Two-level system coupled to phonons: Full analytical solution
Résumé
We propose an analytical procedure to fully solve a two-level system coupled to phonons. Instead of using the common formulation in terms of linear and quadratic system-phonon couplings, we introduce different phonons depending on the system electronic level. We use this approach to recover known results for the linear-coupling limit in a simple way. More importantly, we derive results for the quadratic coupling induced by a phonon frequency change, a problem considered up to now as not analytically solvable. Coupling between a system and its environment is of primordial importance in science and emerging technologies, but its description at the microscopic level is extremely complicated [1-7]. Any consistent study of a quantum system calls for an open description which includes the large, often poorly known environment that interacts with it. A paradigmatic example of open systems is the "spin boson" model [2,8], that describes a two-level system coupled to a vibrational environment. In its simpler version, known as the "independent boson" model [9], the system excited level is suddenly populated or depopulated through its coupling to a photon field. While very simple, this model is not trivial and already allows studying a variety of phenomena that include spectral line shapes [9], electron transfer [10,11], electron-phonon interaction in quantum dots [12,13], and quantum control [1,14-22]. The independent boson model fundamentally deals with the consequences of electronic excitations that induce a spatial shift in the vibrational modes [23-25], and in some materials like aromatic hydrocarbons, a frequency change [26-32]. Instead of using the physically relevant phonons that depend on the occupied level of the electronic system, i.e., a diagonal representation, the common approach to this problem resorts to only using ground phonons-the vibrations physically relevant when the system is in its ground level-even when the system is in its excited level. This off-diagonal representation gives rise to a linear coupling associated with the atom (or molecule) spatial shift and a quadratic coupling associated with the phonon frequency change. These couplings are then commonly eliminated using a polaron transformation [9]. Since then and until now, a plethora of studies follows this polaron procedure to address a diversity of problems, including "open systems," which presently is a very active field. Although the polaron transformation can formally eliminate both linear and quadratic couplings, it involves calculations so tedious that to date, analytical results have been found for the linear limit only. This representation leads to the idea that after its sudden excitation, the electronic system is dressed by a cloud of ground phonons, whereas our treatment suggests interpreting the vibrations as dependent on the system excitation. Approaches relying on cumulant expansion and diagram-matic Green's functions have also been used to study the broadening of the zero-phonon line when the quadratic coupling is included. However, these results were obtained through a weak-coupling expansion [33-36], through a long-time expansion [37-41], or numerically [32]. In this paper, we propose a totally different procedure that relies on level-dependent phonons: the ground phonons and the excited phonons that describe the vibrational environment when the electronic system is, respectively, in its ground and excited level. Compelling evidence demonstrating the strength of this alternative approach is that (i) all the results for the linear limit become easy to derive, and more importantly, (ii) the same straightforward algebras make possible the handling of the quadratic coupling induced by a phonon frequency difference, a configuration considered up to now as not analytically solvable. The hidden difficulty incurred by considering different phonon frequencies comes from the fact that the destruction of an excited phonon not only corresponds to the destruction but also the creation of a ground phonon, making the phonon sub-space associated with the excited level infinitely large when written in terms of ground phonons. We will show that this difficulty is simply accounted for in the excited-phonon vacuum. The two relevant phonon bases. We consider a two-level system coupled to phonons, which for simplicity are taken as one-dimensional harmonic oscillators. When the electronic system is in its ground level |g with energy g , the Hamilto-nian reads in first quantization as H g = g + p 2 x 2m + mω 2 g 2 x 2 |g g|, (1) where m and ω g are the mass and frequency of the oscillator. By introducing the ground-phonon destruction operator b g = mω g 2h x + ip x mω g , (