Dissipative Dynamics of Laser-Induced Torsional Coherences - The

Article pubs.acs.org/JPCC

Dissipative Dynamics of Laser-Induced Torsional Coherences Benjamin A. Ashwell, S. Ramakrishna, and Tamar Seideman* Department of Chemistry, Northwestern University, Evanston, Illinois 60208, United States ABSTRACT: We present a model for strong field coherent control of torsional modes of molecules with a focus on exploring the controllability of molecular torsions subject to dissipative media and understanding how phase information is exchanged between torsional modes and a dissipative environment. Our theory is based on a density matrix formalism, wherein dissipation is accounted for within a multilevel Bloch equation model. Our results point to new and interesting phenomena in wavepacket dissipation dynamics that are unique to torsions and also enrich our general understanding of wavepacket phenomena. In addition, we suggest guidelines for designing torsional control experiments for molecules interacting with a dissipative bath.

1. INTRODUCTION Coherent quantum control in dissipative environments has been a topic of intense scientific interest for over a decade,1−11 in part because most practical applications envisioned will involve a dissipative medium, in part due to our understanding that open systems behave fundamentally differently from closed systems when subject to a control field, and in part due to the conceptual and practical connection of quantum control with quantum information, where maintenance of coherence is critical.12−15 In particular, much effort has been devoted to understanding decoherence processes by developing better models and approximations, with the aim of extending the temporal window between coherent excitation and loss of control due to decoherence. 3 , 7 , 1 6 Low temperatures, optimization schemes,3−5,17−19 and careful system design can all extend the time scale over which coherent control is feasible. Much better studied, and of no less interest than the control problem, are the physics of relaxation and decoherence themselves.13,20,21 From this perspective, torsional coherences offer an interesting model, which is sufficiently complex to exhibit a rich spectrum of phenomena (vide infra) yet offers the simplicity of 1D systems and the transferability of conclusions that characterizes rotational coherences. We follow the conventions of the gas-phase wavepacket dynamics literature, where “dephasing” is the term used for change of the relative phases of the wavepacket components due to anharmonicity of the molecular spectrum, whereas “decoherence” is the term used for loss of phase. The former takes place in the isolated molecule limit, whereas the latter requires collisions or photon emission. It is important to note that 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 reserved for loss of phase information resulting from decay of the overlap between bath wave functions reacting to different system states. © 2013 American Chemical Society

The topic of this paper is the dissipation of torsional coherences subject to a medium and its implications for the controllability of torsional modes in practical environments. Coherent torsional control, first introduced in the theoretical work of ref 22, has been the topic of increased experimental and theoretical interest over the past two years.18,23−26,43 This interest can be attributed to several interesting fundamental properties of torsional wavepackets,24,27,28 which are a result of the richness of torsional spectra (vide infra) and do not have analogs in vibrational, rotational, or electronic spectra. It is also fueled by a wide variety of potential applications in physics, chemistry, materials research, and possibly biology.24 To date, torsional control has been studied in the isolated molecule limit. The controllability of torsional modes subject to dissipative media, and the way in which torsions exchange phase information with an environment, are thus open questions that carry not only fundamental value but also potential practical consequences. The majority of applications envisioned will involve a dissipative environment, such as a solvent, a surface, or a dense gas medium.24 Furthermore, given the unique properties of torsional coherences (see below), one expects torsional dissipation to exhibit new phenomena that have not been studied in the context of other types of wavepackets and that will enhance our general understanding of wavepacket dissipation. We remark here that whereas torsional alignment has been to date experimentally demonstrated only for isolated molecules,23,29,30 the much better studied problem of rotational alignment, which is both conceptually and practically related to torsional alignment, has been realized in a dissipative environment in several laboratories.31−33 Special Issue: Ron Naaman Festschrift Received: March 28, 2013 Revised: June 13, 2013 Published: August 1, 2013 22391

dx.doi.org/10.1021/jp403090u | J. Phys. Chem. C 2013, 117, 22391−22400

The Journal of Physical Chemistry C

Article

1 Ĥ ind = Δαε 2(t ) cos2 β 8

Our objective is to develop a theoretical framework to study torsional alignment subject to dissipation and then apply it to the exploration of the response of torsional coherences to a dissipative bath and the extent to which (and the conditions under which) torsions subject to dissipation are controllable. Our theory is based on a reduced density matrix approach that starts from the quantum Liouville equation, wherein the multilevel Bloch model is used to account for population relaxation and loss of coherence resulting from inelastic processes as well as pure decoherence (arising from elastic processes and often referred to as “pure dephasing”). Inelastic processes are always present in dissipative media, whereas pure decoherence is important in liquids but often found to be negligible in dense gases. In the next section, we outline our model system, our control scheme, and the details of our theory. In Section 3, we present and discuss our results, for clarity, starting with a brief review of torsional alignment in isolated systems (Section 3.1) that will serve as an introduction and a basis for comparison for the study of torsional dissipation in Section 3.2. Finally, in Section 4, we summarize our conclusions.

