Influence of complex exciton-phonon coupling on optical absorption and energy transfer of quantum aggregates
We present a theory that efficiently describes the quantum dynamics of an electronic excitation that is coupled to a continuous, highly structured phonon environment. Based on a stochastic approach to non-Markovian open quantum systems, we develop a dynamical framework that allows to handle realistic systems where a fully quantum treatment is desired yet usual approximation schemes fail. The capability of the method is demonstrated by calculating spectra and energy transfer dynamics of mesoscopic molecular aggregates, elucidating the transition from fully coherent to incoherent transfer.
pacs:33.70.-w, 02.70.Uu, 31.70.Hq, 82.20.Rp
One often encounters the situation where a quantum particle or an elementary excitation couples to a complex environment. Examples range from the classical polaron problem Berciu (2006) over electron-phonon interaction in superconductors Greco and Zeyher (1999), ultracold impurity atoms immersed in a Bose–Einstein condensate and trapped in a tight optical lattice Bruderer et al. (2008), molecular aggregates and crystals Scherer and Fischer (1984); Zhao and Spano (2005); Kobayashi (1996), atoms or molecules in photonic band gap materials Lambropoulos et al. (2000) or light-harvesting units in photosynthesis van Amerongen et al. (2000). We speak of a quantum aggregate (QA), if the excitation can reside on different sites, if it can be handed over from site to site mediated by a matrix element , and, importantly, if the excitation couples to a complex environment. In the following we will use the language of exciton-phonon coupling in molecular aggregates, although this scenario is obviously much more general.
For very small QAs () it may be possible to extract the few most relevant phonon modes and treat them fully quantum mechanically Zhao and Spano (2005); Kelley (2003). Then, however, the overall irreversible nature of the dynamics caused by the existence of many more environmental modes is ignored. Other approaches, like the coherent exciton scattering approximation Eisfeld and Briggs (2006a), are best suited for large aggregates (), yet fail for a small number of monomers. If the overall influence of the vibrational environment is small, a perturbative approach (Redfield) may be appropriate Breuer and Petruccione (2002); May and Kühn (2000).
All these established approaches fail for the QAs we are interested in: a quantum excitation that may reside on a finite number of sites, significantly coupled to a complex environment of phonons that consists of a few distinct vibrations embedded in an overall dissipative bath. We tackle this intricate regime of complex dynamics fully quantum mechanically using ideas from a stochastic description of non-Markovian open quantum systems Diósi and Strunz (1997) (see also Breuer and Petruccione (2002)). We present a dynamical framework based on a Stochastic Schrödinger Equation (SSE) that allows us to determine the quantum dynamics emerging from a significant exciton-phonon coupling in complex QAs. We are able to treat QAs consisting of a very small to a large number of monomers, bridging a gap in previous approaches. Moreover, being non-perturbatively, in a single unified theory we may describe exciton-dynamics ranging from fully coherent, over weakly perturbed, to strongly affected by the vibrations. To demonstrate the capability of the method, it is applied to study optical absorption and the coherent-incoherent transition of energy transfer in ring-shaped molecular aggregates Kobayashi (1996), as they appear e.g. in the Light Harvesting units of some bacteria.
We consider QAs where the wave functions of different monomers do not overlap (tight binding) and each monomer has two electronic states with a transition energy for monomer . A state in which monomer is electronically excited and all other monomers are in their electronic ground state is denoted by . The Holstein model Holstein (1959) includes the crucial influence of (possibly damped) vibrations on each monomer and is given by the Hamiltonian
is the purely electronic part of the Hamiltonian and
describes the collection of phonon modes. Here denotes the annihilation operator of mode of monomer with frequency . For each monomer , the sum over takes into account internal vibrations and their coupling to modes of the local environment. The coupling of electronic excitation to these vibrations is contained in
where the coupling constants are related to the dimensionless Huang-Rhys factor through Medvedev and Osherov (1995). An energy shift is incorporated into the transition energy .
The complex structure of the phonon “bath” of monomer is encoded in the bath correlation function at temperature May and Kühn (2000)
with the spectral density of monomer which is usually replaced by a smooth function to guarantee genuine irreversibility. For simplicity we will restrict ourselves in the following to the zero temperature limit.
We use recently developed ideas from a SSE approach to open quantum system dynamics Diósi and Strunz (1997); Diósi et al. (1998) to treat the model with a complicated continuous and structured phonon distribution fully quantum mechanically. In brevity, the SSE approach amounts to a solution of the full Schrödinger equation for a total Hamiltonian of the type of Eq. (1). It may be derived using a (Bargmann) coherent state basis Bargmann (1961) with for each environmental (vibrational) degree of freedom. Here is the state where no vibrations are excited and is a complex number. Thus (here for a zero temperature environment) the full state of system and environment at all times is written in the form Bargmann (1961)
with representing the collection of coherent state labels . Remarkably, in an approximation to be discussed below (see also section III.B of Yu et al. (1999)), the dynamics of the Holstein model (1) is now captured in the Schrödinger equation
in the small Hilbert space of the electronic degrees of freedom alone – a huge reduction in complexity. In Eq. (7), we use the abbreviations and
Here, represents a -independent operator
in the electronic
Hilbert space, introduced to approximate a functional derivative
that appears in the
with initial condition Yu et al. (1999).
Equation (7) (for the electronic state), together with
Eqs. (8) and (9) (for operators in the
electronic Hilbert space) is the new dynamical framework
which will be used to determine all properties of interest of the QA
The cross-section for absorption of light with frequency in dipole-approximation at zero-temperature turns out to be connected to a simple autocorrelation function:
The state is obtained from Eq. (7) with initial condition where the geometry of the aggregate enters explicitly via the transition dipoles and the polarization of the light . Note that only the projection of the total state onto the vibronic ground state is needed, which is the single solution of Eq. (7) with the choice .
