14: Fluctuations in Spectroscopy

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Here we will describe how fluctuations are observed in experimental observables, as is common to experiments in molecular condensed phases. As our example, we will focus on absorption spectroscopy and how environmentally induced dephasing influences the absorption lineshape. Our approach will be to calculate a dipole correlation function for transition dipole interacting with a fluctuating environment, and show how the time scale and amplitude of fluctuations are encoded in the lineshape. Although the description here is for the case of a spectroscopic observable, the approach can be applied to any such problems in which the deterministic motions of an internal variable of a quantum system are influenced by a fluctuating environment.

We also aim to establish a connection between this problem and the Displaced Harmonic Oscillator model. Specifically, we will show that a frequency-domain representation of the coupling between a transition and a continuous distribution of harmonic modes is equivalent to a time-domain picture in which the transition energy gap fluctuates about an average frequency with a statistical time scale and amplitude given by the distribution of coupled modes. Thus an absorption spectrum is merely a spectral representation of the dynamics experienced by a experimentally probed transition.

14.1: Fluctuations and Randomness - Some Definitions
For chemical problems in the condensed phase we constantly come up against the problem of random fluctuations to dynamical variables as a result of their interactions with their environment. It is unreasonable to think that you will come up with an equation of motion for the internal deterministic variable, but we should be able to understand the behavior statistically and come up with equations of motion for probability distributions. Models of this form are commonly referred to as stochastic.
14.2: Line-Broadening and Spectral Diffusion
The interactions of this chromophore with its environment can shift it resonance frequency. In condensed matter, time-dependent interactions with the surroundings can lead to time-dependent frequency shifts, known as spectral diffusion. How these dynamics influence the line width and lineshape of absorption features depends on the distribution of frequencies available to your system and the time scale of sampling varying environments.
14.3: Gaussian-Stochastic Model for Spectral Diffusion
We will begin with a classical description of how random fluctuations in frequency influence the absorption lineshape, by calculating the dipole correlation function for the resonant transition. This is a Gaussian stochastic model for fluctuations, meaning that we will describe the time-dependence of the transition energy as random fluctuations about an average value through a Gaussian distribution.
14.4: The Energy Gap Hamiltonian
In describing fluctuations in a quantum mechanical system, we describe how an experimental observable is influenced by its interactions with a thermally agitated environment. For this, we work with the specific example of an electronic absorption spectrum and return to the Displaced Harmonic Oscillator model.
14.5: Correspondence of Harmonic Bath and Stochastic Equations of Motion
So, why does the mathematical model for coupling of a system to a harmonic bath give the same results as the classical stochastic equations of motion for fluctuations? Why does coupling to a continuum of bath states have the same physical manifestation as perturbation by random fluctuations? The answer is that in both cases, we really have imperfect knowledge of the behavior of all the particles present. Observing a small subset of particles will have dynamics with a random character.