotational excitation to the J = 4 state. Therefore, the coherence part cos 2 θ c oscillates at the inverse of the rotational frequency 355 cm −1 and the population part cos 2 θ p is close to the isotropic limit. This mapping of the alignment degree corresponds to a heterodyne-detected signal.
Using the same experimental parameters for N 2 gas, a revival pattern of alignment emerges with τ rot ≈ 8.4 ps in Fig. 2(b). Because the odd and even J-states are unevenly populated (even states contribute twice as much as odd states), alignment is also observed at quarter and three-quarter revival times. At 80 K, the highest thermally populated state is J = 4 according to the Boltzmann statistics. As the nonresonant pulse excites the rotational states via the Raman ΔJ = ±2 transitions, the resulting rotational wave packet is significantly broader in the J-space than the equivalent in the case of hydrogen. In the Fourier transform of the signal, peaks at 2B 0 (2J + 3) separated by 4B 0 (neglecting the centrifugal term) would be observed.
At the zero temperature limit, an order of magnitude estimate for the maximum rotational state J max in the wave packet can be given: With short and intense laser pulses, the coupling strength Ω R and pulse duration σ dictate the extent of rotational excitation as J max ∼ σΩ R .
With long and low intensity pulses, a proportionality relation J max ∼ Ω R /2B e is more suitable. 67 The coupling strength, referred as Rabi coupling, is Ω JJ R = 1 4 ε 2 m Δα J| cos 2 θ|J . In addition to the J-dependence of the Rabi coupling, the interaction strength is proportional to field intensity and anisotropic polarizability.
The post-pulse alignment degree is seen to degrade in Fig. 3(a), if longer pump pulses are used for pH 2 . At the adiabatic limit, the alignment degree returns to the isotropic value of 1/3 after the pulse has gone. The adiabatic condition is generally met at σ ≈ τ rot = 1/2Bc, which for pH 2 is 280 fs. Here, however the 94 fs observed period necessitates significantly shorter pulses. At σ = 120 fs, the system is already close to adiabatic behavior. Simultaneous decrease in the expectation value of J(J + 1) is observed in Fig. 3(b). The same simulation for nitrogen shows different behavior, which results from the order-of-magnitude difference in the rotational parameters. The alignment enhances as the interaction time under the pump pulse envelope is elongated, see Fig. 3(c). Here, the molecule experiences an increase in the number of Rabi-type cycles. Along with the increase in alignment degree, the expectation value of the J(J + 1) in the wave packet increases in Fig. 3(d). As the pulse duration is increased, with the chosen pulse intensity, the relation J max = σΩ R gives qualitatively correct behavior for the rotational excitation in nitrogen.

B. Alignment in pH 2 crystals
We investigated the impact of laser pulse properties on the (gas-phase) alignment in the previous section. Here, we take on a different approach, and consider the pump-probe OKE signal with a single set of pulse parameters: a moderate pump length of 100 fs and lower peak intensity of 1.0 TW/cm 2 . We solve Eq. (7) to obtain the wave packet [Eq. (6)]. Due to the low temperature in experiments (4-11 K), we can consider the system initially in a pure only the θ -control. The overall amplitude of the J = 2 excitation is also seen to depend on θ , which modulates the sum ( |c 2|M | | 2 ) normalized to the value at θ = 0 • . The state distribution exhibits mirror symmetry for angles θ higher than shown in Fig. 4.
The initial quantum phase of |20 component is ϕ 0 (0) = π/2, when (cos 2 θ − 0.5 sin 2 θ ) is positive, and −π/2 otherwise. The phases of other J = 2 components develop in parallel during the pulse, thus maintaining the relative difference. The initial phase, however, varies with both the angles according to Eq. (28). The expectation values relevant to alignment degree and signal detection show a variable phase relation in contrast to the gas-phase case.
Since the orientation χ within a pump focus volume cannot be controlled, we scan this angle for the two representative values of θ . For the θ = 54 • case, Fig. 5(a) shows the Fourier transform of signal intensity: see Eq. (47), where the probe envelope function was replaced by δ(t − τ ). Self-heterodyning was obtained by setting C = 0.003, and C 2 was subtracted. Also the homodyne part was     Fig. 7(b), which also reveals a mismatch between the M = 0 spectral weights.
