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Alignment by short intense laser pulses has been the topic of rapidly growing experimental and theoretical activity during the past few years, due to both the fascinating physics involved and the variety of already demonstrated and projected applications (Ad.At.Mol.Opt.Phys.52, 289, (2006) pdf. and Rev. Mod. Phys., 75, 543 (2003) pdf ). In the isolated molecule limit, corresponding to a molecular beam experiment, a moderately intense laser pulse of duration short with respect to the rotational time-scales produces a broad wavepacket of rotational levels. Depending on the pulse duration and intensity, the probability density aligns along the field polarization direction during or after the laser pulse and subsequently dephases, due to the anharmonicity of rotational spectra (J. Chem. Phys., 103, 7887 (1995) pdf). (it is worth elaborating here on the conventions of the gas phase wavepacket dynamics literature that we follow, where “dephasing” is used for change of the relative phases of the wavepacket components due to anharmonicity of the molecular spectrum, whereas “decoherence” is used for loss of phase. The former takes place in the isolated molecule limit whereas the latter requires collisions or photon emission. This usage differs from that standard in the condensed phase literature, where loss of phase information due to collisions is often referred to as “dephasing” while the term “quantum decoherence” is generally reserved for loss of phase information due to decay of the overlap between bath wavefunctions reacting to different system states). Since coherence is maintained in the isolated molecule limit, the wavepacket undergoes periodic revivals and fractional revivals, where the alignment is restored under field-free conditions. For linear and symmetric top molecules the initially created alignment is precisely reconstructed at multiples of the full rotational revival time (Phys.Rev. Lett. 83, 4971 (1999) pdf).
The alignment dynamics and rotational revival structure of asymmetric top molecules is much more interesting – as one would expect since the corresponding classical motion is unstable (an asymmetric top never revisits its original position). As a coherence-based control tool, alignment of asymmetric tops is also more relevant to chemistry, since most molecules are asymmetric tops. We discuss alignment and 3D orientational control of complex isolated systems in Sec. 2A and extend these concepts to dissipative media in Sec. 2B.
Consider first a rigid, linear molecule subject to a linearly polarized laser field whose frequency is tuned near resonance with a vibronic transition. In the weak field limit, if the system has been initially prepared in a rotational level Jo, electric dipole selection rules allow Jo and Jo ± 1 in the excited state. The interference between these levels gives rise to a mildly aligned excited state population, depending in sense on the type of the dipole transition. At nonperturbative intensities, the system undergoes Rabi-type oscillations between the two resonant rotational manifolds, exchanging another unit of angular momentum with the field on each transition. Consequently, a rotationally-broad wavepacket is produced in both states. The associated alignment is considerably sharper than the weak-field distribution, as it arises from the interference of many levels (see J.Chem.Phys. 103, 7887 (1995) pdf; Phys.Rev. Lett. 83, 4971 (1999) pdf). The degree of rotational excitation is determined by either the pulse duration or the balance between the laser intensity and the detuning from resonance. In the limit of a short, intense pulse, , the number of levels that can be excited is roughly the number of cycles the system has undergone between the two states, , where is the pulse duration, Be is the rotational constant, is the Rabi coupling and is the corresponding period. In the case of a long, or lower intensity pulse, , the degree of rotational, excitation is determined by the accumulated detuning from resonance, i.e., by the requirement that the Rabi coupling be sufficient to access rotational levels that are further detuned from resonance as the excitation proceeds,. A quantitative discussion is given in J.Chem.Phys. 115, 5965 (2001) pdf .
A rather similar coherent rotational excitation process takes place at non-resonant frequencies, well below electronic transition frequencies. In this case a rotationally-broad superposition state is produced via sequential Raman-type transitions, the system remaining in the ground vibronic state. The qualitative criteria determining the degree of rotational excitation remain as discussed in connection with the near-resonance case. In the case of near-resonance excitation, however, the Rabi coupling is proportional to the laser electric field, whereas in the case of nonresonant excitation it is proportional to the intensity, i.e., to the square of the field, due to the two-photon nature of the cycles. Consequently, higher intensities are required to achieve a given degree of alignment, as expected for a non-resonant process. It is interesting to note that the dynamics of rotation excitation and the wavepacket properties are essentially identical in the near and non-resonance excitation schemes. The degree of spatial localization and the time-evolution of the alignment are essentially independent of the frequency regime; they are largely controlled by the temporal characteristics of the laser pulse.