where Δα < 0 is the polarizability anisotropy (the difference between the out-of-plane and in-plane body-fixed polarizability components) of an isolated monomer (a single phenyl ring, in the case of biphenyl), ε(t) is the pulse envelope, taken here as a Gaussian function, and we have neglected the mutual interaction of the induced dipoles in the two monomers. Our use of the simplest possible model for the induced Hamiltonian allows us to generate general insights and simplifies the comparison of torsional coherences with the extensively studied topic of rotational coherences. Analytical considerations published elsewhere illustrate that the qualitative features discussed below are not dependent on the details of the field matter interaction, provided that it is able to generate a coherent torsional wavepacket.36 Because the interaction term conserves the symmetry of the field-free Hamiltonian, the field does not couple solutions of different symmetry; in the isolated molecule limit the four symmetry subsets evolve separately. Transferability of the results presented below to other molecules that may have advantageous properties or interesting applications is clearly important. To that end, we introduce a dimensionless interaction parameter

2. THEORY In this section we discuss our model system and theoretical approach. We start in Section 2.1 with the details of the field-free Hamiltonian and field−matter interaction and proceed in Section 2.2 to describe our method of solving the quantum equations of motion subject to the combination of a dissipative medium and a nonperturbative laser field. Hartree atomic units are used throughout. 2.1. Hamiltonian. The complete Hamiltonian, Ĥ , is written as the sum of the field-free molecular Hamiltonian and the field− matter interaction Hamiltonian, which, in the far-off-resonance frequency domain considered here, is approximated as the induced dipole interaction. Coupling between the highfrequency vibrations and the much lower frequency torsional modes is not expected to play an important role at the short time scales we consider in this work,34 and hence the field-free Hamiltonian is approximated by the torsional component Ĥ tor alone 1 d2 Ĥ tor = − + V2 cos(2β) + V4 cos(4β) 2Itor dβ 2

1

Ω=

a = 1, 2, 3, 4

− 8 Δα |ε0|2 barrier to coplanarity

(4)

where ε0 is the field at the pulse peak, which serves as a moleculeindependent measure of the interaction strength. Because the temporal characteristics depend on several system properties, we prefer, in the present study, to use physical rather than dimensionless molecule-independent units for the time variable. 2.2. Time Evolution and Decoherence. To account for dissipative processes, it is necessary to use a density matrix formulation, where the time evolution is governed by the quantum Liouville equation. We thus expand the density operator ρ̂ on the basis of the field-free Whittaker−Hill solutions as ρ (̂ t ) =

∑ ρa b (t )|aj⟩⟨bk| j k

aj , bk

(1)

(5)

where ρajbk(t) defines a density matrix element, and we introduced the shorthand notation

where Itor is the torsional moment of inertia, the cosine series is the π-periodic torsional potential energy surface, and β is the torsional, or dihedral, angle. With eq 1, the field-free torsional time-independent Schrödinger equation is readily seen to map onto the Whittaker−Hill equation, which we solve using the technique described in refs 25 and 35. The solutions of the (fieldfree) torsional Schrödinger equation fall into four symmetry groups wa(j) ,

(3)

|aj⟩ ≡ |wa(j)⟩, a = 1, 2, 3, 4

(6)

The quantum Liouville equation thus assumes the form ⎛ ∂ρ ̂ ⎞ ∂ρ ̂ i = − [Htor + Hind , ρ ]̂ + ⎜ ⎟ ⎝ ∂t ⎠diss ∂t ℏ

(7)

If one ignores the coupling between coherences and populations induced by the dissipative dynamics, then the dissipative operator takes the multilevel Bloch form21

(2)

where the subscript denotes the symmetry species and the superscript gives the torsional quantum number. The w(j) a are used as the basis set for our calculations. The ground state is fourfold degenerate (one from each of the four symmetry solution sets), so for T = 0 K our initial state is an equal mixture of the ground state from each solution set. For T > 0 K, we create the initial state with a Boltzmann distribution. Ref 24 discusses our approximation of the induced Hamiltonian, Ĥ ind

⎧ ⎪ ⎛ ∂ρ ̂ ⎞ 1 ⎜ ⎟ = −⎨∑ ([K cldm|cl⟩⟨cl| , ρ (̂ t )]+ − K cldm|dm⟩ ⎪ ⎝ ∂t ⎠diss ⎩ cldm 2 ⎫ ⎪ × ⟨cl|ρ (̂ t )|cl⟩⟨dm|)⎬ + ⎪ ⎭ 22392

∑ γc(dpd)|cl⟩⟨cl|ρ (̂ t )|dm⟩⟨dm| cldm

l m

(8)