The spectrally dispersed version of the above calculated trace (see Fig. 7) is given in  More detailed account for the effect of quantum phase factors between the components is given in Fig. 9. In the left column, we quantify the effect of angle χ = 0 • -90 • on the phases ϕ M in the roton packet. In the right column, the emergence of quantum beats in the homodyne part of the total signal field is demonstrated. We remind that the wave packet amplitudes in Fig. 4 are not affected by the variation of χ ; the θ = 77 • -set ratio of 0.69:1.44:1 for the |M| values 1:2:0, respectively, remains through the Fig. 9 columns. We shortened the pump duration to 80 fs in order to avoid the reseting of phases at the Rabi period. The probe was negatively chirped as above. The top panel corresponds to the case shown in Fig. 6, where the intensity maximum was seen at τ = 0. The situation is not markedly changed when going to χ = 22.5 • , near the condition of Figs. 7-8, except that the 8 ps period (ω 01 ) in addition to the 17 ps period (ω 21 , ω 02 ) becomes visible. In order to have a shifted first signal peak, higher angles are needed. For example, the χ = 67.5 • panel appears to satisfactorily represent the chosen experimental trace, where the first peak was on the numerical evaluations, the measurements should indeed be near the detection limit with the 140-160 fs long pump pulses. Noting that the rotational period (1/2B 0 c) for pH 2 is ca. 280 fs, the time scale approaches that of the static, adiabatic regime of intense-field alignment. However, the successful detections 27,29 of the roton dynamics emphasize the high Raman gain effect in the solid that compensates for the signal weakness.
It is clear from the above considerations that the appearance of the signal in terms of the relative weights of the M states is coincidental in experiments. The χ orientation is changing from sample to sample and from one laser focus spot or volume to another rendering the state resolution in ambiguity. To unravel wave packet compositions in experimentally observed time traces and their Raman spectral densities, we can search correspondences by varying the angular space. To illustrate the concept, we compared a simulated trace to an experimental signal which was a spectrally selected cut from a dispersed emission. To reproduce the experiment as a whole, the signal simulation must be switched to a timeintegrated, frequency-resolved coherent Raman scattering type, where details of the probe and local oscillator fields have to be considered. For example, the wavelength dependence of the extinction coefficient introduces an intensity minimum for the local oscillator at the probe center. The Raman spectral density shows three frequencies in agreement with the theory of the solid pH 2 , i.e., the levels |M| = 0, 1, 2 are resolved. Another aspect is the ambiguity in the phases, manifest in time domain signals. We introduced a time shift to match with the picked experimental curve. The signal intensity exhibits peaks when the oscillators are rephased, and one might consider estimating the temporal shift in the experiment from expression I hom (τ ) ∝ jk cos(ω 21 τ + ϕ 21 ) + kl cos(ω 02 τ + ϕ 02 ) + jl cos(ω 01 τ + ϕ 01 ) , where the coefficients j, k, l represent the roton amplitudes. For example, at τ = 4.74 ps the signal I hom shows a maximum as in the experiment if ϕ 21 = 10π/7, ϕ 02 = −4π/7, and ϕ 01 = 6π/7. The emergence of M-states in the Fourier transform of the signal is delicate with respect to changes of the crystal orientation, and it is a challenging task to seek for a perfect match. We note that it is easier to reproduce the experimental trace by fitting a suitably parametrized set of cosine functions than by simulating. When the signal is analyzed with damping terms exp(−γ 21 τ ) etc., augmented in Eq. (52), the experiment reveals |M|-specific dephasing times: 30 132, 114, and 82 ps in the order of increasing energy. The simulation and Eq. (28) indicate, however, that all five M components are needed instead of just the three |M|. The roton phase information in Fig. 9 cannot be directly used, since the signal acquires additional phase terms through the detection, see Eq. (46), which is quite complicated to expand. However, the quantum phase and Fig. 9 give the first answer to the question of origin of temporal shifts of the first signal maxima observed in experiments. 29,30 As the outlook we note the importance in unraveling the details in dispersed detection of the birefringence signal from crystalline parahydrogen samples. The experiments resolve the OKE signal into Stokes and anti-Stokes shifted branches of radiation and produce 2D spectrograms. According to the third-order polarization mechanism, the two nonresonant, intramolecular pathways between ket and bra side excitations should show an in-phase response, as verified theoretically and experimentally for CHCl 3 . 52,70 The present evaluations faithfully reproduce the spectrogram images reported for pH 2 . 30 Interestingly, in some samples there were differences between the branches, 71 which calls for a revised theoretical description. With the present investigation we cannot rule out whether the detected differences between Stokes and anti-Stokes branches arise from the properties of the probe pulse or the parahydrogen crystal itself. We included both the chirp of the probe and the amplitude modulation of the local oscillator but did not observe significant phase, amplitude, or signal modulation asymmetry between the pathways. A more probable cause of the effect is that, despite the lower intensity, the Raman cross section is higher for the frequency-doubled probe than for the pump. Thus the probe pulse might not have been just a nonintrusive spectator as it was considered in the simulations. The pump and probe pulse power dependencies and corresponding nonlinear effects are beyond the present analysis. So a challenge remains in explaining whether the Stokes and anti-Stokes shifted branches of the scattered probe can show different dynamics via higher order Raman transitions. Further studies on rotationally equivalent orthodeuterium could be helpful to rule out the solid-state origin of the effect.

VI. CONCLUSIONS
The paper shows how experimental femtosecond pump-probe OKE signals from solid pH 2 can be computationally reproduced. We obtained a functional form of the molecular alignment degree, which reflects the induced birefringence and rotational dynamics. The