The simplest case scenario is that of alignment in a continuous wave field, where dynamical considerations play no role. In practice, intense field experiments are generally carried out in pulsed mode, but, provided that the pulse is long with respect to the rotational period, , each eigenstate of the field-free Hamiltonian is guaranteed to evolve adiabatically into the corresponding state of the complete Hamiltonian during the turn-on, returning to the original (isotropic) field-free eigenstate upon turn-off. Formally, the problem is thus reduced to alignment in a CW field of the peak intensity. The latter problem is formally equivalent to alignment of nonpolar molecules in a strong DC field, since the oscillations of the laser field at the light frequency can be eliminated at both near- and non-resonance frequencies. Thus, in the long pulse limit, laser alignment reduces to an intensively studied problem that is readily understood in classical terms, namely the problem of field-induced pendular states. In this limit, the sole requirement for alignment is that the Rabi coupling be large as compared to the rotational energy of the molecules at a given rotational temperature. Intense laser alignment in the CW limit (termed adiabatic alignment) grew out of research on alignment and orientation in DC fields, and is very similar to DC alignment conceptually and numerically. It shares the advantage of DC field alignment of offering an analytical solution in the linear, rigid rotor case. It shares, however, also the main drawback of DC field alignment, namely, the alignment is lost once the laser pulse has been turned off. For applications one desires field-free aligned molecules.
Short-pulse-induced alignment (termed nonadiabatic, or dynamical alignment) was introduced (see J.Chem.Phys. 103, 7887 (1995) pdf ) at the same time as the adiabatic counterpart but is qualitatively different in concept. This field of research grew out of research on wavepacket dynamics and shares several of the well-studied features of vibrational and electronic wavepackets while exhibiting several unique properties (vide infra). A short laser pulse, , leaves the system in a coherent superposition of rotational levels that is aligned upon turn-off, dephases at a rate proportional to the square of the wavepacket width in J-space, and subsequently revives and dephases periodically in time. As long as coherence is maintained, the alignment is reconstructed at predetermined times and survives for a controllable period. As with other discrete state wavepackets of stable motions, all observables obtained from the wavepacket are periodic in the full revival time, given, in the case of rotational wavepackets, by the rotational period.
In the ultrashort pulse limit, where the interaction is an impulse as compared to rotations, the dynamics following immediately the laser pulse is again readily understood in classical terms. Here the pulse imparts a "kick" to the molecule, which rapidly transfers a large amount of angular momentum to the system. The molecule is rotationally frozen during the interaction, but the sudden "kick" is encoded in the wavepacket rotational composition and gives rise to alignment upon turn-off. The subsequent long time dynamics, by contrast, exhibit strongly quantum mechanical dephasing and rephrasing dynamics that does not have a classical analog. At very low rotational temperatures, the revival structure is rich and dense, differing qualitatively from the generic revival structures of bound state wavepackets. As the rotational temperature increases, however, the structure simplifies drastically. At each multiple of the rotational period the initial alignment is precisely reconstructed. At half the period the molecules align perpendicular to the field. Depending on the spin statistics of the molecule studied, one observes features (termed fractional revivals) also at other fractions of the rotational period (Phys.Rev. Lett. 83, 4971 (1999) pdf).
Our discussion so far has been limited to linear systems, which have been the focus of the vast majority of theoretical and experimental contributions to the field so far. We show in what follows that the case of nonlinear, in particular asymmetric, rotors offers new and fascinating physics.
One new feature that polyatomic molecules bring along is trivial – they rotate about three nonequivalent axes. In the presence of the linearly polarized field, the most polarizable molecular axis is confined to the polarization vector, but the molecule remains free to rotate about it’s own axis as well as about the polarization vector.
Elliptically-polarized fields suggest themselves as a simple, intuitive route to three-dimensional orientational control. In particular, they introduce the possibility of tailoring the field polarization to the molecular polarizability tensor by varying the eccentricity from the linear (best adapted to linear, or prolate symmetric tops) to the circular (best adapted for planar symmetric tops) polarization limit. The theoretical and experimental techniques to spatially localize rotational wavepackets of asymmetric top molecules in the three angular variables were developed already in 2000 but were applied only in the adiabatic domain (see Phys.Rev.Lett. 85, 2470 (2000) pdf ).
The generalization of this method to field-free (post pulse) 3D alignment of a general asymmetric top molecule has been the subject of a growing number of studies in the years that followed, fueled by a host of potential applications. Nonetheless, the question of how to best achieve such alignment remained open for a decade. The combination of a long with a short (with the respect to the rotational period of the molecule) orthogonally polarized pulses discussed below, produces sharp 3D alignment, which can be transformed to field-free alignment by rapidly truncating the long pulse following the short pulse turn-off. In a different regime, a single short pulse or a pair of time-separated short pulses were shown to induce field-free 3D alignment provided that the field is polarized differently in two directions, e.g., a single elliptically polarized pulse or two pulses linearly polarized in orthogonal directions and with a time delay between their peaks. The most efficient way to choose the pulse configuration that produces the best 3D alignment, or enhances the alignment in a particular angle, is by tailoring the pulse parameters to a given molecule through optimization, as discussed below (see Phys.Rev. A. (2010) ).
