Hamiltonian level systems

We now add die field back into the Hamiltonian, and examine the simplest case of a two-level system coupled to coherent, monochromatic radiation. This material is included in many textbooks (e.g. [6, 7, 8, 9, 10 and 11]). The system is described by a Hamiltonian having only two eigenstates, i and with energies = and Define coq = - co. The most general wavefunction for this system may be written as... 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

In this section we shall consider in some detail the mechanism of coherence breakdown due to the bath, in order to clarify the physical assumptions which underlie the concept of rate constant at low temperatures. The particular tunneling model we choose is the two-level system (TLS) with the Hamiltonian... 

Thus for Hamiltonians of finite dimension the effective action functional can be found by immediately integrating a system of ordinary differential equations. The simplest yet very important case is a bath of two-level systems. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The Hamiltonian HSf(a0 = 0) ex describes then the coupling between two undamped two-levels systems, whose excited states may be nonresonant. Introducing the relaxation yG and y5 of the excited states in the representation of Hsf(a0 = 0) ex within the base (105), we obtain... 

Another ambiguity in defining the classical mapping Hamiltonian is related to the fact that different bosonic quantum Hamiltonians may correspond to the same original quantum Hamiltonian H. This problem was already discussed in Section VI.A.2 for A-level systems. In the context of nonadiabatic dynamics, a different version of the mapping Hamiltonian is given by... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

We have generalised these results to the case when the reduction of the Zwanzig Hamiltonian to a 2-level system is not appropriate. We started with the Hamiltorrian... 

We first examine the fast-forward protocol in a two-level system, whose Hamiltonian is represented as... 

A one-level system e) that can exchange its population with the bath states [/) represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states e) and g), respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the RWA, which is alleviated in Section 4.4, the present formalism applies to a relaxing qubit, under the substitutions... 

We focus on two regimes a two-level system coupled to either an AN or PN thermal bath (Figure 4.5). The bath Hamiltonian (in either regime) will be... 

We first consider the AN regime of a two-level system coupled to a thermal bath. We will use off-resonant dynamic modulations, resulting in AC-Stark shifts (Figure 4.5(a)). The Hamiltonians then assume the following form ... 

For a three-level system the Hamiltonian in the interaction picture //, in Rotating Wave Approximation is given in matrix representation by... 

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

We start by re-writing the generic Hamiltonian Eq. (7) for an electronic two-level system coupled to N nuclear modes,... 

Let us consider the trivial example of a two-level system with the Hamiltonian... 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

A sketch of an argument that leads to the adiabatic theorem for the Floquet Hamiltonian of an iV-level system is given in Appendix C. 

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