Two-Dimensional Spectroscopy for Harmonic Vibrational Modes with Nonlinear System-Bath Interactions. I. Gaussian-White Case

Abstract

The quantum Fokker-Planck equation is derived for a system nonlinearly coupled to a harmonic oscillator bath. The system-bath interaction is assumed to be linear in the bath coordinates but quadratic in the system coordinate. The relaxation induced dynamics of a harmonic system are investigated by simulating the higher-order correlation functions of the Raman polarizability and the dipole moment, which represent the nonlinear optical responses of Raman or infrared spectroscopy. The 5th-order Raman response shows that, in addition to the frequency fluctuations induced by the bath, higher-order energy transfer between the system and bath plays a role. The nonlinearity of the system-bath interaction yields also an interesting feature in the 7th-order Raman echo or the 3rd-order infrared photon echo response: The calculations predict a finite signal for the case of a harmonic potential and a linear coordinate dependence of the polarizability or dipole while for linear system-bath coupling this response vanishes completely due to destructive interference of different Liouville space pathways.KEYWORDS: 2D Raman, 2D IR, quantum Fokker-Planck, nonlinear system-bath interaction monly assumes a certain functional form for the Hamiltonian of the medium, which allows for the derivation of closed form expressions that are fit to the experimental results. Such model approach, which is useful to understand the complex molecular interactions involved in optical processes, is complementary to more "realistic" approaches such as Molecular dynamics simulations. A typical example for this strategy is the Brownian oscillator model, which can be used to calculate, e.g., the first-order resonant IR response or the third-order offresonant Raman response. 5,8,9) Since the dipole moment or the polarizability is usually expanded in the power of molecular coordinate Q as μ(Q) = μ 1 Q + μ 2 Q 2 · · · or α(Q) = α 1 Q + α 2 Q 2 · · ·, the first-order IR or the third-order Raman onse in lowest order of Q is nothing but the two-time correlation functions of Brownian resp 3115 * E-mail: tanimura@ims.ac.jp rules for the transitions involved.To calculate the (non)linear optical response one com- §1. IntroductionFemtosecond nonlinear optical spectroscopies have proven to be valuable and versatile tools to obtain information on the dynamic characteristics of condensed phase systems.1-6) Usually, these experiments are described by a response function formalism, 5, 7) which is based on a perturbative expansion of the optical polarization in powers of the applied electric fields. For resonant spectroscopy, which is carried by the Infrared (IR) laser for molecular vibrational modes, the laser interaction is described by μE(t), where μ is the dipole moment of the system. Then, for example, the lowestorder signal is expressed by the dipole correlation function μ(t )μ(t) . For off-resonant Raman spectroscopy, in which resonance arises from a pair of laser pulses through Raman excitation processes, the laser interaction is described by αE 2 (t), where α is ...

Stochastic Liouville, Langevin, Fokker–Planck, and Master Equation Approaches to Quantum Dissipative Systems

“…15) To have equations of motion in a similar hierarchy form as the stochastic Liouville equation, we need to choose L 2 ðtÞ in exponential (Markovian) form. Hence, we assume [15][16][17][18][19]151,[159][160][161][162][163][164][165][166][167] Jð where k 2k=h " are the Matsubara frequencies and …”

“…Wðp; qÞ; ð5:30Þ To write the equations in explicit form, we assume that the system-bath interaction consists of the linear-linear (LL) interaction qX 18,19) and the square-linear (SL) interaction q 2 X, 105,161,162) where X P j c j x j . We then employ 164,165) VðqÞ…”

“…From this approach, the effects of anharmonicity as well as LL and SL interactions were studied for the fifth-and seventh-order 2D Raman, and third-order 2D infrared (IR) processes, respectively. 25,151,[161][162][163][164][165][166] Here, we briefly explain a way to calculate the multibody correlation function and show the wavepacket dynamics involved in the third-order 2D IR response function. Then, we present 2D signals for harmonic and Morse potential systems.…”

“…(5.41) and (5.42), was used to investigate the roles of LL+SL interactions on the fifth-order Raman and third-order IR signals for a harmonic system. [161][162][163][164][165] It has turned out that multi-dimensional spectroscopies are extremely sensitive to the anharmonicity of the potential and system-bath coupling. For example, if the system is harmonic and the polarizability or the dipole moment is the linear function of molecular coordinate, the 2D signals R ð5Þ Raman ðt 2 ; t 1 Þ / h½½q qðt 12 Þ;q qðt 1 Þ;q qi and R ð3Þ IR ðt 3 ; t 2 ; t 1 Þ / h½½½q qðt 123 Þ;q qðt 12 Þ;q qðt 1 Þ;q qi for t 12 t 1 þ t 2 and t 123 t 1 þ t 2 þ t 3 will vanish due to the Gaussian integrals involved in the thermal average for the Raman case and the destructive contribution of the coherence in optical Liouville paths for the IR case.…”

An Approach to Quantum Transport Based on Reduced Hierarchy Equations of Motion: Application to a Resonant Tunneling Diode

“…[11][12][13][14][15] As an example, we employ a model describing the thermal effects in resonant tunneling diodes (RTDs). 16,17) Although the validity of the Caldeira-Leggett Hamiltonian for describing electron transport phenomena has not yet been investigated thoroughly, because it has a firm quantum mechanical foundation, and because we feel that it is necessary to investigate the validity of the previous theoretical results in a fully quantum mechanical treatment, we believe that the present study will be helpful in constructing a general understanding of electron transport phenomena.…”