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previous torsional control experiments and simulations.22,25,29 The larger polarizability and inertia of AAC, along with its substantially lower barrier to coplanarity, make it advantageous for experiments. These advantages are most important in dissipative environments, first because a laser intensity allowing sharp torsional alignment must be kept below the solvent damage threshold and second because a larger inertia translates into longer delay between the pulse peak and the alignment peak. This delay allows for a much extended field-free torsional alignment immediately following the pulse turn-off compared with that for biphenyl. Whereas in isolated molecules field-free torsional alignment is also obtained at the time-delayed partial revivals, in a solvent the only opportunity for sharp torsional alignment under field-free conditions is immediately after the turn-off. Although experimentally and numerically these two systems are quite different, conceptually they exhibit the same characteristics, which are common to other molecules undergoing torsion. In particular, the barriers to coplanarity and coperpendicularity differ in magnitude, with the latter being higher for both molecules, thus introducing a (system-dependent) intensity regime where the molecule can be forced into coplanarity but does not exhibit free internal rotation. The postpulse evolution of wavepackets directly mirrors the eigenvalue spectrum of the field-free molecule and is completely determined by the relative amplitudes (magnitude and phase) of the energy eigenstates comprising the wavepacket and the eigenstate energy level spacing. Likewise, dissipative phenomena are partially determined by the energy level spacing because inelastic processes are subject to energy conservation in the complete (system+bath) system. It is thus germane to precede a discussion of the wavepacket dynamics with an illustration of the energy-level spacings of torsional modes. Figure 1 illustrates a gradual change of the level spacing from vibrational-like for energies well below both torsional barriers to rotational-like for energies just above the barrier to coplanarity, at energy regimes where neither barrier plays a role, reverting back to vibrationallike (with a much lower frequency) as the higher of the barriers is felt, corresponding to bound librational motion in the larger range potential well. Finally, when the energy level exceeds the barrier to coperpendicularity, the eigenstates begin to behave as free 2D rotors and the level spacing reassumes the rotation-like character. Both molecules show the same qualitative energy level spacing, differing only in their density of states. This alternation in the level-spacing structure is unique to torsional motions and leaves a clear signature on the dissipative dynamics. 3.1.2. Light-Controlled Torsional Dynamics. Our focus here is on the case of nonadiabatic torsional alignment, where the pulse duration, τ, is short compared with the torsional period, τtor, introducing coherent wavepacket motion after the pulse turn-off and allowing for field-free manipulation as long as coherence is (at least partially) maintained. The anharmonicity of the torsional spectrum causes coherent dephasing of the wavepacket. The anharmonicity also leads to a shift in the dynamic equilibrium angle of the molecule, detailed in ref 24. These phenomena are analogous to the well-studied, so-called permanent and transient alignment phenomena in rotational wavepackets.34,37 One key difference between torsional and rotational alignment is the presence of a field-free torsional potential energy barrier, because of which full revivals (where the initial field-free torsional alignment is reconstructed) are not expected. Unlike adiabatic alignment, the nonadiabatic case allows us to study field-free decoherence as well as to benefit from the practical advantages of postpulse alignment.

where Kcldm denotes the rate of transition from the state |cl⟩ to the state |dm⟩ and γ(pd) cldm incorporates the pure decoherence between the respective levels. The equation of motion for the density matrix elements is dρa b

j k

dt

⎧ i⎪ = − ⎨(Eaj − Ebk )ρa b − j k ℏ⎪ ⎩

∑ χ(t )(Va c ρc b j l

cl

l k

⎫ ⎧ ⎪ ⎪ 1 − Vclbk ρa c )⎬ − ⎨∑ [K ajcl ρa b + K bkcl ρa b ] j l ⎪ j k j k ⎪ ⎭ ⎩ cl 2 −

∑ K c a ρc c δa ,b l j

cl

l l

j

k

⎫ ⎪ ) + γa(pd − ρ δ (1 ) ajbk ⎬ jbk ajbk ⎪ ⎭

(9)

where the terms in the first set of curly brackets describe the coherent dynamics, the terms in the second set of curly brackets describe the dissipation within the multilevel Bloch model, Vajbk is 2 (k) 2 the matrix element ⟨w(j) a |cos β|wb ⟩, and χ(t) = (1/8)Δaε (t). The rate of transition between quantum states can be modeled, in the weak system−bath coupling limit, as K cldm = |⟨cl|f (β)|dm⟩|2 C(ωcldm)

(10)

where C(ωcldm) is the Fourier transform of the bath correlation function, with ωc1dm = (Ec1−Edm)/ℏ.21 The transition between quantum states is mediated by the bath coupling function, f(β) in eq 10, whose precise form for the problem at hand is derived in Section 3.2.1. The Fourier transform of the bath correlation function is given in terms of the bath spectral density J(ω) as C(ω) = 2π[1 + n(ω)]ω 2(J(ω) − J( −ω))

(11)

−1

where n(ω) = [exp(ℏω/kT) − 1] is the Bose−Einstein distribution function. In the present work we use the Debye spectral density ω 2J(ω) = Θ(ω)

j0 ω 2

ω + ωD2

(12)

where the step function Θ(ω) ensures J = 0 for ω < 0, j0 is a constant, and ωD is the Debye frequency. The spectral function has a single peak at ωD, signifying maximum energy transfer between system and bath at this frequency. The two parameters in the dissipation model, j0 and the Debye frequency, are determined by the exact nature of the bath. In the calculations below they are treated as adjustable parameters.

3. RESULTS AND DISCUSSION In this section, after a brief discussion of the model system and the results it provides for isolated molecules, we explore the dynamics of relaxation and decoherence of torsional wavepackets within the multilevel Bloch model. Torsion offers a rich perspective for the study of dissipation because torsional potential energy surfaces give rise to a strongly nonmonotonic dependence of the energy level spacings on the quantum number, which strongly influences the nature and magnitude of the system−environment coupling. We therefore begin by briefly discussing the structure of torsional spectra. 3.1. Torsional Alignment in Isolated Molecules. 3.1.1. Model System. In ref 24 we introduced 9-[2-(anthracen9-yl)ethynyl]anthracene (AAC) as an alternative to the more standard biphenyl (and its derivatives) used as a model in 22393



