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# Theory of time-resolved spectral function in high-temperature superconductors with bosonic modes 

Jianmin Tao and Jian-Xin Zhu 

Theoretical Division & CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 

(Dated: November 21, 2018) 

We develop a three-temperature model to simulate the time dependence of electron and phonon temperatures in high-temperature superconductors displaying strong anistropic electron-phonon coupling. This model not only takes the tight-binding band structure into account, but also is valid in superconducting state. Based on this model, we calculate the time-resolved spectral func- tion via the double-time Green’s functions. We find that the dip-hump structure evolves with the time delay. More interestingly, new phononic structures are obtained when the phonons are excited by a laser field. This signature may serve as a direct evidence for electron-vibration mode coupling. 

The discovery of high-temperature superconductors (HTSC) has raised an important issue on the mechanism leading to the formation of Cooper pairs, which is still under debate. To address this issue, a number of spec- troscopy techniques [1–3] have been used to study the role and nature of bosonic modes, to which electrons are strongly coupled. In particular, due to technological ad- vances and the improved sample quality, angle-resolved photoemission spectroscopy (ARPES) [1] has been used to probe details of the energy and momentum struc- ture of single-particle excitations via the measurement of photoemission intensity. However, different interpre- tations of the same data may lead to completely different mechanisms [4, 5]. For example, the dip-hump structure observed in ARPES [6–13] could be interpreted [14] as naturnally occurring in the interacting system but having little effect on the superconducting pairing mechanism. It could also be interpreted [15] in terms of phonon modes that could drive d -wave pairing. Recent theoretical anal- ysis [16] on the spectral function of thermally excited electrons in the cuprates has shown that the out-of-plane and out-of-phase buckling mode strongly couples to the electronic states near the anti-nodal M points in the Bril- louin zone, while the in-plane breathing mode couples strongly to the electronic states near the d -wave nodal points. The signature of the anistropic electron-phonon (el-ph) coupling seems to get enhanced in the supercon- ducting state, suggesting the significant role of the el-ph interaction in the superconducting mechanism. 

The information extracted from the conventional ARPES is limited. Time-resolved ARPES offers the capability to simultaneously capture the single-particle (frequency domain) and collective (time domain) infor- mation, thus making it possible to directly probe the link between the collective modes and single-particle states. In this setting, either electrons or lattice vibrational modes can be selectively excited with an ultrafast laser pulse. Recent applications of this technique include the studies of transient electronic structure in Mott insula- tors [17, 18] and high- T$_{c}$ cuprates [19] with optical pump. 

PACS numbers: 74.25.Jb, 74.72.-h, 71.38.-k, 79.60.-i 

Furthermore, direct pumping of vibrational mode has also been realized in manganites [20], though not yet in the cuprates. Motivated by the thrust of this experi- mental technique, in this Letter, we aim to provide a theoretical underpinning of transient electronic structure for HTSC, in a hope to better understand the nature of bosonic modes in these systems. As such, the time evolu- tion of the el-ph coupling is investigated for both normal and superconducting states with the time-resolved spec- tral function. Our theory consists of two parts. First, we develop a three-temperature model to simulate the time dependence of the electron and phonon temperatures. Then, based on the three-temperature model, we calcu- late the time-resolved spectral function. Our results show a kink-structure in the time dependence of the electronic temperature at the superconducting transition temper- ature. Accordingly, we find from the spectral density that the energy position of the phonon mode is offset by a time-dependent gap function. More interestingly, new signatures of the el-ph coupling, which are absent when electrons are excited, can be observed in the case of se- lective excitation of phonons. This signature may serve as a direct evidence for the electron-vibration mode cou- pling as opposed to the coupling between electrons and spin fluctuations. 

Three-temperature model : Consider a two-dimensional superconductor exposed to a laser field. The model Hamiltonian for a vibrational mode ν can be written as 

<img src='content_image/430376.jpg'>

where c † k σ ( b † ν $_{q}$) and c$_{k}$$_{σ}$ ( b$_{ν}$$_{q}$ ) are the creation and anni- hilation operators for an electron with momentum k and spin σ (phonon with momentum q and vibrational mode ν ), A$_{ν}$$_{q}$ = b † − ν $_{q}$+ b$_{ν}$$_{q}$ , the quantity ξ$_{k}$ is the normal-state energy dispersion, µ the chemical potential, Δ$_{k}$ the gap 