On the optimal route to field-free three-dimensional alignment
We consider (Phys.Rev. A. (2010) ) an asymmetric top molecule that is subjected to two perpendicular linearly polarized laser pulses with Gaussian envelops. The parameters of the pulses, i.e., their widths, centers, and amplitudes, are optimized in order to improve 3D alignment at a predefined target time immediately following the field turn-off. Since the alignment is sought in the field-free regime, the optimization algorithm is modified to incorporate a constraint guaranteeing that the optimal field amplitude at the target time will be below a set threshold value. Figure 2A-1 shows the alignment dynamics (quantified in terms of the expectation values of the squared cosines of the three Euler angles) produced by the initial field (dashed curves) and by the optimal field (solid curves). Here, we require the optimal field to produce alignment in and (perfect alignment corresponds to the value of 1) and antialignment in (perfect antialignment corresponds to the value of 0). The initial field is circularly polarized and centered at t=0. Its envelope as well as the envelope of the optimal field are shown in Figure 2A-2 in red and green, respectively. We take advantage of the excellent convergence properties of the Quasi-Newton optimization method. The changes in the field envelope as the optimization progresses are shown in Figures 2A-3a and 2A-3b.
To test the extent to which two time-separated orthogonally polarized pulses provide advantage over a single elliptically polarized pulse in producing field-free 3D alignment, the three envelope parameters, i.e., the width, center, and amplitude, and the field ellipticity are subjected to the same optimization procedure. The resultant optimized pulse with elliptical polarization is shown in Figure 2A-2 in blue. The alignment dynamics induced by the optimal elliptical pulse are almost indistinguishable from those produced by the two-pulse control field (Figure 2A-1, solid curves). As is often the case, increasing the flexibility of the controls, i.e., increasing the number of optimizable parameters, improves the target objective; however, in the present case, the field-free 3D alignment produced by the two-pulse field is only marginally better (less than 1 percent difference) than the alignment produced by the single elliptically polarized pulse.
Holding and Spinning Molecules in Space
In the polyatomic domain, complete control over the rotational motions requires one to go beyond alignment (or orientation). In the limit of micron-sized particles, this task has already been accomplished by optical tweezer type techniques, which allow mesoscopic objects to be held and rotated about an axis of choice. In Phys.Rev.Lett. 99,143602 (2007) pdf we collaborate with the experimental group of H. Stapelfeldt (Aarhus) to introduce a method, based on the combination of nanosecond and femtosecond laser pulses, that represents a step toward achieving this goal for molecules, and illustrate its potential theoretically and experimentally. We use 3,5 difluoroiodobenzene (DFIB) molecules as a model but the approach is general.
Our method uses two linearly polarized laser pulses, one long and one short with respect to the inherent molecular rotational periods. The long pulse tightly aligns the most polarizable molecular axis along its polarization vector. The tight alignment maintained while the long pulse is on, enables us to direct the short pulse onto the molecules with a polarization perpendicular to the most polarizable axis. This perpendicular geometry essentially eliminates the interaction between the short pulse and the aligned molecular axis, and hence the expected rotation of this axis towards the polarization direction of the short pulse is avoided. Instead, the short pulse interacts with the second most polarizable axis, which sets the molecule into controlled rotation about the arrested molecular axis. As a result strong 3D alignment is observed shortly after the short pulse, and is repeated periodically, reflecting periodic revolution about the molecular axis. This represents a quantum mechanical version of an asymmetric rigid rotor spinning around an axis held fixed in space.
The results of numerical simulations for the case of difluorobromobenzene (DFIB) are shown in Figure 2A-4. In the figure, the expectation values of the squared cosines of the Euler angles quantify the extent of alignment, and the expectation values of the squared rotational quantum numbers quantify the extent of the underlying rotational excitation. The molecular Z-axis is defined by the C-I bond, while the X-axis is in the molecular plane, and the Y-axis is out of the plane. The long pulse is taken to be polarized along the space-fixed z-axis (for reasons of numerical efficiency), and the short pulse is polarized along the y-axis. The zero of time is chosen at the center of the long and short pulses. The long pulse produces nearly perfect alignment of the body-fixed Z axis (the C-I bond) to the space-fixed z axis while the short pulse causes the molecule to rotate about the Z axis. The dynamics in the azimuthal angle exhibits a periodic revival structure corresponding to nearly regular, uniform rotation of the ensemble of molecules about the molecular axis. The short pulse produces considerably larger rotational excitation compared to the long pulse. This fact is illustrated in Figure 2A-5, where the initial thermal distribution of rotational states produced by the long pulse (black bars) is shown together with the distribution of rotational states at the end of the simulation (red bars).