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(corresponding to 16 TW·cm−2 for biphenyl and to 3 TW·cm−2 for AAC), taking the pulse duration to be ∼30% of the molecule’s respective fundamental torsional period. (This choice has been shown to maximize the field-free torsional alignment immediately following the pulse for the range of intensities used here.)24 As a transferable, experimentally observable measure of the degree and time dependence of the torsional alignment, we consider the expectation value of cos2 β in the torsional wavepacket. In each panel the left- and right-hand graphs show the short- and long-time dynamics, respectively. In the case of biphenyl, Figure 2a, the coherent dephasing causes wavepacket partial revivals at ∼20 ps intervals, as determined by the anharmonicity of the potential energy surface, whereas the smaller anharmonicity of AAC at the level of excitation considered gives rise to wavepacket revivals at ∼200 ps intervals. The red, green, and blue curves in each panel correspond to rotational temperatures of T = 0, 100, and 300 K, respectively. The initial torsional alignment depends only weakly on the rotational temperature, but the revival pattern is significantly diminished and distorted as the temperature increases. These features are analogues to the temperature dependence of rotational alignment and are a result of the incoherent averaging of signals that revive at slightly displaced times. It is interesting to note that of the simulations shown in Figure 2 the greatest alignment is achieved at T = 100 K for both molecules. Qualitatively, we can explain this as a result of the initial population distribution in a range of temperatures involving levels closer to the coplanar configuration, allowing for greater alignment with the same laser pulse. At even higher temperatures, however (see, e.g., the T = 300 K traces), the initial population distribution includes levels very near the barrier to coplanarity, and hence the laser moves the population into eigenstates with probability densities closer to the coperpendicular configuration. A similar phenomenon, caused by highintensity lasers rather than high temperature, was reported in ref 24. This change in behavior at high temperatures is very general and is associated with population residing in both the initial potential well (corresponding to bound librational motion that does not involve rotation across the barrier to coplanarity) and the next potential well (corresponding to bound librational motion across the barrier to coplanarity but not a full rotation). This behavior always occurs in a certain range of temperatures

Figure 1. (a) Energy level spacing between adjacent eigenstates of biphenyl (solid red, left ordinate) and AAC (dashed blue, right ordinate) for the w1 solution set. Details of the potential energy surfaces and other molecular properties can be found in ref 24. Both molecules exhibit the same level spacing characteristics, differing only in the density of torsional states. The points corresponding to the parallel and perpendicular configurations are marked by ∥ and ⊥, respectively. (b) The two molecules are shown in their equilibrium positions.

Time Evolution. Figure 2 compactly summarizes the torsional alignment dynamics in the isolated molecule limit and is provided as a basis for understanding and as a contrast with the dissipative dynamics discussed below. To that end, we point to the short- and long-time dynamics, the role played by the ratio of the field-induced potential and the field-free barrier height, and the effects of the system density of states and rotational temperature. Specifically, we consider the torsional alignment of biphenyl and AAC in response to Ω = 1 laser pulses

Figure 2. Time evolution of the expectation value ⟨cos2β⟩ for biphenyl (panel a) and AAC (panel b) in response to an Ω = 1 laser pulse. Each panel shows the short time torsional alignment, during which the molecule is subject to the laser field (given as a black curve) on the left and its long time (field-free) evolution on the right. The red, green, and blue curves correspond to rotational temperatures of T = 0 K, T = 100 K, and T = 300 K, respectively. 22394



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greater than that of the AAC-resonant solvent by 20 times the difference between the maximum absorbance frequencies of the biphenyl-resonant and AAC-resonant solvents. By defining the model solvents relative to our model molecules rather than attempting to imitate a specific solvent we are able to study the more general question of how the molecular system decoheres in solvents whose response is detuned by varying degrees with respect to the natural system transitions. The time scale of dissipation is clearly sensitive to the spectral function parameters; hence we are unable to predict experimental time scales for a specific solvent but are able to observe relative time scales for different classes of solvents. As we would expect, Figure 3 shows that biphenyl and AAC decohere most quickly in solvents tailored to their dominant torsional transition frequencies and more slowly in the other solvents. The interplay between the molecular energy level spacing and the bath modes means that different solvents cause different groups of torsional levels in the wavepacket to be more strongly coupled to the bath, resulting in qualitatively different decoherence when compared with other solvents. 3.2.2. Excited-State Lifetimes. By examining the lifetimes of excited torsional states at different temperatures, in different solvents, and for different molecules, we can learn how to create conditions that will extend coherence as long as possible. The lifetime of a given torsional eigenstate is given by