More involved is the determination of transport properties which requires the reduced density operator Tr. It is found by considering the in Eq. (7) to be independent colored stochastic processes with correlations and . The covariance of these processes is connected to the bath correlation function via . It follows that the reduced density operator of the electronic part can be obtained as an ensemble mean over the noises .
Our novel approach is now applied to study optical absorption and transfer properties of molecular aggregates Kobayashi (1996). The spectral density of the monomers is taken to be a sum of Lorentzians, see Fig. 1, resulting in the monomer absorption spectrum shown in Fig. 2a, which nicely resembles that of a typical organic dye Renge and Wild (1997). In the following we will take the width (standard deviation) of this monomer spectrum as the unit of energy (for organic dyes is in the order of 0.1 eV). The vibrational progression due to the high-energy modes with energies around 1.5 in the spectral density (see Fig. 1) is clearly visible. The considerable broadening of this progression mainly stems from the low-energy vibrations below 0.5 in the spectral density.
In the following we focus on aggregates for which the absorption exhibits a narrow band, red-shifted w.r.t. the monomer absorption, the so-called J-band Kobayashi (1996); Walczak et al. (2008). We consider an aggregate of identical monomers arranged equidistantly along a ring with transition dipoles lying in the plane of the ring, such that the angle between the transition dipoles of neighboring monomers is identical for all monomers. In the calculations we have taken into account the interaction between neighboring monomers only. For the chosen geometrical arrangement, without coupling to vibrations, the aggregate absorption would be a single line, shifted by an energy w.r.t. the electronic monomer absorption line.
Fig. 2b-d shows aggregate absorption spectra for for different . We find that the mean of the aggregate spectrum is shifted by the energy w.r.t. to the mean of the monomer spectrum, in accordance with sum rules Briggs and Herzenberg (1970). Furthermore, with increasing the vibrational structure vanishes and the lowest peak (around ) becomes narrower by roughly a factor . This is the well-known effect of motional narrowing which leads to the narrow shape of the J-band of molecular aggregates Walczak et al. (2008); Knapp (1984), obtained here from a fully dynamical calculation.
Upon increasing further the shape of the aggregate spectrum (especially the width) undergoes only very small changes, hardly noticeable even for . Therefore one might assume that also other properties of the QA will only slightly change when increasing .
However this is not the case as we will now show considering energy transfer for the same situation as in Fig. 2d, (i.e. and ). Initially, the electronic excitation is chosen to be localised on monomer number eight. As unit of time we take the typical time of intermonomer electronic excitation transfer Roden et al. (2009). In Fig. 3b we show the time dependent probability to be electronically excited as a function of site number and time (note that the aggregate is ring-shaped).
For reference, in Fig. 3a the case where the electronic transfer does not couple to any vibrational modes is shown Merrifield (1958). While the transfer in Fig. 3a, obtained from a purely electronic theory, exhibits clear excitation maxima over a long period of time, the excitation in Fig. 3b is distributed quickly over all monomers due to the coupling to the vibrational continuum (the transfer shown is averaged over 1000 realisations of the stochastic noise , but was well converged after only 600 realisations). We have found that upon increasing the fast smearing of the excitation in Fig. 3b is suppressed and at about the purely electronic situation of Fig. 3a is reached. This is quite remarkable, showing, that from the width of the J-band alone it is not easily possible to infer the influence of the phonon bath on transfer properties. To gain deeper insight into the nature of the transfer, in Fig. 3c-e three of the 1000 single realisations over which the transfer in Fig. 3b is averaged are shown. In these single realisations the excitation stays localised in a small region (about 3 monomers) and performs a random-walk-like motion.
These considerations show that our dynamical framework based on a SSE in the Hilbert space of electronic excitation allows an efficient and detailed description of properties of QAs, including complex vibrational couplings. Therefore an examination of the approximation underlying Eq. (7) is in order. It is based on a functional expansion of w.r.t. the noise Yu et al. (1999), taking only the lowest order term into account. This approximation has been confirmed to be true in many cases of interest: it is true near the Markov limit (Lindblad), and contains the weak coupling (Redfield) limit de Vega et al. (2005). Moreover, it holds true for many soluble cases, including the case of independent monomers () of this Holstein model. To check the quality of the approximation beyond the usual limits (Markov, Redfield), we investigated the case of a spectral density consisting of a single Lorentzian in more detail. For the dimer () we were able to compare with spectra obtained from full quantum calculations and found overall good agreement. Last but not least, we have confirmed that the sum rules Briggs and Herzenberg (1970); Hemenger (1977) for the first five moments of the absorption spectrum are satisfied.
To conclude, we have developed a new dynamical framework for the determination of optical and transport properties of QA. This method allows a fully quantum treatment with realistic complex vibrational environments. The usefulness has been shown by considering the emergence of the J-band as grows and by capturing the transition from coherent to incoherent energy transfer. The next step is to investigate the much more complicated case of the H-band Eisfeld and Briggs (2006b) where the details of the vibrational structure play a more pronounced role. Since the method is based on the time-propagation of a SSE, it is also ideally suited to include external time-dependent fields. This should enable us to efficiently study coherent control schemes or describe multidimensional spectroscopy using realistic spectral densities. Clearly, as indicated in the introduction, the model we solve here and variants thereof appear in many applications well beyond molecular aggregates which will be a subject of future research.
Acknowledgements.We thank John S. Briggs for many fruitful discussions and for initiating this collaboration.
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