The general idea of the "hold and spin" method can be extended to furnish field-free 3D alignment by rapidly truncating the long pulse following the short pulse turn-off. Figure 2A-6 shows alignment subsequent to truncation of the long pulse (in red) superimposed for comparison on the results of Figure 2A-4 (in blue). The long pulse populates a broad superposition of J states while conserving the space-fixed projection of J. The short pulse, then, tilts the angular momentum vector with respect to the body- and space-fixed frames by coherent excitation of M and K levels. This causes the plane spanned by the two most polarizable molecular axes to align with the plane spanned by the laser polarizations, with the trade-off of diminishing the initial alignment of the symmetry axis. After the short pulse, if the long pulse is not switched off, the symmetry axis alignment peaks again and the plane alignment gradually deteriorates. If the long pulse is truncated, the plane alignment persists, providing field-free 3D alignment.
In another modification to the "hold and spin" method, an elliptically polarized long pulse is combined with a linearly polarized short pulse. The elliptically polarized long pulse provides sharp alignment of the most polarizable molecular axis while aligning the second most polarizable molecular axis along the minor semi-axis of the field polarization ellipse. As a result, the short pulse interacts with a sample of molecules with their planes prealigned, leading to stronger overall alignment.
The motivations for extending the concepts of alignment and 3D alignment to dissipative media are many. First it may be expected (and is illustrated below) that the unique coherence properties of rotationally-broad wavepackets would provide a sensitive probe of the dissipative properties of the environment. In particular, we show that the experimental observables of alignment disentangle decoherence from population relaxation effects (often referred to as T1 and T2 effects), providing independent measures of the relaxation and the decoherence dynamics that go beyond rate measurements. The potential of such measures is evident from several decades of intensive research on pressure broadening and polarization spectroscopy, motivated by the realization that rotationally-inelastic and elastic re-orienting collisions contain unique information about reaction mechanisms and pathways. Until recently, however, means of disentangling decay from decoherence or probing the dynamics of elastic collisions has been lacking. Second, we expect laser alignment to become a versatile tool in chemistry, once the effects of dissipative media on alignment are properly understood. In particular, we anticipate applications of 3D alignment, as induced by elliptically polarized pulses, ranging from control of charge transfer reactions and new designs of molecular switches, thorough enhancement of NMR, EPR and diffraction spectroscopies, to means of controlling molecular assembly in device fabrication. Likewise interesting, and the topic of ongoing study, is the application of rotational wavepackets in the gas cell environment for quantum storage. Further impetus for extending the theory of alignment to dissipative media within a quantum mechanical framework comes from recent experiments on rotational coherence spectroscopy in a dense gas environment. So far interpreted within a weak field approach, experimental work in this area has recently provided ample evidence for strong field effects, calling for a non-perturbative theory. Thus, both the case of alignment in a gas cell and that of alignment in solution are of formal interest and practical relevance.
In Phys. Rev. Lett., 95, 113001 (2005) pdf we extend the concept of short-pulse-induced alignment is to dissipative environments within a quantum mechanical, density matrix formalism. Here we only briefly illustrate the application of alignment as a probe of the dissipative properties of the medium, omitting a discussion of our formalism and of the extent to which, and the method through which, field-free alignment can survive in dense environments, see Phys. Rev. Lett., 95, 113001 (2005) pdf for details. To that end we first partition the alignment measure as
While formally the partitioning of a representation of the density operator into diagonal and off-diagonal terms is always possible, in the context of alignment Eq. (1) carries interesting physical significance and practical consequences. Its power owes to the fact that provides a direct experimental measure of the population elements, whereas measures directly the time evolution of the coherence elements of the density matrix. This property can be illustrated by expressing the matrix elements of in terms of analytical functions, to provide explicit forms for the time-evolution of and . . It follows that presents an experimental measure of the dynamics of population relaxation and is independent of decoherence, whereas responds solely to decohering processes. Each component of the density matrix can be obtained independently from an experimental signal as illustrated in Fig. 2B-1.
Thus, while much insight into the rotational relaxation dynamics has been derived from the combination of measurements and theoretical analysis of rotationally-resolved line-widths, we expect that alignment signals will provide a valuable complementary probe, yielding information that is silent in the conventional observables. Responsible for this effect is the fact that the alignment is a coherence response - it is determined by the relative phases of the rotational components comprising the wavepacket.