and interaction strengths because both contribute to moving population into higher levels. 3.2. Torsional Alignment in Dissipative Media. Many of the most exciting applications of torsional alignment will take place in dissipative media, but the role of dissipation in this context has not been explored. In this section, we discuss the results of the dissipative theory derived in Section 2, focusing on both formal and practical insights from the model. Until Section 3.2.4, we assume that the pure decoherence (defined earlier) is negligible and focus on the inelastic processes. 3.2.1. System−Bath Interactions. The multilevel Bloch equation is very general in its basic formulation and, as discussed in Section 2, allows flexibility in choice of the bath coupling function and the spectral density function (here taken to be the Debye spectral density). Commonly, the bath coupling function is approximated as linear in the coordinate in question, f(x) = x, corresponding to expansion of the (unknown) bath coupling function as a Taylor series and neglect of all but the first order term.21 For the periodic system at hand this form is unsuitable, and hence we tailor the simplest function that possesses the required π-periodicity and satisfies our expectation that the system−bath coupling is weakest at the equilibrium configuration. Therefore, we chose f(β) = sin(2(β − βeq)), where βeq is the field-free equilibrium torsional angle of the molecule. Unlike the induced Hamiltonian, which only couples Whittaker−Hill solutions of the same symmetry, the system− bath interaction couples torsional eigenstates both within symmetry sets and between different symmetry sets. Specifically, the function introduced above couples solution sets 1 and 4 as well as solution sets 2 and 3 and in addition couples them to functions within a given set, allowing population to be exchanged between solution sets with the same periodicity. Other coupling functions that satisfy the required periodicity were examined and were found to generate qualitatively similar results. Thus, the details of the bath coupling function do not appear to play a noticeable role in the dynamics and can be disregarded. Whereas the bath coupling function defines the general form of the molecule-medium interaction, the spectral density of eqs 10 and 11 defines the properties of the solvent.21 Specifically, the Debye spectral density uses two parameters, ωD and j0, to specify the solvent behavior, where ωD is the Debye frequency, which determines the peak of the spectral density (the energy at which the bath most strongly couples to the system), and j0 determines the magnitude of this coupling. The generality of this model allows one to investigate limiting behaviors of the solvent and its response to the system coherences but at the same time makes simulating a specific solvent difficult. With these considerations in mind, we study three generalized solvents, (Table 1), where the ‘biphenyl-resonant’ and ‘AACresonant’ solvents have their peak absorption frequencies (ωD) set equal to their respective dominant ΔEtorsion (Figure 1). The ‘nonresonant’ solvent’s maximum absorbance frequency is

(K̃ aj)−1 = (∑ K aj , bk)−1 b,k

where the Kaj,bk are the quantum state-specific rates of eq 10. The variation of the lifetime with increasing energy eigenvalue can be broken into two distinct regimes, as shown in Figure 4. The lifetime trends, shown in panels a and b, are readily understood by reference to the energy-level structure provided along with the corresponding potential-energy curve in the inset c. Below the barrier to coperpendicularity, the lifetime decreases with increasing energy eigenvalue, rapidly at first and then, as the energy approaches the barrier to coperpendicularity, more gradually. This trend mimics what we expect for vibrational spectra. Above the barrier to coperpendicularity, the lifetime behavior as a function of the eigenenergy depends on the detuning of the torsional energy level spacing for the level in question from the peak absorption of the solvent. In the nonresonant solvent (Figure 4a), for instance, the lifetimes of eigenstates in this regime decrease with increasing quantum level, as their torsional level spacings approach the solvent’s Debye frequency. In the biphenyl-resonant solvent (Figure 4b), by contrast, the lifetimes in this regime increase with increasing quantum level as they become further detuned from the solvent’s peak absorption. Above the barrier to perpendicularity (n ≳ 16) (for biphenyl, cf., the level structure depicted as inset) the eigenfunctions approach the behavior of 2D free rotors and the lifetime changes more slowly. As expected, because of its larger density of states, AAC exhibits shorter-lived states in most solvents (not shown). With increasing temperature the lifetimes of all states (below and above the barrier) decrease. As a consequence of the overall decrease in the lifetime with increasing torsional energy level, high-intensity lasers create short-lived wavepackets. This implies that while increased peak alignment can be often achieved by increasing the laser intensity, a higher intensity may reduce the strength and number of wave packet revivals in dissipative environments.

Table 1. Parameters Used to Define Different Classes of Solventsa solvent type biphenyl-resonant AAC-resonant nonresonant

ωD

j0 −4

2.5 × 10 5 × 10−5 0.00425

(13)

1.5 × 10−9 3 × 10−10 2.55 × 10−8

a

Equations 10 and 11. j0 is chosen so that the maximum of the debye spectral density is 3 × 10−6 ω−1 for all baths. 22395



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Figure 3. Time evolution of a wavepacket subject to differently detuned solvents (see Table 1 for details of the solvents) for biphenyl (panel a) and AAC (panel b). The interaction strength is Ω = 1 and the temperature is T = 300 K. Green = nondissipative environment; blue = nonresonant solvent; red = AAC-resonant solvent; and black = biphenyl-resonant solvent.

Figure 4. Eigenstate-specific lifetimes for biphenyl subject to (a) the nonresonant solvent and (b) the biphenyl-resonant solvent. Red crosses: 50 K; green x's: 100 K; blue asterisks: 300 K. The inset (c) shows the torsional potential energy curve of biphenyl along with the corresponding energy level structure of the w1 symmetry solution set.

In general, |⟨cos2 β⟩| ≫ |⟨cos2 β⟩p|, hence the signal is typically dominated by the transient component. In what follows we first consider the coherence component of ⟨cos2 β⟩ and next turn to the population component. Torsional Decoherence. Because the decoherence is superimposed on highly nonperiodic dynamics associated with the coherent dephasing (which, as the name suggests, is a fully coherent process) and revivals (see, e.g., Figure 3, where maxima in the dissipative simulations do not correspond to maxima in the nondissipative one), disentangling the effects of coherent dephasing from those of decoherence in the experimental observable is an interesting problem. If the modulating envelope that describes the nondissipative dynamics were periodic, then it would be possible to trace the decoherence by inspection,38 but because this is not the case this approach fails to yield a unique trace of the decoherence. Here we disentangle decoherence from coherent dephasing and develop a quantitative measure of the former phenomenon by first transforming the equations of motion to a representation in which the coherent oscillations are eliminated. To that end we introduce the transformation

3.2.3. Torsional Wavepacket Decoherence and Relaxation. The expectation value ⟨cos2 β⟩ can be conveniently partitioned into a decoherence component, which tracks the evolution of the off-diagonal density matrix elements, and a relaxation component, which measures the diagonal elements; see eq 14. This partitioning has also been used for rotational alignment.38 In this subsection, we apply our understanding of the lifetime trends, established above, to analyze the information content of the decoherence and relaxation components of ⟨cos2 β⟩ in the absence of pure decoherence. The case of pure decoherence is addressed in the next section. We proceed by partitioning the expectation value into diagonal and off-diagonal density matrix contributions (corresponding to population and coherences, respectively) as ⟨cos2 β⟩ = Tr {cos2(β)·ρ ̂(t )} =

∑ ρajbk Vaj , bk aj , bk

= ⟨cos2 β⟩p + ⟨cos2 β⟩c

(14)

where ⟨cos β⟩p is the population contribution to the torsional alignment (corresponding to the diagonal elements of the density matrix and giving rise to the permanent alignment), whereas ⟨cos2 β⟩c is the contribution from the coherences (corresponding to the off-diagonal elements that give rise to the transient alignment)37 2

⟨cos2 β⟩p =

∑ ρa ,a Va ,a j

j

j

∑ aj , bk (j ≠ k)

j

dρa̅ , b

j k

dt

(15)

j

(17)

k

1 = − [K̃ aj + K̃ bk]ρa̅ , b (t ) j k 2

j ≠ k, a = b

(18)

where the field-induced term vanishes because we consider times after the pulse turnoff; pure decoherence, as noted above, is neglected in this subsection, and K̃ aj = ΣclKajcl. The transformed

ρa , b Vaj , bk j

k

with which the equation of motion (eq 9) for the off-diagonal elements of the density matrix takes the form

j

aj , aj

⟨cos2 β⟩c =

ρa , b (t ) = ρa̅ , b (t )ei(Ebk − Eaj)t

k

(16) 22396



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density matrix elements ρ̅aj,bk(t) in eq 18 do not exhibit the rapid coherent dephasing and fractional revival dynamics that are contained in the energy spacing phases and thus allow us to focus on the decoherence effects of interest. Although the decay of individual off-diagonal elements of the density matrix is monoexponential, the superposition that determines ⟨cos2 β⟩c exhibits a more complex time evolution through summation of states with different lifetimes. Rewriting eq 16 in terms of the ρ̅aj,bk(t) and differentiating, we find dρa̅ , b d⟨cos2 β⟩c j k = ∑ Vaj , bk dt d t a ,b j

t

⟨cos2 β⟩c (t ) = ⟨cos2 β⟩c (t0)e− ∫t0 Kc(t ′) dt ′

(22)

where t0 is a time past the aligning pulse when the molecule is field-free. Numerical integration of eq 22 thus provides the coherence projection of the torsional alignment in the transformed space, where the decoherence dynamics are disentangled from the (coherent) dephasing phenomenon.

(19)

k

2

(in the absence of decoherence, of eq 18 into eq 19 yields d⟨cos2 β⟩c dt

∑−

=

aj , bk

d⟨cos β⟩ dt

= 0), and substitution

1 ̃ [K aj + K̃ bk]Vaj , bk ρa̅ , b (t ) j k 2

= −Kc(t ) ∑ Vaj , bk ρa̅ , b (t ) j

k

aj , bk

= −Kc(t )⟨cos2 β⟩c (t )

(20)

where Kc(t) is defined as Kc(t ) =

∑a , b j

k

1 [K̃ aj 2

j

∑a , b Vaj , bk ρa̅ , b (t ) j

Figure 6. Rate of decoherence as determined by ⟨cos2 β⟩c (dashed green curves) compared with the observable ⟨cos2β⟩c(t) (red curves) for (a) biphenyl, T = 50 K, far off-resonant solvent; (b) biphenyl, T = 300 K, far off-resonant solvent; and (c) AAC, T = 50 K, AAC-resonant solvent. The interaction strength is Ω = 1.

+ K̃ bk]Vaj , bk ρa̅ , b (t ) k

j

k

k

(21)

Equation 21 defines a rate of decoherence, which can be evaluated numerically for a given system. In the case where the lifetimes, K̃ aj, are similar for all relevant levels, Kc(t) is timeindependent, and decoherence proceeds as a monoexponential process with a lifetime of K̃ aj (similar to the concept of T1 in NMR relaxation). When the K̃ aj that make non-negligible contributions to the wavepacket are dissimilar, by contrast, the rate will be multiexponential. Figure 5 shows the time evolution of Kc(t) for different temperatures and interaction strengths. High temperatures and high laser intensities both tend to create coherences that involve highly excited states with short lifetimes, resulting in a more rapid loss of coherence and stronger variation of Kc(t) with time. Integration of eq 21 gives