In Sec. 4A we discuss the application of optimal control theory both as a coherence spectroscopy, to gain further insights into the system bath interactions and the alignment dynamics, and as a control tool, to structure rotational wavepackets of desired alignment properties in dissipative media.
Atoms and molecules subject to an intense pulse of radiation have been shown to re-emit radiation with frequencies that are many high integer multiples of the driving frequency. This is known as high-harmonic generation (HHG). The mechanism for the process has been understood for some time in terms of the so called three-step model. First, the field drives an electron to tunnel ionize. Second, it accelerates the electron away from the core. Finally, the electron is driven back as the field reverses, and, with a certain probability, recombines with the core, emitting the kinetic energy gained during the journey in the field in the form of high harmonics of the incident radiation.
HHG has attracted much scientific interest over the past few years. From a practical perspective, it may serve as a route to high frequency ultra-short laser pulses (down to attoseconds), a tool that could potentially probe the fastest chemical processes on an electronic timescale. From a fundamental perspective, it has been suggested that the combination of HHG with the nonadiabatic alignment concept (Sec. 2A) has the potential to provide useful information about the electronic structure of molecules. Since 2005 the interest in HHG from nonadiabatically aligned molecules has been rapidly growing worldwide.
Our group’s early work on the problem of HHG from aligned molecules (Phys.Rev.Lett 99 113901 (2007) pdf) pointed to the information content of HHG from aligned molecules regarding the rotational coherences of aligned wavepackets and was able to explain analytically several disturbing experimental observations. Our more recent work unraveled the physics underlying birefringence in HHG from aligned molecules, and illustrated under what conditions birefringence is expected (Phys.Rev. A Rapid Communication 81, 021802(R) (2010) pdf). Currently we develop a new numerical approach to describe the underlying electronic dynamics, which will allow us to match the accuracy of future experiments, to compute detailed observables that are sensitive to the quality of the theory, and to address challenging systems see Figs. 2C-1, 2C-2.
Laser filamentation is a propagation regime restricted to short laser pulses (picoseconds and below), where a low density plasma is left in the wake of the pulse. Laser filamentation is observed when the peak power of intense laser pulses is higher than the critical power for self-focusing in the medium. Under this condition, the Kerr self-focusing will overcome the natural diffraction and the group velocity dispersion, causing the collapse of the pulse. The main mechanism that stops the beam collapse is the diffraction by the plasma that is self-generated through multiphoton/tunnel ionization of the atoms and/or molecules in the medium. As a result of the highly non-linear nature of tunneling ionization, the intensity is clamped to about 5×1013 W/cm2 in the case of filament in air. The diameter of the filament ranges typically from 50 to 100 mm.
Although the discovery of filamentation dates back to 1995, the critical role of nonadiabatic alignment in this field was not realized until 14 years later. It is now clear that, on the one hand, proper understanding and modeling of filamentation requires quantum mechanical account of nonadiabatic alignment and, on the other hand, alignment can serve as a unique control tool for filamentation physics. The underlying theory is developed in a joint experimental-theoretical study with the group of S.-L. Chin (see Optics Communications 283, 2732 (2010) pdf), where we study the combined effects of the electronic response, the rotational revivals, and the plasma response on the time-dependent refractive index.
Intense lasers can serve to manipulate also the translational modes of molecules. The spatial intensity profile of the laser beam defines an attractive potential for the center-of-mass motion, which can be used to focus molecular beams, waveguide them in space, collimate them and even separate them into isotopes, velocity components, rotational levels, or electronic states. Molecular focusing is introduced in J.Chem.Phys. 106, 2881 (1997) pdf. (See also Phys.Rev. A 56, R17 (1997) pdf). The generalization of the molecular lens to a field of molecular optics elements is given in J.Chem.Phys. 107, 10420 (1997) pdf along with some formal and practical details.
We show, in J.Chem.Phys. 111, 4397 (1999) pdf, that by excitation to high Rydberg states it is possible to reverse the sign of the polarizability tensor thus providing a means of reflecting molecules with light (a molecular mirror). It is also possible to combine molecular optics with molecular alignment, bringing a beam of aligned molecules to a focus in a field-free point in space. (See J.Chem.Phys. 111, 4113 (1999) pdf). In Phys.Rev. A 56, R17 (1997) pdf we propose the application of simultaneous alignment and focusing to form orientationally-ordered nanostructures on a substrate, potentially a route to materials with interesting electric and optical properties. An experimental realization of our scheme is discussed in J.App.Phys. 94, 669 (2003) pdf.