Figure 6 illustrates the ⟨cos2 β⟩c(t) of eq 22 for three situations. The low-temperature case of Figure 6a illustrates the case where the wavepacket is dominated by relatively low angular momentum components and hence the Kc(t) (eq 21) exhibit only very weak time dependence (compare Figure 5). Here the decoherence dynamics could be reasonably approximated by a simple exponential fit to the peaks of the (untransformed) ⟨cos2 β⟩c(t) revivals. By contrast, the high-temperature case of Figure 6b illustrates the decoherence dynamics in the more interesting case, where the degree of torsional excitation is high and Kc(t) varies markedly with time. Here the coherent dephasing and decoherence dynamics are strongly entangled in the untransformed ⟨cos β⟩c(t), and hence that latter observable deviates markedly from the proper decoherence measure of eq 22. Finally, in the case of Figure 6c, we see that because the time scale of decoherence is so much faster than that of coherent dephasing in AAC, the decoherence dynamics can again be approximated by a single exponential function. In general, cases where the coherent dephasing and decoherence take place on very different time scales should be amenable to determination of the rate of decoherence by inspection and can be fit with a single exponential. We remark that whereas ⟨cos2 β⟩c(t) is an experimental observable, ⟨cos2 β⟩c(t) relies in general on theoretical analysis and numerical calculations. The decoherence dynamics, as quantified in terms of the ⟨cos2 β⟩c, are next analyzed to yield the decoherence time scales by fitting to a multiexponential decay curve. Interestingly, for all systems tested, a sum of three exponentials accurately describes the decoherence. As the temperature decreases, fewer states are

Figure 5. Kc(t) of eq 21 for different temperatures and interaction strengths. Solid curves correspond to Ω = 1, dashed curves to Ω = 2, red curves to T = 50 K, green curves to T = 100 K, and blue curves to T = 300 K. 22397



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involved in the dynamics, and their lifetimes are similar, and hence Kc(t) approaches a constant value, leading to closer to monoexponential behavior. The above analysis provides a quantitative measure of the decoherence. For the case considered in Figure 3 for the nonresonant solvent, for instance, we find an initial decoherence time of 6 ps, an intermediate time of 16 ps, and a long time decoherence time of 50 ps. These correspond well to the lifetimes of the dominant coherences at various points in the simulation. These results further correspond with our understanding of the physics of decoherence: the higher energy, shorter lived coherences dominate the transient alignment at first but decohere quickly, leaving only the longer lived, lower energy coherences behind. Torsional Population Relaxation. Compared with the decoherence rate, the rate of population relaxation is far slower, often by an order of magnitude. Furthermore, the dynamics of the latter can manifest in counterintuitive ways, as sometimes ⟨cos2 β⟩p increases (becomes better aligned) well after the pulse is off, as shown, for instance, in Figure 7. It is interesting to note

K pin(t ) =

=−

k , bk

∑a , b (Vaj , ajK aj , bk ρa , a (t ) − Vaj , ajK bk , aj ρb ⟨cos2 β⟩p

(t ))

⟨cos2 β⟩p

= − [K pout(t ) − K pin(t )]⟨cos2 β⟩p = K pT (t )⟨cos2 β⟩p

(23)

see eq 14, where K pout(t ) =

∑a Vaj , ajK̃ aj ρa , a (t ) j

j

j

∑a Vaj , aj ρa , a (t ) j

j

j

Kinp

j

(25)

Kout p .

Given the numerically evaluated Kp(t), we can exactly 2 in reproduce the data in Figure 7. When Kout p (t) > Kp (t), ⟨cos β⟩p decreases with time as the population relaxes from levels favoring the coplanar configuration into the equilibrium levels, 2 whereas when Kinp (t) > Kout p (t), ⟨cos β⟩p increases. The latter, at first sight counterintuitive, behavior is expected at high temperature, where the initial thermal population of high torsional states encourages the coherent excitation of levels that favor the perpendicular configuration and is indeed exhibited in the T = 300 K study of Figure 7. Both trends can be seen in Figure 7 at T = 200 K, as the population initially relaxes from the perpendicular states, but this process soon becomes dominated by relaxation from the coplanar states, reversing the rate of change of ⟨cos2 β⟩p. Figure 9 shows KTp (t) for a variety of temperatures and laser intensities, illustrating that at high temperatures the postpulse population includes levels above the barrier to coplanarity, resulting in a negative KTp (t). The above analysis also explains the root of the difference between the torsional and the rotational alignment decay processes; namely, the assumptions made in the model used in refs 37 and 38 result in the source term canceling out, in which case decoherence and population relaxation both occur at the same rate. More recent work shows dynamics similar to those we predict.39,40 Namely, they show that the permanent alignment lasts considerably longer than the transient alignment. In the limit of very low torsional excitation, corresponding to the combination of low-temperature and low-laser intensity, the

(t ))

k , bk

j j

j k

j

k

in Figure 8. Kout p (t) of eq 24 (red, solid curve) and Kp (t) of eq 25 (green, dashed curve) both referred to the left-hand ordinate, along with KTp (t) (blue, dotted line) referred to the right-hand ordinate. The difference in scale between the ordinates highlights the balance between two similarly rapid processes that determine the evolution of ⟨cos2 β⟩p (t).

d ⟨cos2 β⟩p dt j j

k

k

∑a Vaj , aj ρa , a (t )

and = − As in the case of Kc(t), if K̃ aj is similar for all relevant levels, then Kout p (t) is essentially time-independent. The competition between Kpout(t) and Kpin(t) drives population evolution, which in turn gives rise to time evolution of in ⟨cos2 β⟩p. Both Kout p (t) and Kp (t) predict relaxation lifetimes on the same order of magnitude as Kc(t), implying that the two competing rates must be very similar to one another to explain the overall slow rate of population relaxation. Indeed, we found in Kout p (t) and Kp (t) to differ by a few hundredths of a percent, 2 making ⟨cos β⟩p the result of an interplay between two fast processes, as highlighted in Figure 8.

that while loss of coherence could only be described by a multiexponential fit, population relaxation can generally be fit with a single exponential, with the simulations shown in Figure 7 corresponding to lifetimes in the 400−500 ps range. Population relaxation is determined by the balance between two competing processes, corresponding to population flowing into and out of eigenstates that favor the coplanar configuration

aj , bk

j

j

KTp

Figure 7. Time evolution of ⟨cos2 β⟩p(t) for Ω = 1, with the left inset showing the dynamics during the pulse (solid black). Solid red, T = 100 K; dashed green, T = 200 K; dotted blue, T = 300 K.

= − ∑ (Vaj , ajK aj , bk ρa , a (t ) − Vaj , ajK bk , aj ρb

∑a , b Vaj , ajK bk , aj ρb , b (t )

(24) 22398



Article

are concealed, and the information content of torsional control studies is limited.

4. CONCLUSIONS Our goal in the research summarized in the previous sections has been two-fold. It has been to understand the controllability of torsional coherences subject to dissipative environments and hence also the prospect of torsional alignment in realistic open systems and to develop criteria for the choice of the system and experimental parameters that would allow coherent control. We are motivated by the wide variety of potential applications of torsional alignment, the majority of which necessarily involve a dissipative medium and all of which would benefit from the use of design criteria. As detailed in previous work, such applications range from enhanced spectroscopies and novel separation techniques to control of energy transfer, charge transport, and chemical reactivity.24 A second, related goal of this work has been to address the more general problem of wavepacket dissipation using torsion as a simple model that is richer than the previously studied problems of vibrational and rotational coherences. To address these goals we introduced a multilevel Bloch formulation within the Markov approximation, wherein the dissipative properties of the bath are determined by the parameters characterizing the bath spectral density. Our formulation introduces an approach for partitioning the experimental observable quantifying torsional alignment into a component tracing the coherences and one tracing the evolution of populations, with which one can disentangle and clock the phase decoherence and population relaxation processes. The decay of coherences was shown to follow a complex, multiexponential decay dynamic, which we traced to the unusual composition of the torsional wavepacket and ultimately to the torsional potential energy surface. The population relaxation component was shown to exhibit a seemingly very simple, yet counterintuitive structure, evolving on a much longer time scale. These features were shown to result from an interesting interplay between two rapid processes, whose consequences nearly balance one another. Finally, we explored the system and medium attributes that would benefit experimental demonstration of torsional coherent control in dissipative media. In some cases, changes to the system may not be possible or desirable, so a subject of future research will be extending control in dissipative media via pulse trains and optimal control techniques.

Figure 9. KTp (t) of eqs 24 and 25. Solid curves, Ω = 1; dashed curves, Ω = 2; red curves, T = 50 K; green curves, T = 100 K; blue curves, T = 300 K.

above behavior is reversed and the population relaxation rate exceeds the phase decoherence rate. This results from the occupation of only two levels, where, within the Bloch model, the first excited-state decays as ρ1,1(t0)·exp(−K1,1t), where K1,1 is the state-specific relaxation rate of the first excited state, whereas the coherence ρ0,1 decays as ρ0,1(t0)·exp[−(K0,0 + K1,1)t/2] ≈ ρ0,1(t0)·exp[−K1,1t/2] (because K0,0 ≈ 0). As a result, the population relaxation rate exceeds the phase decoherence rate. In this limit, the assumption that pure decoherence is negligible may no longer be valid. 3.2.4. Pure Torsional Decoherence. Pure decoherence, arising from elastic collisions that lead solely to phase (but not energy) transfer between the collision partners are typically assumed to be negligible in dense gases but make an important contribution to the dissipation in the case of a solution.21 The general formulation is discussed in Section 2, and in this subsection we make the assumption that γ(pd) is independent of (pd) the quantum state, γ(pd) (making it analogous to (T*2 )−1 in ajbk = γ NMR terms). Because literature values for γ(pd) for torsions are not available, we draw from a survey of literature on vibrational decoherence. Accordingly, we expect γpd to be strongly temperature-dependent for certain molecules41 and in general to be negligible in the low-temperature limit.42 Within the assumption that γ(pd) is independent of the quantum state, the effect of pure decoherence is very simple, as seen in Figure 10, for

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +1-847-467-4979. Fax: +1-847-467-4996. Notes

The authors declare no competing financial interest.

■

Figure 10. Pure decoherence effects, for the parameters of Figure 2, except T = 300 K. Green, no decoherence; red, γpd = (10 ps)−1; blue, γpd = (2 ps)−1.

ACKNOWLEDGMENTS We are grateful to the Department of Energy (Grant number DEFG02-04ER15612) and the Army Research Office (Grant number W911NF-11-0297) for support.

■

several values that are not so strong as to be instantaneous but not so weak as to be negligible. Here we focus on the isolated effect of pure decoherence, ignoring to that end the dissipative processes discussed in the previous subsections. Clearly, in the limit where pure decoherence dominates over other dissipative processes, the rich decay dynamics discussed in previous sections

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Dissipative Dynamics of Laser-Induced Torsional Coherences - The

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